# Axial curvatures for corank 1 singular $n$-manifolds in $\mathbb{R}^{n+k}$

P. Benedini Riul, J. L. Deolindo-Silva, R. Oset Sinha  Departamento de
Estatística, Física e Matemática, Universidade Federal de São João Del-Rei -
UFSJ, Campus Alto Paraopeba, 36497-899, Ouro Branco, Brazil
<EMAIL_ADDRESS>Departamento de Matemática. Universidade Federal de Santa
Catarina - UFSC, 89036-004 - Blumenau-SC, Brazil<EMAIL_ADDRESS>Departament de Matemàtiques, Universitat de València, Campus de Burjassot,
46100 Burjassot, Spain<EMAIL_ADDRESS>
###### Abstract.

For singular $n$-manifolds in $\mathbb{R}^{n+k}$ with a corank 1 singular
point at $p\in M^{n}_{\operatorname{sing}}$ we define up to $l(n-1)$ different
axial curvatures at $p$, where $l=\min\\{n,k+1\\}$. These curvatures are
obtained using the curvature locus (the image by the second fundamental form
of the unitary tangent vectors) and are therefore second order invariants. In
fact, in the case $n=2$ they generalise all second order curvatures which have
been defined for frontal type surfaces. We relate these curvatures with the
principal curvatures in certain normal directions of an associated regular
$(n-1)$-manifold contained in $M^{n}_{\operatorname{sing}}$. We obtain many
interesting geometrical interpretations in the cases $n=2,3$. For instance,
for frontal type 3-manifolds with 2-dimensional singular set, the Gaussian
curvature of the singular set can be expressed in terms of the axial
curvatures. Similarly for the curvature of the singular set when it is
1-dimensional. Finally, we show that all the umbilic curvatures which have
been defined for singular manifolds up to now can be seen as the absolute
value of one of our axial curvatures.

###### Key words and phrases:

axial curvature, singular manifolds in Euclidean spaces, second order
geometry, umbilic curvature, frontals

###### 2000 Mathematics Subject Classification:

Primary 57R45; Secondary 53A07, 58K05

Work of J. L. Deolindo-Silva partially supported by FAPESP Grant number
2019/07316-0.

Work of R. Oset Sinha partially supported by Grant PGC2018-094889-B-100 funded
by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”

##  1\. Introduction

In the last 15 years the study of the differential geometry of singular
surfaces has flourished to be an area of great interest for researchers from
many different backgrounds. These objects are cherished by differential
geometers as much as by singularists. Even contact topologists are
encountering singular objects when studying wave fronts and frontals, and
Gauss-Bonnet type theorems provide a link to the geometry. Singularity Theory
has proved to be the ideal framework to study these objects and its approach
for regular surfaces (see the recent book [10]) can be adapted for the
singular case. Ways of studying the geometry of the singular surface are
relating it to the geometry of a regular surface from which it is obtained by
an orthogonal projection ([2, 16, 20]), or studying its contact with planes
and spheres ([18, 21]). Another way is to define curvatures which give
information about the surface. For example, in [22] a singular and a limiting
normal curvature were defined for certain frontal type singularities and a
Gauss-Bonnet theorem was proved using the former. In [9] the intrinsity of
this kind of invariants is studied and in [23] the limiting normal curvature
is interpreted as a principal curvature.

In [12] the authors defined the curvature parabola at a singular corank 1
point in a surface in $\mathbb{R}^{3}$. This is the image by the second
fundamental form of the unitary tangent vectors and plays an analogous role to
the curvature ellipse defined by Little in [11]. Using this parabola they
defined an umbilic curvature which captures the round geometry of the surface
and generalises the limiting normal curvature defined for cuspidal edges in
[13]. In [19], the third author and K. Saji, using the curvature parabola,
defined for any corank 1 singular surface an axial curvature which generalises
the singular curvature defined in [22]. The axial curvature was defined using
the properties of the parabola, so it was not clear until now how to
generalise this to higher dimensions.

For $p\in M^{n}_{\operatorname{sing}}$ a corank 1 singular point in an
$n$-manifold in $\mathbb{R}^{n+k}$, the curvature locus can have many
different topological types and can even have singularities. These types of
loci have been studied in [3] for $n=2$ and $k=2$ and in [2, 4, 5] for $n=3$
and $k=2$.

In this paper, using the curvature locus, we define axial curvatures for any
corank 1 singular $n$-manifold in $\mathbb{R}^{n+k}$ which can be seen as
principal curvatures. These curvatures generalise the umbilic and axial
curvatures for surfaces in $\mathbb{R}^{3}$. We define a special adapted frame
of axial vectors $\\{v_{a}^{1},\ldots,v_{a}^{l}\\}$, where
$l=\min\\{n,k+1\\}$, in the normal space $N_{p}M$ and for each axial vector we
define the axial curvatures as the critical values of the projection of the
curvature locus onto the direction of the corresponding axial vector.

In Section 3, we give the main definitions and show that there can be up to
$l(n-1)$ different axial curvatures at a point $p\in
M^{n}_{\operatorname{sing}}$. Section 4 is devoted to the particular case of
surfaces in $\mathbb{R}^{n}$. We give formulas for the axial curvatures and
give geometrical interpretations for them. For example, we show that for
certain surfaces there is a distinguished curve on them with curvature
$\kappa$ which satisfies
$\kappa^{2}=(\kappa_{a_{1}})^{2}+(\kappa_{a_{2}})^{2}$, where $\kappa_{a_{1}}$
and $\kappa_{a_{2}}$ are the primary and secondary axial curvatures which
generalise the axial and umbilic curvatures from [19] and [12], respectively.
In Section 5 we study 3-manifolds in $\mathbb{R}^{3+k}$. For $k=1,2$ we define
adapted frames and prove the following elegant relation: For a corank 1
singular manifold $M^{n}_{\operatorname{sing}}\subset\mathbb{R}^{n+k}$, it is
possible to take a Monge form

(1)
$f(x_{1},\ldots,x_{n})=(x_{1},\ldots,x_{n-1},f_{1}(x_{1},\ldots,x_{n}),\ldots,f_{k+1}(x_{1},\ldots,x_{n})),$

where $\frac{\partial f_{\ell}}{\partial x_{i}}=0$ for $\ell=1,\ldots,k+1$ and
$i=1,\ldots,n$. We show that the axial curvatures corresponding to the axial
vector $v_{a}^{i}$ coincide with the $v_{a}^{i}$-principal curvatures of the
regular $(n-1)$-manifold $M_{\operatorname{reg}}^{n-1}$ given by
$f(x_{1},\ldots,x_{n-1},0)$. In this sense, the axial curvatures can be
understood as principal curvatures of singular manifolds. Using this relation
we obtain interesting geometrical interpretations. For instance, for frontal
type 3-manifolds with a smooth 2-dimensional singular set, the Gaussian
curvature of the singular set can be expressed in terms of the axial
curvatures. The same can be done for the curvature of the singular set when it
is 1-dimensional. This opens countless directions in which to study the
differential geometry of higher dimensional frontals. Finally, in Section 6 we
give an overview of all the different umbilic curvatures which have been
defined up to now (both in the regular and singular setting), which are
related to the umbilical focus of centers of spheres with degenerate contact
with the manifold, and prove that all of them can be obtained as the absolute
value of an axial curvature.

##  2\. The geometry of singular $n$-manifolds in $\mathbb{R}^{n+k}$

In this section we review the basic definitions and results related to the
second order geometry of corank 1 singular $n$-manifolds in
$\mathbb{R}^{n+k}$. For more details see [2, 3, 5, 12].

Let $M^{n}_{\operatorname{sing}}\subset\mathbb{R}^{n+k}$ be a $n$-manifold
with a singularity of corank 1. We can consider $M^{n}_{\operatorname{sing}}$
as the image of a smooth map $g:\tilde{M}\to\mathbb{R}^{n+k}$ from a smooth
regular $n$-manifold $\tilde{M}$ whose differential map has rank $\geq n-1$ at
any point such that $g(q)=p$. Consider $\phi:U\rightarrow\mathbb{R}^{n}$ a
local coordinate system defined in an open neighborhood $U$ of $q$ at
$\tilde{M}$. Using this construction, we may consider a local parametrisation
$f=g\circ\phi^{-1}$ of $M^{n}_{\operatorname{sing}}$ at $p$ (see the diagram
below).

$\textstyle{\mathbb{R}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$-------$\textstyle{U\subset\tilde{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\phi}$$\textstyle{M^{n}_{\operatorname{sing}}\subset\mathbb{R}^{n+k}}$

Considering $dg_{q}:T_{q}\tilde{M}\rightarrow T_{p}\mathbb{R}^{n+k}$ the
differential map of $g$ at $q$, the tangent space,
$T_{p}M^{n}_{\operatorname{sing}}$, at $p$ is given by $\mbox{Im}\ dg_{q}$
that degenerates to a $(n-1)$-space and the normal space at $p$,
$N_{p}M^{n}_{\operatorname{sing}}$, is the $k+1$-space of orthogonal
directions to $T_{p}M^{n}_{\operatorname{sing}}$ in $\mathbb{R}^{n+k}$ such
that $T_{p}M^{n}_{\operatorname{sing}}\oplus
N_{p}M^{n}_{\operatorname{sing}}=T_{p}\mathbb{R}^{n+k}$.

The first fundamental form of $M^{n}_{\operatorname{sing}}$ at $p$,
$I:T_{q}\tilde{M}\times T_{q}\tilde{M}\rightarrow\mathbb{R}$ is given by
$I(u,v)=\langle dg_{q}(u),dg_{q}(v)\rangle$, $\forall\ u,v\in T_{q}\tilde{M}$.
This induces a pseudometric in $T_{q}\mathbb{R}^{n}$ since the image of non-
zero vectors can be zero. The second fundamental form of
$M^{n}_{\operatorname{sing}}$ at $p$, $II:T_{q}\tilde{M}\times
T_{q}\tilde{M}\rightarrow N_{p}M^{n}_{\operatorname{sing}}$ is given by
$II(u,v)=\pi_{2}(d^{2}f_{\phi(q)}(d\phi(u,v))),$ where
$\pi_{2}:T_{p}\mathbb{R}^{n+k}\rightarrow N_{p}M^{n}_{\operatorname{sing}}$ is
the orthogonal projection and is extended to the whole space uniquely as a
symmetric bilinear map.

Given a normal vector $\nu\in N_{p}M^{n}_{\operatorname{sing}}$, the second
fundamental form along $\nu$, $II_{\nu}:T_{q}\tilde{M}\times
T_{q}\tilde{M}\rightarrow\mathbb{R}$ is given by $II_{\nu}(u,v)=\langle
II(u,v),\nu\rangle$, for all $u,v\in T_{q}\tilde{M}$.

Let $C_{q}=\\{u\in T_{q}\tilde{M}\;|\;I(u,u)=1\\}$ be the subset of unit
tangent vectors and let $\eta:C_{q}\rightarrow
N_{p}M^{n}_{\operatorname{sing}}$ be the map given by $\eta(u)=II(u,u)$. The
_curvature locus_ of $M^{n}_{\operatorname{sing}}$ at $p$, denoted by
$\Delta_{p}$, is the subset $\eta(C_{q})$. The curvature locus does not depend
on the choice of the local coordinates of $\tilde{M}$. Define $Aff_{p}$ as the
affine space of minimal dimension which contains $\Delta_{p}$.

###  2.1. Singular surfaces in $\mathbb{R}^{2+k}$, $k\geq 1$

Let $M^{2}_{\operatorname{sing}}$ be a corank $1$ surface at $p$. If
$\\{\partial_{x},\partial_{y}\\}$ is a basis for $T_{q}\tilde{M}$ and using
the parametrisation $f$, the coefficients of the first fundamental form are:
$E(q)=I(\partial_{x},\partial_{x})=\langle f_{x},f_{x}\rangle(q)$,
$F(q)=I(\partial_{x},\partial_{y})=\langle f_{x},f_{y}\rangle(q)$ and
$G(q)=I(\partial_{y},\partial_{y})=\langle f_{y},f_{y}\rangle(q)$ and taking
$u=a\partial_{x}+b\partial_{y}\in T_{q}\mathbb{R}^{2}$,

$I(u,u)=a^{2}E(q)+2abF(q)+b^{2}G(q).$

The second fundamental form of $M^{2}_{\operatorname{sing}}$ at $p$,
$II:T_{q}\tilde{M}\times T_{q}\tilde{M}\rightarrow
N_{p}M^{2}_{\operatorname{sing}}$ is given by

$\begin{array}[]{c}II(\partial_{x},\partial_{x})=\pi_{2}(f_{xx}(\phi(q))),\
II(\partial_{x},\partial_{y})=\pi_{2}(f_{xy}(\phi(q))),\
II(\partial_{y},\partial_{y})=\pi_{2}(f_{yy}(\phi(q))).\end{array}$

The coefficients of $II_{\nu}$ with respect to the basis
$\\{\partial_{x},\partial_{y}\\}$ of $T_{q}\tilde{M}$ are

$\begin{array}[]{cc}l_{\nu}(q)=\langle\pi_{2}(f_{xx}),\nu\rangle(\phi(q)),\
m_{\nu}(q)=\langle\pi_{2}(f_{xy}),\nu\rangle(\phi(q)),\\\
n_{\nu}(q)=\langle\pi_{2}(f_{yy}),\nu\rangle(\phi(q)).\end{array}$

Thus, if $u=\alpha\partial_{x}+\beta\partial_{y}\in T_{q}\tilde{M}$ and fixing
an orthonormal frame $\left\\{\nu_{1},\ldots,\nu_{k+1}\right\\}$ of
$N_{p}M^{2}_{\operatorname{sing}}$, the second fundamental form is given by

$\begin{array}[]{c}II(u,u)=\displaystyle\sum_{i=1}^{k+1}II_{\nu_{i}}(u,u)\nu_{i}=\sum_{i=1}^{k+1}(\alpha^{2}l_{\nu_{i}}(q)+2\alpha\beta
m_{\nu_{i}}(q)+\beta^{2}n_{\nu_{i}}(q))\nu_{i}.\\\ \end{array}$

It is possible to take a coordinate system $\phi$ and make rotations in the
target in order to obtain a local parametrisation in the Monge form as in (1).

Taking an orthonormal frame $\\{\nu_{1},\ldots,\nu_{k+1}\\}$ of
$N_{p}M^{2}_{\operatorname{sing}}$, the curvature locus $\Delta_{p}$ can be
parametrised by

$\eta(y)=\sum_{i=1}^{k+1}(l_{\nu_{i}}+2m_{\nu_{i}}y+n_{\nu_{i}}y^{2})\nu_{i},$

where each parameter $y\in\mathbb{R}$ corresponds to a unit tangent direction
$u=\pm\partial_{x}+y\partial_{y}=(\pm 1,y)\in C_{q}$. We denote by
$y_{\infty}$ the parameter corresponding to the tangent direction given by
$u=\partial_{y}=(0,1)$.

