# The normal map as a vector field

Thomas Waters∗ Matthew Cherrie  School of Mathematics and Physics, University
of Portsmouth, England PO13HF<EMAIL_ADDRESS>
###### Abstract

In this paper we consider the normal map of a closed plane curve as a vector
field on the cylinder. We interpret the critical points geometrically and
study their Poincaré index, including the points at infinity. After projecting
the vector field to the sphere we prove some counting theorems regarding the
winding and rotation index of the curve and its evolute. We finish with a
description of the extension to focal sets of surfaces.

##  1 Introduction

The envelope/caustic/focal set of the normal lines to a plane curve is known
as the ‘evolute’ and is a classical topic in Differential Geometry (see Figure
1 for an example). Evolutes have been studied by many authors over the years:
most well known texts follow a traditional approach ([6],[4],[19]), some treat
the topic in terms of contact and singularity ([5],[26],[13]) and others take
a more abstract approach ([2], [8], [9], [1]). While known and studied as far
back as Huygens [31] (or even earlier [11]), one reason for interest in
evolutes is as a Euclidean analogue for the conjugate locus ([25], [23], [28],
[27]); indeed Jacobi’s original formulation of his last geometric statement,
recently proved by Itoh and Kiyohara [12], was that the conjugate locus of a
generic point on the triaxial ellipsoid has “die gestalt der evolute der
ellipse”[14].

Figure 1: On the left in blue is a plane curve of the type described at the
beginning of Section 2, in red its evolute and the normal segments with
$\rho<0$ and $0<\rho<1/k$ in green and red repsectively. On the right the
normal map of the same curve (with origin at $A$) as a vector field; in red
the curve $\rho=1/k$ and the dots are critical points (axes are $s,\rho$).

This paper will take a novel approach: we view the normal map of a closed
plane curve as a vector field, whose phase space is the cylinder. Critical
points of this vector field have a Poincaré index with a clear geometrical
meaning, and the index at infinity can be found in terms of the rotation index
of the original curve. Then by projecting to the 2-sphere we prove certain
counting results concerning the winding index of the plane curve and its
evolute. Before we begin we will fix some terminology and notation:

The degree of a map is a central notion in Differential Topology [24], and we
are primarily interested in the degree of a map from $\mathbb{S}^{1}$ to
$\mathbb{S}^{1}$. There are two ways to view this degree: the sum of
orientation-defined signs attached to the preimages of a regular point (see
for example [22]), or the total change in the lifted angle made by the map in
the sense of covering spaces (see for example [6] or [3]); but intuitively if
$f:\mathbb{S}^{1}\to\mathbb{S}^{1}$ then $deg(f)$ is the number of times the
codomain is covered as we traverse the domain once, taking orientation into
account. There are three main examples, all of which we will use in this work:

  1. 1.

Let $\mbox{\boldmath$\alpha$}:\mathbb{S}^{1}\to\mathbb{R}^{2}$ be a closed,
regular, smooth, positively oriented plane curve. Then the winding index of
$\alpha$ about the origin, denoted $w^{\alpha}_{o}$, is
$deg(\mbox{\boldmath$\alpha$}/|\mbox{\boldmath$\alpha$}|)$.

  2. 2.

With $\alpha$ as defined above, if $\dot{\mbox{\boldmath$\alpha$}}$ denotes
the derivative w.r.t. some appropriate parameter, then the rotation index of
$\alpha$, denoted $r^{\alpha}$, is
$deg(\dot{\mbox{\boldmath$\alpha$}}/|\dot{\mbox{\boldmath$\alpha$}}|)$.

  3. 3.

If $\mbox{\boldmath$v$}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is a smooth vector
field with an isolated critical point at $p$, i.e.
$\mbox{\boldmath$v$}(p)=\mbox{\boldmath$0$}$, let $\mbox{\boldmath$v$}^{*}$
denote $v$ restricted to a smooth simple positively oriented closed curve that
contains $p$ and no other critical point of $v$. Then the Poincaré index of
$p$, denoted $i_{p}$, is
$deg(\mbox{\boldmath$v$}^{*}/|\mbox{\boldmath$v$}^{*}|)$.

All of these definitions have various extensions which we may need to make use
of in the text, in particular it is customary to consider the Poincaré index
of a simple closed positively oriented curve in the domain of a vector field,
which is equal to the sum of the Poincaré indices of the critical points
contained by the curve [16].

In the next section we will introduce the normal map of a plane curve with
positive curvature, its connection with the evolute and its representation as
a vector field on the cylinder. We describe the meaning and structure of the
critical points of this vector field, and how they respond to moving the curve
around in the plane. In Section 3 we project the vector field onto a sphere
and prove certain counting theorems concerning the rotation and winding index
of the plane curve and its evolute, in terms of quantities which are easily
read off the vector field picture. In Section 4 we extend the discussion to
the normal map of a surface in $\mathbb{R}^{3}$.

##  2 Finite critical points

Suppose $\alpha$ is a closed, regular, smooth plane curve, with $s$ as arc-
length parameter. The exposition of what follows is clearest is we assume the
curvature $k$ has one sign; we therefore orient $\alpha$ such that $k>0$ if
$(\mbox{\boldmath$t$},\mbox{\boldmath$n$})$ (the unit tangent and normal
respectively) are right-handed. At any point of $\alpha$, we may travel a
distance $\rho$ in the direction of $n$, to arrive at the point in the plane
$\mbox{\boldmath$\alpha$}+\rho\mbox{\boldmath$n$}$. This map,

$F(s,\rho):\mathbb{S}^{1}\times\mathbb{R}\to\mathbb{R}^{2}:(s,\rho)\to\mbox{\boldmath$\alpha$}(s)+\rho\mbox{\boldmath$n$}(s)$

is known as the normal map. As well as providing a coordinate system adapted
to the curve, the normal map is of interest since its singular set (the set of
points where the map is not invertible) defines the evolute of $\alpha$ (or
rather its image under $F$ does). To see this we find the Jacobian of $F$:

$\displaystyle
DF=\begin{vmatrix}\dot{\mbox{\boldmath$\alpha$}}+\rho\dot{\mbox{\boldmath$n$}}\\\
\mbox{\boldmath$n$}\end{vmatrix}=\begin{vmatrix}\mbox{\boldmath$t$}+\rho(-k\mbox{\boldmath$t$})\\\
\mbox{\boldmath$n$}\end{vmatrix}=1-\rho k$ (1)

and therefore the image of the singular set of this map is the curve
$\mbox{\boldmath$\beta$}(s)=\mbox{\boldmath$\alpha$}+(1/k)\mbox{\boldmath$n$}$;
since $k>0$ then $\beta$ is closed and bounded. See Figure 1 on the left for
an example.

The main idea of this paper is to view the normal map as a vector field whose
domain is $\mathbb{S}^{1}\times\mathbb{R}$, the cylinder. An example is in
Figure 1 on the right. The first thing to notice is the existence of critical
points (those with $\rho<\infty$ we will call ‘finite critical points’);
geometrically a critical point at $(s^{*},\rho^{*})$ means the normal line to
$\alpha$ at $s^{*}$ passes through the origin, after travelling a distance
$\rho^{*}$ in the direction of the positively oriented unit normal. The nature
of the critical points can be classified as follows:

######  Theorem 