In the case of $k=1$, the curvature locus $\Delta_{p}$ is a planar parabola
that may degenerate into a line, a half-line or a point.

When $\Delta_{p}$ degenerates to a line, a half-line or a point, a special
adapted frame $\\{\nu_{2},\nu_{3}\\}$ of $N_{p}M^{2}_{\operatorname{sing}}$
was defined in [12], and this definition was extended for the case when
$\Delta_{p}$ is a non-degenerate parabola in [19], where $\nu_{2}$ was called
the axial vector, $v_{a}$. With this frame and $u\in C_{q}$,
$II(u,u)=II_{v_{a}}(u,u)\nu_{2}+II_{\nu_{3}}(u,u)\nu_{3}$. When $\Delta_{p}$
degenerates to a line, a half-line or a point $II_{\nu_{3}}(u,u)$ does not
depend on $u$ up to sign and the umbilic curvature of
$M^{2}_{\operatorname{sing}}$ at $p$ is defined in [12] by
$\kappa_{u}=|\langle II(u,u),\nu_{3}\rangle|=|II_{\nu_{3}}(u,u)|.$ On the
other hand the axial curvature is defined in [19] as
$\kappa_{a}(p)=min\\{K_{v_{a}}(u):u\in
C_{q}\\}=min\\{\langle\eta(y),v_{a}\rangle:y\in\mathbb{R}\\}$ where
$K_{v_{a}}(u)=\langle II(u,u),v_{a}\rangle=II_{v_{a}}(u,u)$ is the axial
normal curvature function.

For $k=2$, $\Delta_{p}$ is again a planar parabola that may degenerate into a
line, a half-line or a point, however, this parabola now lies on a plane in
$\mathbb{R}^{3}$. In [3], the umbilic curvature was defined even when
$\Delta_{p}$ is a non-degenerate parabola as the height of this plane.

###  2.2. Singular $3$-manifold in $\mathbb{R}^{k+3}$

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{k+3}$ be a 3-manifold with
a singularity of corank 1 at $p\in M^{3}_{\operatorname{sing}}$. Let
$\mathcal{B}=\\{\partial_{x},\partial_{y},\partial_{z}\\}$ be a basis for
$T_{q}\tilde{M}$ and
$u=\alpha\partial_{x}+\beta\partial_{y}+\gamma\partial_{z}\in T_{q}\tilde{M}$.
The coefficients and images of the first and second fundamental forms are
defined analogously to the case of surfaces. In particular, for each normal
vector $\nu\in N_{p}M^{3}_{\operatorname{sing}}$, the coefficients of
$II_{\nu}$ in terms of local coordinates $(x,y,z)$ are:

$\begin{array}[]{c}l_{\nu}(q)=\langle\pi_{2}(f_{xx}),\nu\rangle(\phi(q)),\,m_{\nu}(q)=\langle\pi_{2}(f_{xy}),\nu\rangle(\phi(q)),\,n_{\nu}(q)=\langle\pi_{2}(f_{yy}),\nu\rangle(\phi(q)),\\\
p_{\nu}(q)=\langle\pi_{2}(f_{zz}),\nu\rangle(\phi(q)),\,q_{\nu}(q)=\langle\pi_{2}(f_{xz}),\nu\rangle(\phi(q)),\,r_{\nu}(q)=\langle\pi_{2}(f_{yz}),\nu\rangle(\phi(q)),\\\
\end{array}$

Fixing an orthonormal frame $\\{\nu_{1},\ldots,\nu_{k+1}\\}$ of
$N_{p}M^{3}_{\operatorname{sing}}$, the second fundamental form can be written
as

$\begin{array}[]{c}II(u,u)=\displaystyle\sum_{i=1}^{k+1}II_{\nu_{i}}(u,u)\nu_{i}.\end{array}$

Taking a coordinate system $\phi$ and making rotations in the target, it is
possible to obtain a local parametrisation in the Monge form as in (1). Hence,
the subset of unit tangent vectors $C_{q}$ is the cylinder given by
$\\{(\alpha,\beta,\gamma)\in T_{q}\tilde{M}\;:\;\alpha^{2}+\beta^{2}=1\\}$
parallel to the $z$-axis. Taking an orthonormal frame
$\\{\nu_{1},\ldots,\nu_{k+1}\\}$ of $N_{p}M^{3}_{\operatorname{sing}}$, the
curvature locus $\Delta_{p}$ can be parametrised by
$(\alpha,\beta,\gamma)\mapsto$

$\sum_{i=1}^{k+1}(\alpha^{2}l_{\nu_{i}}(q)+2\alpha\beta
m_{\nu_{i}}(q)+\beta^{2}n_{\nu_{i}}(q)+\gamma^{2}p_{\nu_{i}}(q)+2\alpha\gamma
q_{\nu_{i}}(q)+2\beta\gamma r_{\nu_{i}}(q))\nu_{i}$

with $\alpha^{2}+\beta^{2}=1$.

Throughout the paper, $\mathscr{A}=Diff(\mathbb{R}^{n},0)\times
Diff(\mathbb{R}^{n+k},0)$, i.e. changes of coordinates in source and target,
and $\mathscr{A}^{2}$ represents the 2-jets of elements in $\mathscr{A}$. All
our results are local in the sense that we are considering germs of manifolds
at a certain point.

##  3\. Definition of axial curvatures for $M^{n}_{\operatorname{sing}}$ in
$\mathbb{R}^{n+k}$

For the case of corank 1 surfaces
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{3}$ the parametrisation of $f$
can be given at the origin in Monge form

(2)
$j^{2}f(0)=(x,\frac{1}{2}(a^{1}_{20}x^{2}+2a^{1}_{11}xy+a^{1}_{02}y^{2}),\frac{1}{2}(a^{2}_{20}x^{2}+2a^{2}_{11}xy+a^{2}_{02}y^{2})).$

In [19], when $\Delta_{p}$ is a non-degenerate parabola or a half-line the
axial vector is defined using the direction perpendicular to the directrix of
the parabola or the direction of the half-line and is given by
$v_{a}=\frac{1}{\sqrt{(a^{1}_{02})^{2}+(a^{2}_{02})^{2}}}(a^{1}_{02},a^{2}_{02})$.
The axial curvature is given by

$\kappa_{a}(p)=\frac{1}{||(a^{1}_{02},a^{2}_{02})||}\left((a^{1}_{02}a^{1}_{20}+a^{2}_{02}a^{2}_{20})-\frac{(a^{1}_{02}a^{1}_{11}+a^{2}_{02}a^{2}_{11})^{2}}{||(a^{1}_{02},a^{2}_{02})||^{2}}\right).$

For the case when the curvature parabola is a line or half-line, in [12] the
authors define the image by $\eta$ of the direction $y_{\infty}\in
T_{q}\tilde{M}$ as the direction in which the curvature locus is not bounded.
This was generalized for all $\mathscr{A}^{2}$-orbits in [19]. The following
result which relates this direction with the axial vector, despite natural and
not surprising, had been unnoticed so far.

For $M^{n}_{\operatorname{sing}}\subset\mathbb{R}^{n+k}$ considering the
pseudo-metric in $T_{q}\tilde{M}$, we call _null tangent direction_ the
unitary tangent direction $u_{\infty}\in T_{q}\tilde{M}$ such that
$I(u_{\infty},u_{\infty})=0$ (it corresponds to $y_{\infty}$ in the notation
of [12] for $M^{2}_{\operatorname{sing}}$ in $\mathbb{R}^{3}$).

######  Proposition 3.1.

Let $M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{3}$ be such that
$\Delta_{p}$ is a non-degenerate parabola or a half-line. Let $u_{\infty}\in
T_{q}\tilde{M}$ be the null tangent direction, then
$\frac{II(u_{\infty},u_{\infty})}{||II(u_{\infty},u_{\infty})||}=v_{a}$.

###### Proof.

Consider $M^{2}_{\operatorname{sing}}$ given by the image of $f$ in Monge form
with $j^{2}f$ as in (2), then $E(q)=1$, $F(q)=G(q)=0$, and $u_{\infty}=(0,1)$
corresponds to the null tangent direction. Then
$II(u_{\infty},u_{\infty})=(a_{02}^{1},a_{02}^{2})$. ∎

This result suggests how the axial vector can be defined for higher
dimensions. However the idea of projecting the curvature locus to a certain
direction in order to obtain meaningful curvatures should not be restricted to
the axial vector. We shall define several axial vectors and, consequently,
several axial curvatures.

######  Definition 3.2.

Consider $M^{n}_{\operatorname{sing}}\subset\mathbb{R}^{n+k}$, then $\dim
T_{q}\tilde{M}=n$ and $\dim N_{p}M^{n}_{\operatorname{sing}}=k+1$. Let
$l=min\\{n,k+1\\}$. Define the _axial space_ $Ax_{p}\subset
N_{p}M^{n}_{\operatorname{sing}}$ as $Aff_{p}$ if $dimAff_{p}=l$ and as any
$l$-vector space containing $Aff_{p}$ if $dimAff_{p}<l$. In the second case
$Ax_{p}$ contains $Aff_{p}$ and $p$ and in both cases $\dim Ax_{p}=l$ (as an
affine or vector space respectively).

Our goal is to define an adapted frame as in [19] for $Ax_{p}$. We start with
a partial definition.

######  Definition 3.3.

Let $u_{\infty}\in T_{q}\tilde{M}$ be the null tangent vector, and suppose
$II(u_{\infty},u_{\infty})\neq 0$. Then the primary axial vector is

$v_{a}^{1}:=\displaystyle\frac{II(u_{\infty},u_{\infty})}{||II(u_{\infty},u_{\infty})||}.$

In order to obtain an adapted frame for $Ax_{p}$ we add normal vectors in such
a way that $\\{v_{a}^{1},\ldots,v_{a}^{l}\\}$ is a positively oriented
orthonormal frame.

How to complete the basis and how to define an adapted frame when
$II(u_{\infty},u_{\infty})=0$ will be defined separately for the cases $n=2,3$
in the next sections.

Although Definition 3.3 needs to be completed, we are already in position for
our main definition.

######  Definition 3.4.

Given an adapted frame $\\{v_{a}^{1},\ldots,v_{a}^{l}\\}$ of $Ax_{p}\subset
N_{p}M^{n}_{\operatorname{sing}}$, the _$i$ -ary normal curvature function_ is
given by

$K_{v_{a}^{i}}(w)=\langle II(w,w),v_{a}^{i}\rangle=II_{v_{a}^{i}}(w,w),$

and the _$i$ -ary axial curvatures_ are the numbers

$\kappa_{a_{i}}(p)=\text{critical values of }K_{v_{a}^{i}}(w)\text{ where
}w\in C_{q}.$

Taking a Monge form as in (1) for a corank 1 singular manifold
$M^{n}_{\operatorname{sing}}\subset\mathbb{R}^{n+k}$, the subset of unit
tangent vectors is the cylinder given by $C_{q}=\\{(x_{1},\ldots,x_{n})\in
T_{q}\tilde{M}:x_{1}^{2}+\ldots+x_{n-1}^{2}=1\\}\subset\mathbb{R}^{n}$
parallel to the $x_{n}$-axis. Under these conditions, we can prove the
following.

######  Proposition 3.5.

There are at most $l(n-1)$ axial curvatures.

###### Proof.

Fix a certain axial vector $v_{a}^{i}$. To study the critical points of
$K_{v_{a}^{i}}(w)$ when $w\in C_{q}$ we want all the $2\times 2$ minors of the
following matrix to be 0:

$\left(\begin{array}[]{cccc}\frac{\partial K_{v_{a}^{i}}}{\partial
x_{1}}&\ldots&\frac{\partial K_{v_{a}^{i}}}{\partial x_{n-1}}&\frac{\partial
K_{v_{a}^{i}}}{\partial x_{n}}\\\ 2x_{1}&\ldots&2x_{n}&0\\\
\end{array}\right),$

where in the second row we have the gradient of the equation for $C_{q}$. This
is a $2\times n$ matrix with linear entries in $n$-variables. The solutions to
the system given by the minors is a homogeneous algebraic variety which is
generically a collection of lines through the origin. The intersection of this
lines with the cylinder $C_{q}$ give the critical points of
$K_{v_{a}^{i}}(w)$. Since the second fundamental form is quadratic
homogeneous, antipodal points in the cylinder have the same image, so there
are at most as many critical values as lines in the solution to the system.

On the other hand in Lemma 5.5 of [7], there is a formula for the multiplicity
of the ideal generated by the $2\times 2$ minors. Applying this formula to our
situation we obtain that the multiplicity is $n$, i.e. generically there can
be up to $n$ lines through the origin of multiplicity 1 as a solution to our
system. However, notice that $x_{1}=\ldots=x_{n-1}=0$ is always a solution,
but the line $(0,\ldots,0,x_{n})$ does not intersect the cylinder. Therefore,
there are at most $n-1$ critical points and so at most $n-1$ critical values.

Since we have $l$ axial vectors, the result follows. ∎

######  Remark 3.6.

  1. i)

When there is more than 1 axial curvature for each $i$ we will denote them by
$\kappa_{a_{i}}^{j}$, $1\leq j\leq n-1$. On the other hand, it is possible
that no axial curvature exists for a certain $i$. Notice that in this case,
saying that an axial curvature exists is equivalent to saying it is finite.

  2. ii)

In the particular case of $M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{3}$,
the primary axial curvature coincides with the axial curvature $\kappa_{a}$
defined in [19] and the absolute value of the secondary axial curvature
coincides with the umbilic curvature $\kappa_{u}$ defined in [12] (see Section
2).

For a corank 1 singular $n$-manifold parametrised in Monge form as above, the
null tangent vector is given by $u_{\infty}=(0,\ldots,0,1)$. Consider the
immersed $(n-1)$-manifold in $\mathbb{R}^{n+k}$ given by
$f(x_{1},\ldots,x_{n-1},0)$. Then
$T_{q}M^{n-1}_{\operatorname{reg}}=u_{\infty}^{\perp}=T_{q}\tilde{M}\cap\\{x_{n}=0\\}$,
and the pseudo-metric induces a metric here. Let
$A_{\nu}:T_{q}M^{n-1}_{\operatorname{reg}}\to
T_{q}M^{n-1}_{\operatorname{reg}}$ be the associated shape operator along the
normal vector field $\nu$ such that $\langle A_{\nu}(w),w\rangle=\langle
II(w,w),\nu\rangle$, where $w\in u_{\infty}^{\perp}$. There exists an
orthonormal basis $\\{e_{1},\ldots,e_{n-1}\\}$ of
$T_{q}M^{n-1}_{\operatorname{reg}}$ of eigenvectors of $A_{\nu}$ and the
corresponding eigenvalues $\kappa_{1}^{\nu},\ldots,\kappa_{n-1}^{\nu}$ are the
$\nu$-principal curvatures.

We will show in Section 5 that, for $n=3$ at least, when $n-1$ $i$-ary axial
curvatures are finite they coincide with the eigenvalues of $A_{v_{a}^{i}}$,
i.e. the $v_{a}^{i}$-principal curvatures of regular $(n-1)$-dimensional
manifold $M^{n-1}_{\operatorname{reg}}$.

##  4\. Axial curvatures for $M^{2}_{\operatorname{sing}}$ in
$\mathbb{R}^{2+k}$

We start by defining the adapted frame for $Ax_{p}$. Here $\dim Ax_{p}=l=2$ so
it is enough to define the primary axial vector in order to obtain an adapted
frame. For surfaces, as in [12] and [3], the curvature locus can be a non-
degenerate parabola, a half-line, a line or a point. Given the null tangent
vector $u_{\infty}\in T_{q}\tilde{M}$, when $II(u_{\infty},u_{\infty})\neq 0$,
$v_{a}^{1}$ is defined as in Definition 3.3 and we can complete the basis in a
unique way such that $\\{v_{a}^{1},v_{a}^{2}\\}$ is a positively oriented
orthonormal frame of $Ax_{p}$. This includes the cases where $\Delta_{p}$ is a
non-degenerate parabola or a half-line.

When $II(u_{\infty},u_{\infty})=0$, define $v_{a}^{1}$ as the direction of
$\Delta_{p}$ when it is a line and if $\Delta_{p}$ is a point $y\neq p$, then
$v_{a}^{1}$ is such that $v_{a}^{2}=y/|y|$. If $y=p$, then any orthonormal
frame is an adapted frame.

Given the nature of the curvature locus for singular surfaces (a non-
degenerate parabola, a half-line, a line or a point), there will only be one
axial curvature for each $i=1,2$. When $\Delta_{p}$ is a line $\kappa_{a_{1}}$
is not defined (there is no critical point of the primary axial normal
curvature function). When $\Delta_{p}$ is a point $\kappa_{a_{1}}=0$. When
$\Delta_{p}$ is a non-degenerate parabola, $\kappa_{a_{2}}$ is not defined
(there is no critical point of the secondary axial normal curvature function).
When $\Delta_{p}$ is a point $\kappa_{a_{2}}=||p||$. In general we can write

$\kappa_{a_{1}}(p)=min\\{K_{v_{a}^{1}}(w):w\in C_{q}\\},$
$\kappa_{a_{2}}(p)=K_{v_{a}^{2}}(w)\text{ for any }w\in C_{q}.$

See Figure 1 for the case when $\Delta_{p}$ is a half-line.

Figure 1. Axial curvatures for
$M_{\operatorname{sing}}^{2}\subset\mathbb{R}^{3}$ when $\Delta_{p}$ is a
half-line.

######  Proposition 4.1.

Let $p$ be the origin in $\mathbb{R}^{2+k}$ and $M^{2}_{\operatorname{sing}}$
be given by the image of $f$ in Monge form such that

(3) $j^{2}f(0)=(x,j^{2}f_{1}(0),\ldots,j^{2}f_{k+1}(0)),$

where
$j^{2}f_{\ell}(0)=\frac{1}{2}(a^{\ell}_{20}x^{2}+2a^{\ell}_{11}xy+a^{\ell}_{02}y^{2})$
for $\ell=1,\ldots,k+1$. Denote ${\bf
a_{ij}}=(a^{1}_{ij},\ldots,a^{k+1}_{ij})$, then
$II(u_{\infty},u_{\infty})={\bf a_{02}}$, where $u_{\infty}$ is the null
tangent direction.

  * (a)

If ${\bf a_{02}}\neq 0$ (i.e. $\Delta_{p}$ is a non-degenerate parabola or a
half-line) then

$\kappa_{a_{1}}(p)=\frac{1}{||{\bf a_{02}}||}\left(\langle{\bf a_{02}},{\bf
a_{20}}\rangle-\frac{\langle{\bf a_{02}},{\bf a_{11}}\rangle^{2}}{||{\bf
a_{02}}||^{2}}\right).$

Furthermore, if ${\bf a_{02}}\times{\bf a_{11}}=0$ (i.e. $\Delta_{p}$ is a
half-line), then

$\kappa_{a_{2}}(p)=\frac{||{\bf a_{02}}\times{\bf a_{20}}||}{||{\bf
a_{02}}||}.$

  * (b)

If ${\bf a_{02}}=0$ and ${\bf a_{11}}\neq 0$ (i.e. $\Delta_{p}$ is a line),
then

$\kappa_{a_{2}}(p)=\frac{||{\bf a_{20}}\times{\bf a_{11}}||}{||{\bf
a_{11}}||}.$

###### Proof.

The proof for $\kappa_{a_{1}}$ follows the same idea as the proof of
Proposition 4.3 in [19]. When $f$ is given as above, the curvature locus is
parameterised by

$\eta(y)=(a^{1}_{20}+2a^{1}_{11}y+a^{1}_{02}y^{2},\ldots,a^{k+1}_{20}+2a^{k+1}_{11}y+a^{k+1}_{02}y^{2})$

When ${\bf a_{02}}\neq 0$, the primary axial vector is given by
$v^{1}_{a}=\displaystyle\frac{{\bf a_{02}}}{||{\bf a_{02}}||}$, so
$K_{v_{a}^{1}}(y)=\langle\eta(y),v_{a}^{1}\rangle$. A direct computation shows
that $\displaystyle y_{0}=-\frac{\langle{\bf a_{02}},{\bf
a_{11}}\rangle}{||{\bf a_{02}}||}$ is the minimal critical point of
$K_{v^{1}_{a}}$ and the primary axial curvature is given by
$\kappa_{a_{1}}(p)=K_{v^{1}_{a}}(y_{0})$.

Moreover, as ${\bf a_{02}}\neq 0$ we can suppose without loss of generality
that $a^{k+1}_{02}\neq 0$. Using smooth changes of coordinates in the source
and isometries in the target we can reduce $j^{2}f$ to the form

(6)
$\displaystyle\begin{array}[]{l}j^{2}f_{\ell}(q)=\frac{1}{2}(\bar{a}^{\ell}_{20}x^{2}+2\bar{a}^{\ell}_{11}xy)\mbox{
for $\ell=1,\ldots,k$}\\\
j^{2}f_{k+1}(q)=\frac{1}{2}(\bar{a}^{k+1}_{20}x^{2}+2\bar{a}^{k+1}_{11}xy+\bar{a}^{k+1}_{02}y^{2}).\end{array}$

If ${\bf a_{02}}\times{\bf a_{11}}=0$ , we can reduce (6), using smooth
changes of coordinates in the source and isometries in the target, to the form
$j^{2}f_{\ell}(q)=0$ for $\ell=1,\ldots,k-1$,
$j^{2}f_{k}(q)=\frac{1}{2}\tilde{{a}}^{k}_{20}x^{2}$, and
$j^{2}f_{k+1}(q)=\frac{1}{2}(\tilde{{a}}^{k+1}_{20}x^{2}+2\tilde{{a}}^{k+1}_{11}xy+\tilde{{a}}^{k+1}_{02}y^{2})$,
where

$\tilde{{a}}^{k}_{20}=-\frac{||{\bf a_{20}}\times{\bf a_{02}}||}{||{\bf
a_{02}}||}.$

We have $v^{1}_{a}=(0,\ldots,0,1)$, $v_{a}^{2}=(0,\ldots,0,-1,0)$ and

$\eta(y)=(0,\ldots,0,\tilde{{a}}^{k}_{20},\tilde{{a}}^{k+1}_{20}+2\tilde{{a}}^{k+1}_{11}y+\tilde{{a}}^{k+1}_{02}y^{2}),$

Therefore $\kappa_{a_{2}}=-\tilde{{a}}^{k}_{20}$.

(b) When ${\bf a_{02}}=0$ and ${\bf a_{11}}\neq 0$, then $II(u,u)=0$. We can
suppose without loss of generality that $a^{1}_{11}\neq 0$ then using smooth
changes of coordinates in the source and isometries in the target, we can
reduce $j^{2}f$ to the form
$j^{2}f_{1}(q)=\frac{1}{2}(\bar{a}^{1}_{20}x^{2}+2\bar{a}^{1}_{11}xy)$,
$j^{2}f_{2}(q)=\frac{1}{2}\bar{a}^{2}_{20}x^{2}$, and $j^{2}f_{\ell}(q)=0$ for
$\ell=3,\ldots,k+1$, where

$\bar{a}^{2}_{20}=\frac{||{\bf a_{20}}\times{\bf a_{11}}||}{||{\bf
a_{11}}||}.$

Here the primary axial vector is $v^{1}_{a}=(1,0,\ldots,0)$ we obtain
$v_{a}^{2}=(0,1,0\ldots,0)$ and
$k_{a_{2}}(p)=K_{v_{a}^{2}}(w)=\bar{a}^{2}_{20}$.

∎

######  Remark 4.2.

  * i)

In the previous proof the isometries may change the orientation of the basis
of $N_{p}M^{2}_{\operatorname{sing}}$. On the other hand, the adapted frame is
constructed using the locus. So the sign of $\kappa_{a_{2}}$ may change if the
new orientation of the basis of $N_{p}M^{2}_{\operatorname{sing}}$ does not
coincide with the positive orientation of the adapted frame.

  * ii)

The previous formulas confirm what was announced in Remark 3.6, i.e. for the
particular case of $M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{3}$
$\kappa_{a_{1}}$ and $\kappa_{a_{2}}$ recover the axial and umbilic curvatures
in [19] and [12]. In fact, the formulas for $\kappa_{a_{2}}$ give explicit
formulas for $\kappa_{u}$ which were not given in [12].

We now give some geometrical interpretations for the axial curvatures.

######  Proposition 4.3.

Let $f$ be the parametrisation of a corank 1 singular surface in
$\mathbb{R}^{4}$ such that $j^{2}f\sim_{\mathscr{A}^{2}}(x,y^{2},0,0)$. Then
the curvature $\kappa$ of the regular curve $\gamma(t)=f(t,0)$ at the origin
satisfies $\kappa^{2}=(\kappa_{a_{1}})^{2}+(\kappa_{a_{2}})^{2}$.

###### Proof.

As $j^{2}f\sim_{\mathcal{A}}(x,y^{2},0,0)$ we can take $f$ parametrised by

$(x,y)\mapsto\left(x,\frac{a^{1}_{20}}{2}x^{2}+y^{2}+p(x,y),\frac{a_{20}^{2}}{2}x^{2}+q(x,y),r(x,y)\right)$

where $p,q,r\in\mathcal{M}_{2}^{3}$ (see Lemma 3.7 in [3]). Here the curvature
parabola is parameterised by $\eta(y)=(a_{20}^{1}+2y^{2},a_{20}^{2},0)$, then
$\kappa_{a_{2}}=a_{20}^{2}$ and by Proposition 4.1 we get
$\kappa_{a_{1}}=a_{20}^{1}$. Furthermore, when $y=0$ the curvature of the
regular curve $\gamma(t)=f(t,0)$ satisfies the desired result. ∎

######  Remark 4.4.

This formula generalizes the formula given in [13, p. 455] for the case of
frontals in $\mathbb{R}^{3}$ where it was proved that
$\kappa^{2}=\kappa_{s}^{2}+\kappa_{\nu}^{2}$, where $\kappa$ is the curvature
of the cuspidal edge curve, $\kappa_{s}$ is the singular curvature and
$\kappa_{\nu}$ is the limiting normal curvature.

######  Example 4.5.

Consider the singular surface parameterised by

$f(x,y)=\left(x,\frac{a_{20}^{1}}{2}x^{2}+y^{2},\frac{a^{2}_{20}}{2}x^{2},\frac{a^{3}_{20}}{2}x^{2}+y^{3}\right)$

which can be seen as a surface in $\mathbb{R}^{4}$ with a cuspidal edge. The
curvature parabola is a half-line parameterised by
$\eta(y)=(a_{20}^{1}+2y^{2},a_{20}^{2},a_{20}^{3})$. By Proposition 4.1,
$\kappa_{a_{1}}(p)=a_{20}^{1}$ and
$\kappa_{a_{2}}(p)=\sqrt{(a_{20}^{2})^{2}+(a_{20}^{3})^{2}}$. The regular
curve $\gamma(t)=f(t,0)$ has curvature $\kappa=||{\bf a_{20}}||$, and hence
$\kappa^{2}=(\kappa_{a_{1}})^{2}+(\kappa_{a_{2}})^{2}$.

Given $M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ a singular corank one
surface at $p$, one can associate a regular surface
$M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{4}$. In [3, p. 782] the authors
provide such construction and prove that their second order geometries are
strongly related to each other, since they have the same second fundamental
form (see Theorem 4.14 in [3]). The singular surface can be obtained as a
projection of a regular surface
$N^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}$ in a tangent direction, via
the map $\xi:N^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}\rightarrow
M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$. The regular surface
$N^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}$ can be taken, locally, as
the image of an immersion $i:\tilde{M}\rightarrow
N^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}$, where $\tilde{M}$ is the
regular surface from the construction in Section 2 ($k=n=2$).

Hence, $M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{4}$ is the regular
surface locally obtained by projecting
$N^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}$ via the map p into the four
space given by
$T_{\xi^{-1}(p)}N^{2}_{\operatorname{reg}}\oplus\xi^{-1}(E_{p})$, where
$E_{p}$ is a plane through $p$ parallel to $Aff_{p}$ (see the following
diagram).

$\textstyle{N^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$p$\scriptstyle{\xi}$$\textstyle{\mathbb{R}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$-----$\textstyle{\tilde{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{g}$$\scriptstyle{i}$$\textstyle{M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}}$$\textstyle{M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{4}}$

######  Proposition 4.6.

Let $M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ a corank 1 surface at
$p$ and $M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{4}$ its associated
regular surface. Suppose $M^{2}_{\operatorname{sing}}$ locally parametrised as
in (3), with ${\bf a_{02}}\neq 0$. The Gaussian curvature of
$\pi(M^{2}_{\operatorname{reg}})\subset\mathbb{R}^{3}$, the regular surface
obtained projecting $M^{2}_{\operatorname{reg}}$ along $(v_{a}^{1})^{\perp}$,
at the respective point, is given by $K=\kappa_{a_{1}}(p)$.

###### Proof.

Since ${\bf a_{02}}\neq 0$, we can change the local parametrisation of
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ to the form (6), where
$k=2$. Furthermore, using a rotation, it is possible to eliminate
$\bar{a}^{\ell}_{20}$, for $\ell=1$ or 2. Without loss of generality, suppose
$\bar{a}^{1}_{20}=0$. Finally, we take the change of coordinates in the source
given by $(x,y)\mapsto(x,\frac{1}{||{\bf a_{02}}||}y)$. Therefore, $E_{p}$ is
the plane generated by the last two coordinate axes,
$M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{4}$ is locally given by
$(x,y,j^{2}f_{1}(q),j^{2}f_{2}(q))$ and $v_{a}^{1}=(0,0,0,1)$. The Gaussian
curvature of $\pi(M^{2}_{\operatorname{reg}})\subset\mathbb{R}^{3}$, at the
respective point, is given by
$K=\bar{a}^{1}_{20}\bar{a}^{1}_{02}-(\bar{a}^{1}_{11})^{2}=\kappa_{a_{1}}(p)$.
∎

Proposition 5.2 in [19] can be easily extended for $\mathbb{R}^{n}$.

######  Proposition 4.7.

If $f$ satisfies that $\Delta_{p}$ is a non-degenerate parabola or a half-
line, the $\mathscr{A}$-singularities of $h_{v_{a}^{1}}$, the height function
in the direction $v_{a}^{1}$, are

  1. (1)

$A^{+}_{1}$ if and only if $\kappa_{a_{1}}(p)>0$,

  2. (2)

$A^{-}_{1}$ if and only if $\kappa_{a_{1}}(p)<0$,

  3. (3)

$A_{\geq 2}$ if and only if $\kappa_{a_{1}}(p)=0$.

###### Proof.

The proof is analogous to the proof in [19]. Following Lemma 3.6 and Lemma 3.7
in [19], for any smooth map $f:\mathbb{R}^{2}\to\mathbb{R}^{k+2}$ with
$q\in\mathbb{R}^{2}$ a corank 1 singular point of $f$ there exists a
coordinate system $(x,y)$ which satisfies $f_{x}(q)\neq 0$, $f_{y}(q)=0$,
$|f_{x}(q)|=|f_{xx}(q)|=1$ and $\langle f_{x}(q),f_{yy}(q)\rangle=0$. In such
a coordinate system

$\kappa_{a_{1}}(p)=(\langle f_{xx},f_{yy}\rangle-\langle
f_{xy},f_{yy}\rangle^{2})(q).$

The right hand side of the formula does not depend on the coordinate system as
long as it satisfies the above conditions, so in particular one can chose $f$
such that $j^{2}f(0)=(x,j^{2}f_{1}(0),\ldots,j^{2}f_{k+1}(0)),$ where

$j^{2}f_{\ell}(0)=\frac{1}{2}\left(a^{\ell}_{20}x^{2}+\frac{2a^{\ell}_{11}}{\sqrt{||{\bf
a_{02}}||}}xy+\frac{a^{\ell}_{02}}{||{\bf a_{02}}||}y^{2}\right)$

for $\ell=1,\ldots,k+1$ ($\bf a_{02}\neq 0$ because $\Delta_{p}$ is a non-
degenerate parabola or a half-line). Now consider contact with the plane
orthogonal to $v_{a}^{1}$. This contact is measured by the
$\mathscr{A}$-singularity of the height function $h_{v_{a}^{1}}(f(q))=\langle
f(q),v_{a}^{1}\rangle$. Direct computation shows

$\kappa_{a_{1}}(p)=\operatorname{Hess}(h_{v_{a}^{1}}(f(q)))$

and the result follows. ∎

We will give a geometrical interpretation for $\kappa_{a_{2}}$ in Section 6.

##  5\. Axial curvatures for $M^{3}_{\operatorname{sing}}$ in
$\mathbb{R}^{3+k}$

###  5.1. Curvature loci and adapted frames

As in the previous section, we start by defining the adapted frame for
$Ax_{p}$. When $k=1$, then $l=2$, if $k>1$, then $l=3$ and so we must
distinguish these two possibilities.

Given a smooth map $f:\mathbb{R}^{3}\to\mathbb{R}^{3+k}$ with
$q\in\mathbb{R}^{3}$ a corank 1 singular point of $f$, there exists a
coordinate system in Monge form such that

(7) $\displaystyle f(x,y,z)=(x,y,f_{1}(x,y,z),\ldots,f_{k+1}(x,y,z))$

where
$j^{2}f_{\ell}(q)=\frac{1}{2}(a^{\ell}_{200}x^{2}+2a^{\ell}_{110}xy+a^{\ell}_{020}y^{2}+2a^{\ell}_{101}xz+2a^{\ell}_{011}yz+a^{\ell}_{002}z^{2})$
for $\ell=1,\ldots,k+1$. Consider the notation ${\bf
a_{pqr}}=(a^{1}_{pqr},\ldots,a^{k+1}_{pqr})$ with $p,q,r=0,1,2$ and the matrix

(8) $A=\left[\begin{array}[]{ccc}{\bf a_{101}}&{\bf a_{011}}&{\bf
a_{002}}\end{array}\right].$

We start with the case of corank 1 3-manifolds in $\mathbb{R}^{4}$. In order
to define the adapted frame we need to understand the types of curvature locus
that can appear. For this we classify first the 2-jet orbits under
$\mathscr{A}$-equivalence. We denote by $J^{2}(3,3+k)$ the subspace of 2-jets
$j^{2}f(0)$ of map germs $f:(\mathbb{R}^{3},0)\to(\mathbb{R}^{3+k},0)$ and by
$\Sigma^{1}J^{2}(3,3+k)$ the subset of 2-jets of corank 1.

######  Proposition 5.1.

There are five $\mathscr{A}^{2}$-orbits in $\Sigma^{1}J^{2}(3,4)$:

$(x,y,xz,z^{2}),\ (x,y,xz,yz),\ (x,y,z^{2},0),\ (x,y,xz,0)\ \mbox{and}\
(x,y,0,0).$

###### Proof.

Let $f:(\mathbb{R}^{3},0)\rightarrow(\mathbb{R}^{4},0)$ be a corank one map
germ given by Monge form as in (7) with $k=1$. Take coordinates $(X,Y,Z,W)$ in
the target. The change in the target given by $X^{\prime}=X$, $Y^{\prime}=Y$,
$Z^{\prime}=Z-a_{200}^{1}X^{2}-a_{020}^{1}Y^{2}-a_{110}^{1}XY$ and
$W^{\prime}=W-a_{200}^{2}X^{2}-a_{020}^{2}Y^{2}-a_{110}^{2}XY$ removes the
terms with $x^{2}$, $y^{2}$ and $xy$ of the last two coordinates. Hence,

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,a^{1}_{101}xz+a^{1}_{011}yz+a^{1}_{002}z^{2},a^{2}_{101}xz+a^{2}_{011}yz+a^{2}_{002}z^{2}).$

Here the matrix $A$ is given by

(9) $A=\left(\begin{array}[]{ccc}a^{1}_{101}&a^{1}_{011}&a^{1}_{002}\\\
a^{2}_{101}&a^{2}_{011}&a^{2}_{002}\\\ \end{array}\right).$

and consider the $2\times 2$ minors
$\alpha=a_{101}^{1}a_{011}^{2}-a^{1}_{011}a^{2}_{101}$,
$\beta=a^{1}_{011}a^{2}_{002}-a^{1}_{002}a^{2}_{011}$ and
$\gamma=a^{1}_{101}a^{2}_{002}-a^{1}_{002}a^{2}_{101}$. We will consider the
case $\operatorname{rank}(A)=2$. The remaining cases are analogous. If $||{\bf
a_{002}}||\neq 0$, suppose $a^{1}_{002}\neq 0$. After the change in the target
$W^{\prime\prime}=-W^{\prime}+\frac{a^{2}_{002}}{a^{1}_{002}}Z^{\prime}$
(keeping the others coordinates unchanged), in the source
$\tilde{z}=z-\frac{a^{1}_{101}}{2a^{1}_{002}}x-\frac{a^{1}_{011}}{2a^{1}_{002}}y$
and eliminating the terms with $x^{2}$, $y^{2}$ and $xy$ of the last two
coordinates, we obtain

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,a^{1}_{002}z^{2},\frac{\gamma}{a^{1}_{002}}xz+\frac{\beta}{a^{1}_{002}}yz)\sim_{\mathscr{A}^{2}}(x,y,z^{2},\gamma
xz+\beta yz).$

Notice that $(\beta,\gamma)\neq(0,0)$, otherwise since
$(a^{1}_{002},a^{2}_{002})\neq(0,0)$, we would have also $\alpha=0$. The
change in the source given by $x^{\prime}=x-\frac{\beta}{\gamma}y$ (assuming
$\gamma\neq 0$) followed by the change in the target
$\tilde{X}=X^{\prime}+\frac{\beta}{\gamma}Y^{\prime}$,
$\tilde{W}=\frac{1}{\gamma}W^{\prime\prime}$ provides us that
$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,z^{2})$. However, if
$(a^{1}_{002},a^{2}_{002})=(0,0)$, $\alpha\neq 0$ and
$(a^{1}_{101},a^{2}_{101})\neq(0,0)$. Considering $a^{1}_{101}\neq 0$, the
change in the target $W^{\prime}=W-\frac{a^{2}_{101}}{a^{1}_{101}}Z$ provides
that

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,a^{1}_{101}xz+a^{1}_{011}yz,\frac{\alpha}{a^{1}_{101}}yz)\sim_{\mathscr{A}^{2}}(x,y,xz+\frac{a^{1}_{011}}{a^{1}_{101}}yz,yz).$

Finally, the change in the target
$Z^{\prime}=Z-\frac{a^{1}_{011}}{a^{1}_{101}}W$ provides
$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,yz)$. ∎

######  Remark 5.2.

Considering $f:(\mathbb{R}^{3},0)\rightarrow(\mathbb{R}^{4},0)$ a corank one
map germ given in Monge form as in (7) and the the matrix $A$ given as in (9),
Table 1 presents conditions on the coefficients to identify when the $2$-jet
is equivalent to one of the five normal forms of Proposition 5.1.

Table 1. Conditions over the coefficients of the $2$-jet for the $\mathscr{A}^{2}$-classification of corank 1 map germ $(\mathbb{R}^{3},0)\rightarrow(\mathbb{R}^{4},0)$. $\mathscr{A}^{2}$-normal form | Conditions
---|---
$(x,y,xz,z^{2})$ | $\operatorname{rank}(A)=2\,\,\,\mbox{and}\,\,\,||{\bf a_{002}}||>0$
$(x,y,xz,yz)$ | $\operatorname{rank}(A)=2\,\,\,\mbox{and}\,\,\,||{\bf a_{002}}||=0$
$(x,y,z^{2},0)$ | $\operatorname{rank}(A)=1\,\,\,\mbox{and}\,\,\,||{\bf a_{002}}||>0$
$(x,y,xz,0)$ | $\operatorname{rank}(A)=1\,\,\,\mbox{and}\,\,\,||{\bf a_{002}}||=0$
$(x,y,0,0)$ |  $\operatorname{rank}(A)=0$.

######  Lemma 5.3.

Let $M^{3}_{\operatorname{sing}}$ be a corank 1 surface in $\mathbb{R}^{3+k}$
given by the image of Monge form $f$ as (7). If $||{\bf a_{002}}||\neq 0$ then
using smooth changes of coordinates in the source and isometries in the
target, we can reduce $j^{2}f$ to the form

(10)
$j^{2}f_{\ell}(q)=\frac{1}{2}(\bar{a}^{\ell}_{200}x^{2}+2\bar{a}^{\ell}_{110}xy+\bar{a}^{\ell}_{020}y^{2}+2\bar{a}^{\ell}_{101}xz+2\bar{a}^{\ell}_{011}yz)\mbox{
for $\ell=1,\ldots,k$}$

and

(11)
$j^{2}f_{k+1}(q)=\frac{1}{2}(\bar{a}^{k+1}_{200}x^{2}+2\bar{a}^{k+1}_{110}xy+\bar{a}^{k+1}_{020}y^{2}+z^{2}),$

where

$\begin{array}[]{l}\displaystyle\bar{a}^{k+1}_{200}=\frac{1}{||{\bf
a_{002}}||}\left(\langle{\bf a_{002}},{\bf a_{200}}\rangle-\frac{\langle{\bf
a_{002}},{\bf a_{101}}\rangle^{2}}{||{\bf a_{002}}||^{2}}\right),\\\
\displaystyle\bar{a}^{k+1}_{020}=\frac{1}{||{\bf a_{002}}||}\left(\langle{\bf
a_{002}},{\bf a_{020}}\rangle-\frac{\langle{\bf a_{002}},{\bf
a_{011}}\rangle^{2}}{||{\bf a_{002}}||^{2}}\right),\\\
\displaystyle\bar{a}^{k+1}_{110}=\frac{1}{||{\bf a_{002}}||}\left(\langle{\bf
a_{002}},{\bf a_{110}}\rangle-\frac{\langle{\bf a_{002}},{\bf
a_{101}}\rangle\langle{\bf a_{002}},{\bf a_{011}}\rangle}{||{\bf
a_{002}}||^{2}}\right).\end{array}$

Moreover, when $rank(A)=1$, where $A$ is given as in (8), the coefficients
$\bar{a}^{\ell}_{101}$ and $\bar{a}^{\ell}_{011}$ are zero for
$\ell=1,\ldots,k$.

###### Proof.

Consider $||{\bf a_{002}}||\neq 0$. Suppose, without loss of generality, that
$a^{k+1}_{002}\neq 0$. Taking the rotation in $\mathbb{R}^{3+k}$ of angle
$\gamma=\displaystyle\arctan\left(\frac{a^{1}_{002}}{a^{k+1}_{002}}\right)$ we
can eliminate the coefficient of $z^{2}$ of $f_{1}$. In this case, we denote
by $\tilde{f}$ the new normal form and by $\tilde{a}^{\ell}_{ijk}$ the
coefficients of its 2-jet. After successive rotations in $\mathbb{R}^{k+3}$
with angle
$\gamma=\displaystyle\arctan\left(\frac{\tilde{a}^{m}_{002}}{\tilde{a}^{k+1}_{002}}\right)$,
$m=2,\ldots,k$, we eliminate all the coefficients of $z^{2}$ of the normal
form, except in the last coordinate. So the 2-jet is

$\frac{1}{2}(\tilde{a}^{\ell}_{200}x^{2}+2\tilde{a}^{\ell}_{110}xy+\tilde{a}^{\ell}_{020}y^{2}+2\tilde{a}^{\ell}_{101}xz+2\tilde{a}^{\ell}_{011}yz)$

in coordinates $\ell=1,\ldots,k$ and

$\frac{1}{2}(\tilde{a}^{k+1}_{200}x^{2}+2\tilde{a}^{k+1}_{110}xy+\tilde{a}^{k+1}_{020}y^{2}+2\tilde{a}^{k+1}_{101}xz+2\tilde{a}^{k+1}_{011}yz+\tilde{a}^{k+1}_{002}z^{2})$

in the last coordinate. Considering the changes of coordinates in the source
$\displaystyle
z=z^{\prime}-\frac{\tilde{a}^{k+1}_{101}}{2\tilde{a}^{k+1}_{002}}x-\frac{\tilde{a}^{k+1}_{011}}{2\tilde{a}^{k+1}_{002}}y$
and then $z^{\prime\prime}=\frac{1}{\sqrt{||{\bf a_{002}}||}}z^{\prime}$ we
obtain the desired normal form.

Moreover, when $rank(A)=1$ we have that $\displaystyle
a^{\ell}_{101}a^{k+1}_{002}-a^{\ell}_{002}a^{k+1}_{101}=0$ and $\displaystyle
a^{\ell}_{011}a^{k+1}_{002}-a^{\ell}_{002}a^{k+1}_{011}=0$ for
$\ell=1,\ldots,k$. Since the coefficients $\bar{a}^{\ell}_{101}$ and
$\bar{a}^{\ell}_{011}$ contains these components as a factor in their
expressions, they are 0 for $\ell=1,\ldots,k$. ∎

######  Remark 5.4.

Taking normal sections of $M^{3}_{\operatorname{sing}}$ with the normal form
from Lemma 5.3 we obtain a singular surface
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{k+3}$ with primary axial vector
given by $v^{1}_{a}=(0,\ldots,0,1)$. Notice that the coefficients
$\bar{a}^{k+1}_{200}$ and $\bar{a}^{k+1}_{020}$ coincide with the primary
axial curvatures of the normal sections obtained by $\\{y=0\\}$ and
$\\{x=0\\}$ respectively (see Proposition 4.1).

An analysis of the conditions in Table 1 can shred some light on the type of
loci we can have in each orbit by following the ideas in the proof of Theorem
3.9 in [5], however, there is a more geometrical way of doing this. Consider a
tangent direction $u\in T_{p}M^{3}_{\operatorname{sing}}$, we call the
singular surface $M^{3}_{\operatorname{sing}}\cap\\{u=0\\}$ the normal section
of $M^{3}_{\operatorname{sing}}$ in the direction $u$. Following the proof of
Theorem 3.3 in [2], we have that the curvature locus of
$M^{3}_{\operatorname{sing}}$ is generated by the union of the curvature loci
of the normal sections. All the normal sections are corank 1 singular surfaces
in $\mathbb{R}^{3}$ and the type of locus which can appear have been studied
in [12]. We can use this information to get the following.

######  Proposition 5.5.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ be parametrised by $f$,
then

  1. i)

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,z^{2})$ if and only if $\Delta_{p}$ is
a planar region,

  2. ii)

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,yz)$ if and only if $\Delta_{p}$ is a
plane,

  3. iii)

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,z^{2},0)$ if and only if $\Delta_{p}$ is
a half-strip (which may degenerate to a half-line),

  4. iv)

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,0)$ if and only if $\Delta_{p}$ is a
strip (which may degenerate to a line),

  5. v)

$j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,0,0)$ if and only if $\Delta_{p}$ is the
curvature locus of a regular surface in $\mathbb{R}^{4}$ (ellipse, segment or
point).

###### Proof.

When $j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,z^{2})$, all normal sections of
type $y=\lambda x$ give corank one surfaces whose 2-jet is
$\mathscr{A}^{2}$-equivalent to $(x,xz,z^{2})$. The curvature locus of these
sections is a non-degenerate parabola with axial vector $(0,1)$ in the normal
plane. The normal section $x=0$ gives a 2-jet $\mathscr{A}^{2}$-equivalent to
$(y,0,z^{2})$, whose curvature locus is a half line in the direction $(0,1)$.
The union of all these curvature loci gives a “parabolic” planar region. This
region is not the whole plane because the 2-jet of the last component of the
parametrisation of the curvature locus by Lemma 5.3 can be taken to
$a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2}+z^{2}$ where
$x^{2}+y^{2}=1$, which is a bounded function plus $z^{2}$, which is positive,
so it is bounded on the bottom.

When $j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,yz)$, the normal section $x=0$
gives a 2-jet equivalent to $(y,0,yz)$, whose curvature locus is a line in the
direction $(0,1)$ and the section $y=0$ gives a 2-jet equivalent to
$(x,xz,0)$, whose curvature locus is a line in the direction $(1,0)$. The rest
of normal sections give lines in any direction between $(0,1)$ and $(1,0)$, so
the curvature locus of the 3-manifold is the whole plane.

When $j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,z^{2},0)$, all normal sections have
a half-line in the direction $(1,0)$ as curvature loci. By Lemma 5.3, the
curvature locus of the 3-manifold can be taken to
$(a^{1}_{200}x^{2}+2a^{1}_{110}xy+a^{1}_{020}y^{2}+z^{2},a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2})$
where $x^{2}+y^{2}=1$. The curvature locus is bounded in the direction $(0,1)$
because the last component of the curvature locus is a bounded function. On
the other hand the component in the direction $(1,0)$ is a bounded function
plus $z^{2}$, which is positive, so it is bounded on the left. We therefore
have a strip bounded on the left, which can degenerate to a half-line.

When $j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,xz,0)$, all normal sections have
lines in the direction $(1,0)$ as curvarutre loci, except for the section
$x=0$, whose curvature locus is a point. The curvature locus of the 3-manifold
is a strip unbounded in the direction $(1,0)$. By item e) in Proposition 4.13
in [5] adapted for $k=1$ (the proof is the same), the last component of the
curvature locus can be taken to
$a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2}$ where $x^{2}+y^{2}=1$, so
the curvature locus is bounded in the direction $(0,1)$.

When $j^{2}f(0)\sim_{\mathscr{A}^{2}}(x,y,0,0)$, by item f) in Proposition
4.13 in [5] adapted for $k=1$ (the proof is the same), the curvature locus
coincides with the curvature locus of a parametrisation of type
$(x,y,\frac{1}{2}(a^{1}_{200}x^{2}+2a^{1}_{110}xy+a^{1}_{020}y^{2}),\frac{1}{2}(a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2}))$,
and so the curvature locus can be any type of curvature locus of a regular
surface in $\mathbb{R}^{4}$, i.e. a non-degenerate ellipse, a segment or a
point. ∎

######  Example 5.6.

We shall present examples of curvature locus for each possibility in
Proposition 5.5. Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ be
locally parametrised by $f:(\mathbb{R}^{3},0)\rightarrow(\mathbb{R}^{4},0)$.

  1. i)

Taking
$f(x,y,z)=(x,y,\frac{3}{2}x^{2}+xy+\frac{1}{2}y^{2}+\frac{1}{2}z^{2},x^{2}+\frac{5}{2}y^{2}+\frac{1}{2}xz)$,
$\Delta_{p}$ is a planar region (Figure 2 Planar region);

  2. ii)

Taking
$f(x,y,z)=(x,y,\frac{3}{2}x^{2}+\frac{1}{2}xy+\frac{1}{2}y^{2}+\frac{1}{2}yz,x^{2}+\frac{5}{2}y^{2}+\frac{1}{2}xz)$,
$\Delta_{p}$ is a plane (Figure 2 Plane);

  3. iii)

For
$f(x,y,z)=(x,y,\frac{3}{2}x^{2}+xy+\frac{1}{2}y^{2}+\frac{1}{2}z^{2},x^{2}+\frac{5}{2}y^{2})$,
$\Delta_{p}$ is a half-strip (Figure 2 Half-strip);

  4. iv)

For
$f(x,y,z)=(x,y,\frac{3}{2}x^{2}+\frac{1}{2}xy+\frac{1}{2}y^{2},x^{2}+\frac{5}{2}y^{2}+\frac{1}{2}xz)$
$\Delta_{p}$ is a strip (Figure 2 Strip);

  5. v)

Finally, taking
$f(x,y,z)=(x,y,\frac{3}{2}x^{2}+xy+\frac{1}{2}y^{2},x^{2}+\frac{5}{2}y^{2})$,
$\Delta_{p}$ is an ellipse (Figure 2 Ellipse).

The curvature loci are not completely depicted in Figure 2, the planar region
and the half-strip should be extended infinitely on the right, the figure
called plane should be extended infinitely in every direction and the strip
should be extended infinitely up and down.

Figure 2. Curvature loci for the different orbits for
$M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$.

Now we can define our adapted frame for $Ax_{p}$. Here $l=2$, so it is enough
to define the primary axial vector.

  1. i)

When $II(u_{\infty},u_{\infty})\neq 0$, $v_{a}^{1}$ is defined as in
Definition 3.3 and we can complete the basis in a unique way such that
$\\{v_{a}^{1},v_{a}^{2}\\}$ is a positively oriented orthonormal frame of
$Ax_{p}$. This includes the first and third orbits in Proposition 5.1.

  2. ii)

If $II(u_{\infty},u_{\infty})=0$, then $f$ is either in the second, fourth or
fifth orbit in Proposition 5.1. In the orbit $(x,y,xz,yz)$ the curvature locus
is a plane and we take any orthonormal frame as an adapted frame.

  3. iii)

For the orbit $(x,y,xz,0)$ we take as $v_{a}^{1}$ the direction in which
$\Delta_{p}$ is not bounded, i.e. the direction of the strip.

  4. iv)

Finally, for $(x,y,0,0)$, if $\Delta_{p}$ is a non-degenerate ellipse, take as
$v_{a}^{1}$ and $v_{a}^{2}$ the unitary vectors in the directions of the semi-
major and semi-minor axes, respectively, such that $\\{v_{a}^{1},v_{a}^{2}\\}$
is a positively oriented orthonormal frame of $Ax_{p}$. If $\Delta_{p}$ is a
segment, take as $v_{a}^{1}$ the direction of the segment and complete to
obtain an orthonormal basis. If $\Delta_{p}$ is a point $y\neq p$ take
$v_{a}^{1}$ such that $v_{a}^{2}=y/|y|$. If $y=p$ any orthonormal frame is an
adapted frame.

When $n=5$, $l=3$, so we have to define an adapted frame with 3 axial vectors.
In [5] there is a result similar to Proposition 5.1.

######  Proposition 5.7.

([5]) There are six $\mathscr{A}^{2}$-orbits in $\Sigma^{1}J^{2}(3,5)$:

$(x,y,xz,yz,z^{2}),\ (x,y,z^{2},yz,0),\ (x,y,xz,yz,0),\ (x,y,z^{2},0,0),$
$(x,y,xz,0,0)\ \mbox{and}\ (x,y,0,0,0).$

With a discussion similar to Proposition 5.5 we can see what type of curvature
locus there is in each orbit. The normal sections for corank 1 3-manifolds in
$\mathbb{R}^{5}$ are corank 1 singular surfaces in $\mathbb{R}^{4}$, which
have been studied in [3].

  1. i)

For the orbit $(x,y,xz,yz,z^{2})$, the primary axial vector $v_{a}^{1}$ can be
defined as in Definition 3.3. By Lemma 5.3 the parametrisation of the first
two components of the curvature locus can be taken to
$a^{1}_{200}x^{2}+2a^{1}_{110}xy+a^{1}_{020}y^{2}+2a^{1}_{101}xz+2a^{1}_{011}yz$
and
$a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2}+2a^{2}_{101}xz+2a^{2}_{011}yz$,
where $x^{2}+y^{2}=1$ and $z\in\mathbb{R}$. Since in this orbit
$a^{1}_{101}\neq 0\neq a^{2}_{011}$, these are two unbounded functions so we
can choose any orthonormal basis of this plane to complete the adapted frame
$\\{v_{a}^{1},v_{a}^{2},v_{a}^{3}\\}$.

  2. ii)

For the orbit $(x,y,z^{2},yz,0)$, $v_{a}^{1}$ can be defined as in Definition
3.3. By item b) in Proposition 4.13 in [5] there is a direction perpendicular
to $v_{a}^{1}$ such that the curvature locus is unbounded in both directions
and a direction in which it is bounded. Choose this unbounded direction as
$v_{a}^{2}$. We complete with a third vector to obtain our orthonormal adapted
frame.

  3. iii)

For the orbit $(x,y,xz,yz,0)$, the curvature locus is unbounded in a plane, so
we choose any orthonormal frame to be $\\{v_{a}^{1},v_{a}^{2}\\}$ and complete
in a unique way to obtain $v_{a}^{3}$. Notice that in the direction of
$v_{a}^{3}$ the curvature locus is bounded.

  4. iv)

For the orbit $(x,y,z^{2},0,0)$, $v_{a}^{1}$ can be defined as in Definition
3.3. By Lemma 5.3 (or Proposition 4.13 in [5]), any germ in this orbit can be
taken by changes of coordinates in the source and rotations in the target to
the form
$(x,y,\frac{1}{2}(a^{1}_{200}x^{2}+2a^{1}_{110}xy+a^{1}_{020}y^{2}+a^{1}_{002}z^{2}),\frac{1}{2}(a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2}),\frac{1}{2}(a^{3}_{200}x^{2}+2a^{3}_{110}xy+a^{3}_{020}y^{2}))$.
The curvature locus is given by
$(a^{1}_{200}x^{2}+2a^{1}_{110}xy+a^{1}_{020}y^{2}+a^{1}_{002}z^{2},a^{2}_{200}x^{2}+2a^{2}_{110}xy+a^{2}_{020}y^{2},a^{3}_{200}x^{2}+2a^{3}_{110}xy+a^{3}_{020}y^{2})$
where $x^{2}+y^{2}=1$. For $z=0$ we get the curvature ellipse (maybe
degenerate) of the regular surface given by $f(x,y,0)$. For any other constant
$z_{0}$, $II(C_{q}\cap\\{z=z_{0}\\})$ is the same curvature ellipse translated
by $(z_{0},0,0)$, i.e. a translation in the direction of the primary axial
vector. This means that the curvature locus is a half-strip contained in a
plane. Choose $v_{a}^{2}$ to be the orthogonal vector to $v_{a}^{1}$ in this
plane. Finally, $v_{a}^{3}$ is the vector perpendicular to this plane. When
the strip degenerates to a half-line choose the plane that contains the locus
and the origin in order to define $v_{a}^{2}$, and $v_{a}^{3}$ follows as
above. If the origin lies in the line that contains the half-line, choose any
plane that contains the curvature locus and proceed as above.

  5. v)

For the orbit $(x,y,xz,0,0)$, take $v_{a}^{1}$ as the direction in which the
curvature locus is unbounded. Arguing as for the previous orbit, the curvature
locus is a strip contained in a plane unbounded in the direction of
$v_{a}^{1}$. Choose $v_{a}^{2},v_{a}^{3}$ as in the previous case.

  6. vi)

For the orbit $(x,y,0,0,0)$, the curvature locus can be any curvature locus of
a regular surface in $\mathbb{R}^{5}$ (an ellipse, a segment or a point).
Choose $v_{a}^{1}$ to be the semi-major axis if it is an ellipse, the
direction of the segment if it is a segment, or any direction perpendicular to
the point otherwise. Consider the plane that contains the ellipse, the plane
that contains the locus and the origin if it is a segment, or any plane that
contains $v_{a}^{1}$, the locus and the origin if it is a point, and define
$v_{a}^{2},v_{a}^{3}$ as in the previous two orbits. If the line that contains
the segment contains the origin, choose any plane that contains the segment.
If the point is the origin, choose any plane.

######  Example 5.8.

Similarly to Example 5.6, we can illustrate some types of curvature loci for
corank 1 3-manifolds in $\mathbb{R}^{5}$. Let
$M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{5}$ be locally parametrised by
$f:(\mathbb{R}^{3},0)\rightarrow(\mathbb{R}^{5},0)$.

  * i)

Let $f(x,y,z)=\frac{1}{2}(x,y,x^{2}+z^{2},xy+xz,3x^{2}+y^{2}+yz)$, whose 2-jet
is $\mathscr{A}^{2}$-equivalent to $(x,y,xz,yz,z^{2})$. The curvature locus is
showed in Figure 3;

  * ii)

Consider $f(x,y,z)=\frac{1}{2}(x,y,xz,yz,x^{2}+3xy+y^{2})$. In this case,
$j^{2}f(0)$ is $\mathscr{A}^{2}$-equivalent to $(x,y,xz,yz,0)$ and the
curvature locus is showed in Figure 4;

  * iii)

Finally, consider
$f(x,y,z)=\frac{1}{2}(x,y,3x^{2}+2xy+y^{2}+z^{2},2x^{2}+5y^{2},x^{2}+2y^{2})$,
whose 2-jet is $\mathscr{A}^{2}$-equivalent to $(x,y,z^{2},0,0)$. In this
case, Figure 5 illustrates the curvature locus, a half-strip.

Figure 3. Different views of $\Delta_{p}$ in Example 5.8 (i). Figure 4.
Different views of $\Delta_{p}$ in Example 5.8 (ii). Figure 5. Different views
of $\Delta_{p}$ in Example 5.8 (iii).

###  5.2. Geometrical interpretations

######  Theorem 5.9.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$ be given in Monge
form (7). Consider the regular surface
$M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{3+k}$ given by $f(x,y,0)$. If
$II(u_{\infty},u_{\infty})=||{\bf a_{002}}||\neq 0$, then the primary axial
curvatures at $p\in M^{3}_{\operatorname{sing}}$ coincide with the
$v_{a}^{1}$-principal curvatures at $p\in M^{2}_{\operatorname{reg}}$.

###### Proof.

Following the discussion at the end of Section 3,
$T_{q}M^{2}_{\operatorname{reg}}$ is identified with
$T_{q}\tilde{M}\cap\\{z=0\\}=u_{\infty}^{\perp}$ and the unitary tangent
vectors are given by $C_{q}\cap u_{\infty}^{\perp}=\\{(x,y)\in
T_{q}M^{2}_{\operatorname{reg}}:x^{2}+y^{2}=1\\}\equiv\mathbb{S}^{1}$.
Therefore, the curvature ellipse of
$M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{3+k}$ at $p$ is contained in the
curvature locus of $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$ at
$p$.

Given the primary normal curvature function $K_{v_{a}^{1}}(w)=\langle
II(w,w),v_{a}^{1}\rangle$ seen as a function from $\mathbb{R}^{3}$ to
$\mathbb{R}$, the primary axial curvatures are given by the critical values of
$K_{v_{a}^{1}}|_{C_{q}}$. The critical points of this function are given by
the $2\times 2$ minors of the following matrix

$\left(\begin{array}[]{ccc}\frac{\partial K_{v_{a}^{1}}}{\partial
x}&\frac{\partial K_{v_{a}^{1}}}{\partial y}&\frac{\partial
K_{v_{a}^{1}}}{\partial z}\\\ 2x&2y&0\\\ \end{array}\right),$

where in the second row we have the gradient of the equation of $C_{q}$, i.e.
$x^{2}+y^{2}=1$. So the primary axial curvatures are given by the solutions of
the system $2(\frac{\partial K_{v_{a}^{1}}}{\partial x}y-\frac{\partial
K_{v_{a}^{1}}}{\partial y}x)=0,2x\frac{\partial K_{v_{a}^{1}}}{\partial z}=0$
and $2y\frac{\partial K_{v_{a}^{1}}}{\partial z}=0$.

On the other hand, the eigenvalues of the shape operator
$A_{v_{a}^{1}}:T_{q}M^{2}_{\operatorname{reg}}\to
T_{q}M^{2}_{\operatorname{reg}}$ are given by the critical values of $\langle
A_{v_{a}^{1}}(w),w\rangle=\langle II(w,w),v_{a}^{1}\rangle$ where
$w\in\mathbb{S}^{1}\subset T_{q}M^{2}_{\operatorname{reg}}$ are the unitary
tangent vectors. So these are the critical values of
$K_{v_{a}^{1}}|_{\mathbb{S}^{1}}=K_{v_{a}^{1}}|_{C_{q}\cap
u_{\infty}^{\perp}}$, which are given by the determinant of

$\left(\begin{array}[]{ccc}\frac{\partial K_{v_{a}^{1}}}{\partial
x}&\frac{\partial K_{v_{a}^{1}}}{\partial y}&\frac{\partial
K_{v_{a}^{1}}}{\partial z}\\\ 2x&2y&0\\\ 0&0&1\end{array}\right),$

where the last row is the gradient of the equation for $C_{q}\cap
u_{\infty}^{\perp}$, i.e. $z=0$. Therefore, the $v_{a}^{1}$-principal
curvatures of $M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{3+k}$ at $p$ are
given by the solutions to $2(\frac{\partial K_{v_{a}^{1}}}{\partial
x}y-\frac{\partial K_{v_{a}^{1}}}{\partial y}x)=0$.

Now, if $II(u_{\infty},u_{\infty})\neq 0$, then $||{\bf a_{002}}||\neq 0$, so
we can use the normal form in Lemma 5.3. Now $v_{a}^{1}=(0,\ldots,0,1)$ and
$K_{v_{a}^{1}}(w)=\bar{a}^{k+1}_{200}x^{2}+2\bar{a}^{k+1}_{110}xy+\bar{a}^{k+1}_{020}y^{2}+z^{2}$
with $x^{2}+y^{2}=1$, so $\frac{\partial K_{v_{a}^{1}}}{\partial z}=2z$ and
this is 0 if and only if $z=0$.

In conclusion, we have that the critical points of $K_{v_{a}^{1}}|_{C_{q}}$
coincide with the critical points of $K_{v_{a}^{1}}|_{C_{q}\cap
u_{\infty}^{\perp}}$. ∎

######  Theorem 5.10.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$, $k=1,2$ and suppose
that $f$ lies in the orbit $(x,y,z^{2},0)$ or $(x,y,z^{2},0,0)$ (depending on
$k=1$ or $2$), then the secondary axial curvatures at $p\in
M^{3}_{\operatorname{sing}}$ coincide with the $v_{a}^{2}$-principal
curvatures at $p\in M^{2}_{\operatorname{reg}}$.

###### Proof.

Consider the Monge form as in Lemma 5.3. We prove it for $k=1$, the proof for
$k=2$ is analogous. In the orbit $(x,y,z^{2},0)$, the curvature locus is given
by
$2(a^{1}_{200}x^{2}+a^{1}_{110}xy+a^{1}_{020}y^{2},a^{2}_{200}x^{2}+a^{2}_{110}xy+a^{2}_{020}y^{2}+z^{2})$
where $x^{2}+y^{2}=1$. For $z=0$ we get the curvature ellipse (maybe
degenerate) of the regular surface given by $f(x,y,0)$. For any other constant
$z_{0}$, $II(C_{q}\cap\\{z=z_{0}\\})$ is the same curvature ellipse translated
by $(z_{0},0)$, i.e. a translation in the direction of the primary axial
vector. Therefore the critical values of $K_{v_{a}^{2}}|_{C_{q}}$ coincide
with the critical values of $K_{v_{a}^{2}}|_{C_{q}\cap u_{\infty}^{\perp}}$. ∎

######  Remark 5.11.

When $k>2$, similarly to Proposition 5.1 there is an orbit of type
$(x,y,z^{2},0,\ldots,0)$. Here the secondary axial curvatures coincide with
the $v_{a}^{2}$-principal curvatures. Notice also that in this case there may
exist 3-ary axial curvatures, but the tangent space of
$M^{2}_{\operatorname{reg}}$ is 2-dimensional.

######  Theorem 5.12.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$, $k\geq 1$ and
suppose that $f$ lies in the orbit $(x,y,0,\ldots,0)$, then the primary and
secondary axial curvatures at $p\in M^{3}_{\operatorname{sing}}$ coincide with
the $v_{a}^{1}$ and $v_{a}^{2}$-principal curvatures at $p\in
M^{2}_{\operatorname{reg}}$ (resp.).

###### Proof.

In this orbit the curvature locus of the 3-manifold is precisely the curvature
locus of the regular manifold so the result follows. In this case, with any
other choice of adapted frame the result would still hold. ∎

######  Remark 5.13.

Similar results to Theorems 5.9, 5.10 and 5.12 can be proven for singular
corank 1 $n$-manifolds in general. In fact, we believe that when $i\leq n-1$,
then, in the orbits where $n-1$ $i$-ary axial curvatures are finite they
coincide with the $v_{a}^{i}$-principal curvatures of the associated regular
$(n-1)$-manifold. However, our proofs depend on the type of curvature locus
and certain normal forms, so we do not have a general proof at the moment.

######  Example 5.14.

Consider
$f(x,y,z)=(x,y,\frac{3}{2}x^{2}+xy+\frac{1}{2}y^{2}+\frac{1}{2}yz,x^{2}+\frac{5}{2}y^{2}+\frac{1}{2}xz)$,
which lies in the orbit $(x,y,xz,yz)$. Here the curvature locus is the whole
plane and we can choose the adapted frame given by $v_{a}^{1}=(1,0)$ and
$v_{a}^{2}=(0,1)$. One can check that $K_{v_{a}^{1}}(\theta,\phi)$ has a
critical point with critical value $\kappa_{a_{1}}=3$. However, the
$v_{a}^{1}$-principal curvatures of $f(x,y,0)$ are given by $2\pm\sqrt{2}$ and
do not coincide with $\kappa_{a_{1}}$.

It is possible to find an adapted frame such that at least one of the
principal curvatures in the direction of one of the vectors of the frame of
the regular manifold coincides with an axial curvature, but it seems unlikely
to be able to obtain a general result as Theorem 5.9. However, we have the
following partial result.

######  Proposition 5.15.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$, $k=1,2$, if the
curvature locus is unbounded on both sides in the direction of a certain axial
vector $v_{a}^{i}$, then there is exactly 1 axial curvature in that direction.
In particular, if there is exactly 1 direction $v_{a}^{i}$ in which the
curvature locus is unbounded on both sides and $f$ is given in Monge form such
that $v_{a}^{i}$ is one of the coordinate axes, then the corresponding
component can be taken to

$j^{2}f_{i}(x,y,z)=\frac{1}{2}({a}^{i}_{200}x^{2}+2{a}^{i}_{110}xy+{a}^{i}_{020}y^{2}+2{a}^{i}_{101}xz),$

and the unique axial curvature in that direction is given by
$\kappa_{a_{i}}={a}^{i}_{020}$. Furthermore, in this case, if
${a}^{i}_{110}=0$ then it coincides with 1 $v_{a}^{i}$-principal curvature of
the associated regular surface.

###### Proof.

First suppose there is a unique direction corresponding to $v_{a}^{i}$ such
that the curvature locus is unbounded on both sides. This is the case of the
orbits $(x,y,z^{2},xz)$ and $(x,y,xz,0)$ when $k=1$ and the orbits
$(x,y,z^{2},xz,0)$ and $(x,y,xz,0,0)$ when $k=2$. By Proposition 4.13 in [5],
for $k=2$ the component corresponding to $v_{a}^{i}$ can be taken, by changes
of variable in the source and isometries in the target, to

$j^{2}f_{i}(x,y,z)=\frac{1}{2}({a}^{i}_{200}x^{2}+2{a}^{i}_{110}xy+{a}^{i}_{020}y^{2}+2{a}^{i}_{101}xz).$

The proof of Proposition 4.13 is valid for the two orbits in $k=1$ too. The
normal curvature function in the unbounded direction is given by

$K_{v_{a}^{i}}(w)=\langle\eta(w),v_{a}^{i}\rangle={a}^{i}_{200}\cos^{2}\theta+2{a}^{i}_{110}\cos\theta\sin\theta+{a}^{i}_{020}\sin^{2}\theta+\cos\theta\frac{\cos\phi}{\sin\phi}.$

Taking the partial with respect to $\phi$ equal to 0 we get $\cos\theta=0$ and
so $\sin\theta=\pm 1$. Therefore the unique critical value is
$\kappa_{a_{i}}={a}^{i}_{020}$.

On the other hand, if ${a}^{i}_{110}=0$ the end points of the projection of
the curvature ellipse of the associated regular surface in the direction
$v_{a}^{i}$ (i.e. the $v_{a}^{i}$-principal curvatures) are given by
${a}^{i}_{200}$ and ${a}^{i}_{020}$.

Now suppose that there are more than one axial vectors in which the curvature
locus is unbounded on both sides. This is the case of the orbits $(x,y,xz,yz)$
when $k=1$ or $(x,y,xz,yz,z^{2})$ and $(x,y,xz,yz,0)$ when $k=2$. In the plane
in which the curvature locus is unbounded we can chose any orthonormal frame
to be part of the adapted frame. Take a vector $v_{a}^{i}=(\alpha,\beta)$ (if
$k=1$, $(\alpha,\beta,0)$ if $k=2$) and, by Proposition 4.13 in [5] (the proof
is also valid for the orbit $(x,y,xz,yz)$), we can consider the normal
curvature function $K_{v_{a}^{i}}(w)=\langle\eta(w),v_{a}^{i}\rangle=$

$=\alpha({a}^{i}_{200}\cos^{2}\theta+2{a}^{i}_{110}\cos\theta\sin\theta+{a}^{i}_{020}\sin^{2}\theta+\cos\theta\frac{\cos\phi}{\sin\phi}+2a_{011}^{i}\sin\theta\frac{\cos\phi}{\sin\phi})+$
$+\beta({a}^{i+1}_{200}\cos^{2}\theta+2{a}^{i+1}_{110}\cos\theta\sin\theta+{a}^{i+1}_{020}\sin^{2}\theta+2a_{011}^{i+1}\sin\theta\frac{\cos\phi}{\sin\phi}).$

The partial derivative with respect to $\phi$ is 0 if and only if
$\alpha\cos\theta+2(\alpha a_{011}^{i}+\beta
a_{011}^{i+1})\sin\theta=\langle(\alpha,2(\alpha a_{011}^{i}+\beta
a_{011}^{i+1})),(\cos\theta,\sin\theta)\rangle=0$, so there are two values of
$\theta$ for which we may have critical points. Substituting
$\alpha\cos\theta+2(\alpha a_{011}^{i}+\beta a_{011}^{i+1})\sin\theta=0$ in
$K_{v_{a}^{i}}(w)$ we can see that the critical value does not depend on
$\phi$. The critical points are two lines of antipodal points in the cylinder
$C_{q}$. Since the second fundamental form is quadratic homogeneous, the image
of antipodal points is the same, so the image of the two lines is the same and
there is only 1 critical value. ∎

######  Corollary 5.16.

For $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$ there are at least 2
and at most 4 axial curvatures when $k=1$ and at least 4 and at most 5 axial
curvatures when $k=2$, which may coincide in degenerate cases.

###### Proof.

From Proposition 3.5, $l(n-1)$ is a higher bound for the number of axial
curvatures.

For $k=1$, $l=2$ and so there are at most 4 axial curvatures. This higher
bound is attained in orbits such as $(x,y,z^{2},0)$ or $(x,y,0,0)$, where the
axial curvatures coincide with principal curvatures of the associated regular
surfaces (by Theorems 5.9, 5.10, and 5.12). On the other hand, there are at
most 2 directions in which the curvature locus is unbounded on both sides
(i.e. in the orbit $(x,y,xz,yz)$) and by Proposition 5.15, there will be
exactly 1 axial curvature in each.

For $k=2$, $l=3$. However, the higher bound 6 is not attained since by the way
the adapted frame is chosen, whenever there are 2 primary and 2 secondary
axial curvatures, there is only 1 3-ary axial curvature ($v_{a}^{3}$ is
perpendicular to the plane that contains the curvature locus, and so the
projection to this direction gives only 1 value). On the other hand, there are
at most two directions in which the curvature locus is unbounded on both sides
of the direction of the axial vector (in the orbits $(x,y,xz,yz,z^{2})$ and
$(x,y,xz,yz,0)$) by Proposition 5.15 there is only 1 axial curvature in each
of these directions. In the remaining direction there will be 2 axial
curvatures.

In degenerate cases, when the curvature locus is a segment, for example, two
$i$-ary curvatures might coincide. In this case, there might be only 3
different axial curvatures when $k=2$. ∎

As corollaries of Theorems 5.9, 5.10, 5.12 and Proposition 5.15 we get some
interesting geometrical interpretations.

######  Corollary 5.17.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ and suppose that $f$
lies in the orbits $(x,y,z^{2},0)$ or $(x,y,0,0)$, the Gaussian curvature of
the regular surface given by $f(x,y,0)$ is given by

$K=\kappa_{a_{1}}^{1}\kappa_{a_{1}}^{2}+\kappa_{a_{2}}^{1}\kappa_{a_{2}}^{2}.$

In particular, this includes when $f$ is a frontal and $f(x,y,0)$ is a
cuspidal surface or a cuspidal surface with a curve of cuspidal cross-caps.

###### Proof.

Given a regular surface in $\mathbb{R}^{4}$ and $\\{v_{1},v_{2}\\}$ an
orthonormal basis of the normal plane, by Theorem 1 in [1] (which can be found
as Theorem 7.1 in [10]), the Gaussian curvature is $K=K_{1}+K_{2}$, where
$K_{i}$ is the Gaussian curvature of the regular surface in $\mathbb{R}^{3}$
given by the projection in the normal direction orthogonal to $v_{i}$. Given
the adapted frame $\\{v_{a}^{1},v_{a}^{2}\\}$ of $N_{p}M$, then $K_{i}$ is the
product of the $v_{a}^{i}$-principal curvatures, $i=1,2$. Therefore, by
Theorems 5.9, 5.10 and 5.12, we get
$K=\kappa_{a_{1}}^{1}\kappa_{a_{1}}^{2}+\kappa_{a_{2}}^{1}\kappa_{a_{2}}^{2}.$
∎

######  Corollary 5.18.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ and suppose that $f$ is
such that the normal curvature tensor $R_{D}(p)=0$, then the curvature locus
of the associated $M^{2}_{\operatorname{reg}}$ is a segment with extremal
points $P_{1}=(\kappa_{a_{1}}^{1},\kappa_{a_{2}}^{1})$ and
$P_{2}=(\kappa_{a_{1}}^{2},\kappa_{a_{2}}^{2})$.

###### Proof.

For a regular $n$-manifold in $\mathbb{R}^{n+2}$, when $R_{D}(p)=0$ there is a
unique orthonormal basis of eigenvectors $\\{X_{1},\ldots,X_{n}\\}$ of the
shape operator $A_{v}$ for all $v$ normal vectors. In [17] it is shown that,
in this case the curvature locus is an $n$-polygon (a convex polygon with $n$
sides) such that the vertices are given by $(\lambda_{i},\mu_{i})$ where
$\lambda_{i}$ and $\mu_{i}$ are the eigenvalues corresponding to
$\\{X_{1},\ldots,X_{n}\\}$ for $v_{1}$ and $v_{2}$, respectively. The result
follows from Theorems 5.9 and 5.10 ∎

######  Corollary 5.19.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ and suppose that $f$
lies in the orbits $(x,y,xz,z^{2})$ or $(x,y,xz,0)$. If $f$ is given in Monge
form and $a_{020}^{2}=0$,then the curvature of the curve $f(0,y,0)$ is equal
to $\kappa_{a_{2}}$ if $f$ lies in the first orbit and equal to
$\kappa_{a_{1}}$ otherwise. In particular, this is the curvature of the curve
of cross-caps when $f$ is in the orbit $(x,y,xz,z^{2})$ and it is the
curvature of the curve of swallowtail points for the corresponding example in
the orbit $(x,y,xz,0)$.

###### Proof.

The 2-jet of the curve $f(0,y,0)$ is given by $(0,y,a_{020}^{1}y^{2},0)$. The
result now follows by Proposition 5.15. ∎

The following result gives a formula to calculate the axial curvatures.

######  Proposition 5.20.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$ be given by the image
of $f$ in Monge form as in Lemma 5.3. Suppose $\kappa^{1}_{a_{1}}(p)$ and
$\kappa^{2}_{a_{1}}(p)$ are defined (i.e. finite), then:

  1. (1)

When $(\bar{a}^{k+1}_{020}-\bar{a}^{k+1}_{200})\neq 0$. If
$\bar{a}^{k+1}_{110}\neq 0$, there exist two primary axial curvatures given by

$\kappa^{i}_{a_{1}}(p)=K_{v^{1}_{a}}\left(\theta_{i},\frac{\pi}{2}\right),$

where $\left(\theta_{i},\frac{\pi}{2}\right)$ are two critical points of

$K_{v^{1}_{a}}(w)=\langle\eta(w),v_{a}\rangle=\bar{a}^{k+1}_{200}\cos^{2}\theta+\bar{a}^{k+1}_{110}\sin(2\theta)+\bar{a}^{k+1}_{020}\sin^{2}\theta+\cot^{2}{\phi}.$

In particular, if $\bar{a}^{k+1}_{110}=0$, then

$\kappa^{1}_{a_{1}}(p)=\bar{a}^{k+1}_{200}\mbox{ and
$\kappa^{2}_{a_{1}}(p)=\bar{a}^{k+1}_{020}$}.$

  2. (2)

When $\bar{a}^{k+1}_{020}=\bar{a}^{k+1}_{200}$,

$\kappa^{1}_{a_{1}}(p)=\kappa^{2}_{a_{1}}(p)=\bar{a}^{k+1}_{200}-|\bar{a}^{k+1}_{110}|.$

###### Proof.

By Lemma 5.3, we can consider the primary axial vector as
$v_{a_{1}}=(0,\ldots,0,1)$ and

$\eta(w)=(*,\ldots,*,\bar{a}^{k+1}_{200}\alpha^{2}+2\bar{a}^{k+1}_{110}\alpha\beta+\bar{a}^{k+1}_{020}\beta^{2}+\gamma^{2}),$

where $w=(\alpha,\beta,\gamma)\in C_{q}$. Then the axial normal curvature at
the direction $v_{a_{1}}$ is given by

$K_{v_{a}^{1}}(w)=\langle\eta(w),v_{a}^{1}\rangle=\bar{a}^{k+1}_{200}\alpha^{2}+2\bar{a}^{k+1}_{110}\alpha\beta+\bar{a}^{k+1}_{020}\beta^{2}+\gamma^{2}$

As $\alpha^{2}+\beta^{2}=1$, we can take $\alpha=\cos\theta$,
$\beta=\sin\theta$, $\gamma=\mathrm{cotan}\phi$, then

$K_{v_{a}^{1}}(w)=\langle\eta(w),v_{a}^{1}\rangle=\bar{a}^{k+1}_{200}\cos^{2}\theta+2\bar{a}^{k+1}_{110}\cos\theta\sin\theta+\bar{a}^{k+1}_{020}\sin^{2}\theta+\mathrm{cotan}^{2}\phi$

The critical points of $K_{v_{a}^{1}}$ are the points $(\theta,\frac{\pi}{2})$
such that $\theta$ satisfies the equation

$(\bar{a}^{k+1}_{020}-\bar{a}^{k+1}_{200})\sin(2\theta)+2\bar{a}^{k+1}_{110}\cos(2\theta)=0.$

We have the following cases:

  1. (1)

When $(\bar{a}^{k+1}_{020}-\bar{a}^{k+1}_{200})\neq 0$, then
$\sin(2\theta)=\frac{2\bar{a}^{k+1}_{110}}{(\bar{a}^{k+1}_{200}-\bar{a}^{k+1}_{020})}\cos(2\theta)$.
Thus, if $\cos(2\theta)\neq 0$, we obtain
$\theta=\frac{1}{2}\arctan\left(\frac{2\bar{a}^{k+1}_{110}}{\bar{a}^{k+1}_{200}-\bar{a}^{k+1}_{020}}\right)$.
Note that there are four possibles values of $\theta$, which we call
$\theta_{i}$, $i=1,2,3,4$, such that
$K_{v_{a}^{1}}(\theta_{i},\pi/2)=\bar{a}^{k+1}_{020}+(\bar{a}^{k+1}_{200}-\bar{a}^{k+1}_{020})\cos^{2}(\theta_{i}))+\bar{a}^{k+1}_{110}\sin(2\theta_{i}))$
is a critical value, but there are only two critical values. Here the axial
curvatures are given by

$\kappa_{a_{1}}^{j}=\\{K_{v_{a}^{1}}\left(\theta_{i},\frac{\pi}{2}\right)|\mbox{
for $i=1,2,3,4$}\\}.$

In particular, $\bar{a}^{k+1}_{110}=0$ iff $\sin(2\theta)=0$, then
$(0,\frac{\pi}{2}),(\pi,\frac{\pi}{2})$ and
$(\frac{\pi}{2},\frac{\pi}{2}),(\frac{3\pi}{2},\frac{\pi}{2})$ are the
critical points, so

$\kappa^{1}_{a_{1}}(p)=\bar{a}^{k+1}_{200}\mbox{ and
$\kappa^{2}_{a_{1}}(p)=\bar{a}^{k+1}_{020}$}.$

  2. (2)

If $\bar{a}^{k+1}_{020}=\bar{a}^{k+1}_{200}$ and $\bar{a}^{k+1}_{110}\neq 0$,
we get $\cos(2\theta)=0$. Thus, if $\bar{a}^{k+1}_{110}>0$,
$(\frac{3\pi}{4},\frac{\pi}{2})$ and $(\frac{7\pi}{4},\frac{\pi}{2})$ are
critical points and if $\bar{a}^{k+1}_{110}<0$, the critical points are
$(\frac{\pi}{4},\frac{\pi}{2})$ and $(\frac{5\pi}{4},\frac{\pi}{2})$, in both
cases

$\kappa_{a_{1}}^{1}(p)=\kappa^{2}_{a_{1}}(p)=\bar{a}^{k+1}_{200}-|\bar{a}^{k+1}_{110}|.$

In particular, when $\bar{a}^{k+1}_{110}=0$, $(\theta,\pi/2)$ are critical
points of $K_{v_{a}^{1}}$ for all $\theta\in\left[0,2\pi\right)$ and the axial
curvature is
$\kappa_{a_{1}}^{1}=\kappa_{a_{1}}^{2}=\bar{a}^{k+1}_{020}=\bar{a}^{k+1}_{200}$.

∎

######  Corollary 5.21.

Let $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ be given by the image
of $f$ in Monge form as in Lemma 5.3 when $rank(A)=1$, i.e. $f$ lies in the
orbit $(x,y,z^{2},0)$, then:

  1. (1)

When $(\bar{a}^{1}_{020}-\bar{a}^{1}_{200})\neq 0$. If $\bar{a}^{1}_{110}\neq
0$, there exist at most two secondary axial curvatures given by

$\kappa^{i}_{a_{2}}(p)=K_{v^{2}_{a}}\left(\theta_{i}\right),$

where $\theta_{i}$ are critical points of

$K_{v^{2}_{a}}(w)=\langle\eta(w),v^{2}_{a}\rangle=-(\bar{a}^{1}_{200}\cos^{2}\theta+\bar{a}^{1}_{110}\sin(2\theta)+\bar{a}^{1}_{020}\sin^{2}\theta).$

In particular, if $\bar{a}^{1}_{110}=0$, then

$\kappa^{1}_{a_{2}}(p)=-\bar{a}^{1}_{200}\mbox{ and
$\kappa^{2}_{a_{2}}(p)=-\bar{a}^{1}_{020}$}.$

  2. (2)

When $\bar{a}^{1}_{020}=\bar{a}^{1}_{200}$,

$\kappa^{1}_{a_{2}}(p)=\kappa^{2}_{a_{2}}(p)=-(\bar{a}^{1}_{200}-|\bar{a}^{1}_{110}|).$

###### Proof.

The proof follows from the fact that $v^{1}_{a}=(0,1)$ and $v_{a}^{2}=(0,-1)$
and analogous arguments to the proof of Proposition 5.20. ∎

######  Remark 5.22.

The obstruction to generalizing the above result for
$M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{3+k}$ with $k>1$ is the fact
that, although with the normal form of Lemma 5.3 $v_{a}^{1}=(0,\ldots,0,1)$,
the secondary axial vector may not coincide with one of the axes in that
coordinate system.

######  Example 5.23.

  1. i)

Consider the frontal 3-manifold given by
$(x,y,\frac{1}{2}(a^{1}_{200}x^{2}+a^{1}_{020}y^{2}+z^{2}),\frac{1}{2}(a^{2}_{200}x^{2}+a^{2}_{020}y^{2}+yz^{3})).$
Here $(x,0,\frac{1}{2}a^{1}_{200}x^{2},\frac{1}{2}a^{2}_{200}x^{2})$ is a
curve of cuspidal cross-cap points. Let $\kappa$ be the curvature of this
curve, then, by Proposition 5.20 and Corollary 5.21
$\kappa^{2}=(\kappa_{a_{1}}^{1})^{2}+(\kappa_{a_{2}}^{1})^{2}$.

  2. ii)

Consider the frontal 3-manifold given by
$(x,y,\frac{1}{2}(a^{1}_{200}x^{2}+a^{1}_{110}xy+a^{1}_{020}y^{2}+2xz+4z^{3}),\frac{1}{2}(a^{2}_{200}x^{2}+a^{2}_{110}xy+xz^{2}+3z^{4})).$
Here $(0,y,\frac{1}{2}a^{1}_{020}y^{2},0)$ is a curve of swallowtail points
and by Proposition 5.15 the curvature of this curve is given by
$\kappa_{a_{1}}$.

######  Example 5.24.

Consider $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ given by

$f(x,y,z)=(x,y,\frac{1}{2}x^{2}+\frac{7}{2}y^{2},\frac{3}{2}x^{2}+xy+\frac{1}{2}y^{2}+\frac{1}{2}z^{2})$

Observe that $f$ lies in the orbit $(x,y,z^{2},0)$ and the primary axial
vector is $v^{1}_{a}=(0,1)$. Direct calculation shows
$(\frac{\pi}{8},\frac{\pi}{2}),(\frac{5\pi}{8},\frac{\pi}{2})$ are critical
points of $K_{v^{1}_{a}}$ and
$\kappa^{i}_{a_{1}}(p)=1+\sin(2\theta_{i})+2\cos^{2}(\theta_{i})$, so for
$\theta=\frac{\pi}{8},\frac{5\pi}{8}$ $\kappa^{1}_{a_{1}}(p)=2+\sqrt{2}$ and
$\kappa^{2}_{a_{1}}(p)=2-\sqrt{2}$, respectively. For the secondary axial
vector $v^{2}_{a}=(-1,0)$, we get $\theta=0,\frac{\pi}{2}$ are critical points
of $K_{v^{2}_{a}}$. So $\kappa^{1}_{a_{2}}(p)=-1$ and
$\kappa^{2}_{a_{2}}(p)=-7$. It follows from Corollary 5.17 that the Gaussian
curvature of the regular surface $f(x,y,0)$ is

$K=\kappa_{a_{1}}^{1}\kappa_{a_{1}}^{2}+\kappa_{a_{2}}^{1}\kappa_{a_{2}}^{2}=(2+\sqrt{2})\cdot(2-\sqrt{2})+(-1)\cdot(-7)=2+7=9.$

######  Example 5.25.

Consider $M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{5}$ given by

$f(x,y,z)=(x,y,x^{2},y^{2},\frac{3}{2}x^{2}+xy+\frac{1}{2}y^{2}+\frac{1}{2}z^{2})$

The primary axial vector is $v^{1}_{a}=(0,0,1)$ and
$(\frac{\pi}{8},\frac{\pi}{2}),(\frac{5\pi}{8},\frac{\pi}{2})$ are the
critical points of $K_{v^{1}_{a}}$. The axial curvatures at the origin are
given by $\kappa^{i}_{a}(p)=1+\sin(2\theta)+2\cos^{2}(\theta)$, where
$\theta=\frac{\pi}{8},\frac{5\pi}{8}$. Now, consider $u$ a tangent direction
in $T_{p}M^{3}_{\operatorname{sing}}$ parametrised by the angle
$\gamma\in\left[0,2\pi\right)$, then the normal section of
$M^{3}_{\operatorname{sing}}$ along $u$ is a corank 1 surface
$M^{2}_{\operatorname{sing}}=M^{3}_{\operatorname{sing}}\cap\\{u=Y-\tan(\gamma)X=0\\}$
that is locally parametrised by

$(x,z)\to(x,\tan(\gamma)x,x^{2},\tan^{2}(\gamma)x^{2},\frac{1}{2}\tan^{2}(\gamma)x^{2}+\tan(\gamma)x^{2}+\frac{3}{2}x^{2}+\frac{1}{2}z^{2})$

By a rotation of angle $\gamma\neq\pi/2,3\pi/2$ in the target and the change
of coordinates in the source,
$(x,z)\mapsto\left(\frac{x}{\tan^{2}(\gamma)+1},z\right)$ we obtain the
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ given by $(x,z)\mapsto$

$(x,0,\frac{x^{2}}{\tan^{2}(\gamma)+1},\frac{\tan^{2}(\gamma)}{\tan^{2}(\gamma)+1}x^{2},\\\
\frac{1}{2}\frac{(\tan^{2}(\gamma)+2\tan(\gamma)+3)x^{2}}{\tan^{2}(\gamma)+1}+\frac{1}{2}z^{2}).$

Here, the curvature locus is
$\eta(z)=(*,*,\frac{(\tan^{2}(\gamma)+2\tan(\gamma)+3)}{\tan^{2}(\gamma)+1}+z^{2})$
and the primary axial vector for this singular surface is $v^{1}_{a}=(0,0,1)$.
Then, using Proposition 4.1, the primary axial curvature of
$M^{2}_{\operatorname{sing}}$ at the origin is
$k_{a_{1}}(p)=\frac{(\tan^{2}(\gamma)+2\tan(\gamma)+3)}{\tan^{2}(\gamma)+1}=1+\sin(2\gamma)+2\cos^{2}(\gamma)$.
Note that, the primary axial curvature of $M^{2}_{\operatorname{sing}}$
coincides with the primary axial curvature of $M^{3}_{\operatorname{sing}}$
when $\gamma=\theta$.

Based on the above example it is natural to ask whether there is a relation
between the axial curvatures at $p\in
M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{5}$ and at $p\in
M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ obtained as a normal section
of $M^{3}_{\operatorname{sing}}$.

######  Theorem 5.26.

Let $M^{3}_{\operatorname{sing}}$ be a corank 1 3-manifold in $\mathbb{R}^{5}$
and consider $M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ the normal
section of $M^{3}_{\operatorname{sing}}$ along $u\in
T_{p}M^{3}_{\operatorname{sing}}$. The $i$-ary axial curvatures at $p\in
M^{3}_{\operatorname{sing}}$ (whenever they are defined) are given by the
critical values of the $i$-ary axial curvatures of the normal sections varying
$u\in T_{p}M^{3}_{\operatorname{sing}}$.

###### Proof.

Following Theorem 3.3 in [2] the curvature locus of
$M^{3}_{\operatorname{sing}}$ is generated by the union of the curvature
parabolas of the normal sections. This is due to the fact that the unitary
tangent vectors $C_{q}^{\prime}$ of the normal section are given by $C_{q}\cap
dg^{-1}(u)$, where $g:\tilde{M}\to M^{3}_{\operatorname{sing}}$ is the corank
1 map germ and $C_{q}$ is the unitary tangent vectors in $T_{q}\tilde{M}$.
This way the primary axial vector is the same for the 3-manifold and any
normal section, and so we have the relation between the axial curvatures. In
fact, the $i$-ary normal axial curvature function $K_{a_{i}}(\theta,\phi)$ at
$\theta=\gamma_{0}$, where $\gamma_{0}$ is the angle which parametrises $u$,
is precisely the $i$-ary normal curvature function of the normal section given
by the angle $\gamma_{0}$, $K_{a_{i}}^{\gamma_{0}}(\phi)$. So the critical
values of $K_{a_{i}}(\theta,\phi)$ are the critical values of the function
$h:[0,2\pi[\to\mathbb{R}$ given by $h(\gamma)=(\text{critical value of
}K_{a_{i}}^{\gamma}(\phi)\text{ where }\phi\in[0,2\pi[)$ when $\gamma$ varies
in $[0,2\pi[$. ∎

##  6\. Relation of axial curvatures and umbilic curvatures

Throughout the literature the umbilic curvature has been defined in many
different contexts, both for regular and singular manifolds. It was first
defined by Montaldi in [15] for semi-umbilic points in
$M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{4}$. Then it was defined by
Mochida, Romero Fuster and Ruas in [14] for
$M^{2}_{\operatorname{reg}}\subset\mathbb{R}^{5}$. More recently, Deolindo-
Silva and Oset Sinha in [8] defined it for
$M^{3}_{\operatorname{reg}}\subset\mathbb{R}^{6}$ (in [6] Binotto, Costa and
Romero Fuster mention focal spheres but do not define the umbilic curvature).
It has also been defined for singular manifolds, namely, Martins and Nuño-
Ballesteros in [12] defined it for
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{3}$ (it was also defined in the
frontal context by Martins and Saji in [13] where they called it the limiting
normal curvature) and finally, Benedini Riul, Oset Sinha and Ruas for
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$ in [3]. The name “umbilic”
is not casual, this curvature defines the centre of a sphere with degenerate
contact with the manifold at $p$, also known as an umbilical focus and
umbilical focal hyperspheres. For surfaces it is a sphere with corank 2
contact (i.e. corank 2 singularity of the distance squared function), for
3-manifolds it is a sphere with corank 3 contact. This was proved in the
previous references when it was defined, except for the case
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$, where it was proved by
Deolindo-Silva and Oset Sinha in [8].

The umbilic curvature has been defined in many different ways but all
definitions can be unified in the following way. Given
$M^{n}\subset\mathbb{R}^{n+k}$ (regular or singular) the umbilic curvature is
given by $\kappa_{u}(p)=d(p,Aff_{p})$. If $\dim N_{p}M>n$ this definition
makes sense always, if $\dim N_{p}M\leq n$, this is only defined when the
curvature locus is degenerate. For example, Montaldi only defined this
curvature when the curvature locus is a segment. Similarly, Martins and Nuño-
Ballesteros also defined the umbilic curvature when the curvature locus is a
degenerate parabola. However, Benedini Riul, Oset Sinha and Ruas defined it
even for non-degenerate parabolas, following the ideas of Mochida, Romero
Fuster and Ruas. Another way of defining this curvature is by projecting to a
direction perpendicular to $Aff_{p}$ (since $d(p,Aff_{p})$ is the shortest
distance from $p$ to points in $Aff_{p}$).

Focusing now in the singular case, in most cases, the umbilic curvature is one
of our axial curvatures.

######  Proposition 6.1.

If $n\geq\dim N_{p}M=k+1$, when $\kappa_{u}$ is defined,
$\kappa_{u}=|\kappa_{a_{i}}|$ for some $i=2,\ldots,l$. More precisely, if
$Aff_{p}$ has codimension $r<k+1$ in $N_{p}M$, then
$\kappa_{u}=|\kappa_{a_{l-r+1}}|$. When $r=k+1$, the curvature locus is a
point and $\kappa_{u}=|\kappa_{a_{2}}|$ by definition.

###### Proof.

We have defined our adapted frame only in $Ax_{p}$, and so we only have $l$
axial vectors. If $n\geq\dim N_{p}M=k+1$, then $l=k+1$, and so
$Ax_{p}=N_{p}M$. In this case, $\kappa_{u}$ is only defined when the curvature
locus is degenerate, i.e. $Aff_{p}$ is not the whole $N_{p}M$. So $\kappa_{u}$
is the projection of the curvature locus onto a direction perpendicular to
$Aff_{p}$. The way in which the adapted frame has been defined, this direction
is one of the axial vectors. In fact, when $Aff_{p}$ is a hyperplane, this
direction is $v_{a}^{l}$ and so $\kappa_{u}=|\kappa_{a_{l}}|$. If $Aff_{p}$
has codimension 2 in $N_{p}M$ then this direction is $v_{a}^{l-1}$,
$\kappa_{u}=|\kappa_{a_{l-1}}|$ and $\kappa_{a_{l}}=0$. This goes on until
codimension $k$. When $r=k+1$, the curvature locus is a point and
$\kappa_{u}=|\kappa_{a_{2}}|$ by definition (there is no need for the absolute
value here because $\kappa_{a_{2}}$ is defined as norm in this case). ∎

If $n<\dim N_{p}M=k+1$ there are less axial vectors than the dimension of the
normal space, and so the umbilic curvature may be given by a projection to a
normal direction which is not in $Ax_{p}$. However, consider the
$(k+1-l)$-vector space of orthogonal directions to the directions in $Ax_{p}$.
The intersection of this vector space and $Ax_{p}$ is a point. Consider the
direction between this point and $p$. We call the unitary vector in this
direction $v_{a}^{l+1}$. Notice that the projection of the curvature locus
onto this direction is constant. We call this constant $\kappa_{v_{a}^{l+1}}$.

######  Proposition 6.2.

If $n<\dim N_{p}M=k+1$ then $\kappa_{u}=|\kappa_{v_{a}^{l+1}}|$ when $\dim
Aff_{p}=l$ and $\kappa_{u}=|\kappa_{a_{i}}|$ for some $i=2,\ldots,l$
otherwise.

###### Proof.

In this case, $l=n$. When $\dim Aff_{p}<l$, then $Ax_{p}$ contains $Aff_{p}$
and $p$, so, similarly to the above Proposition, $\kappa_{u}=|\kappa_{a_{i}}|$
for some $i=2,\ldots,l$. When $\dim Aff_{p}=l$, then $Ax_{p}=Aff_{p}$ and so
$d(p,Aff_{p})$ coincides with $|\kappa_{v_{a}^{l+1}}|$ by definition. ∎

######  Example 6.3.

In [3] the umbilic curvature $\kappa_{u}$ was defined for
$M^{2}_{\operatorname{sing}}\subset\mathbb{R}^{4}$. Here $Ax_{p}$ is a plane
with adapted frame $\\{v_{a}^{1},v_{a}^{2}\\}$. When the curvature parabola is
degenerate (a half-line, a line or a point) $\kappa_{u}=|\kappa_{a_{2}}|$.
When the curvature parabola is non-degenerate, $\kappa_{u}$ is the height of
$Aff_{p}=Ax_{p}$ and so $\kappa_{u}=|\kappa_{v_{a}^{3}}|$ as defined above.

######  Remark 6.4.

In the same way as for all other situations where the umbilic curvature has
been defined, a geometric interpretation of the umbilical curvature for
$M^{3}_{\operatorname{sing}}\subset\mathbb{R}^{5}$ can be given. For example,
when $\dim Aff_{p}=2$ (i.e. the curvature locus is a planar region), then
there exists a unique umbilical focus at $p$ given by

$a=p+\frac{1}{\kappa_{u}}v_{a}^{3}.$

In fact, here $\kappa_{u}=|\kappa_{v_{a}^{3}}|$. Similar results can be
obtained when $\dim Aff_{p}<2$, although instead of one umbilical focus there
may be a line of umbilical foci. The proof relies on the analysis of the
Hessian of the distance squared function and is analogous to the proof of
Proposition 6.6 in [8].

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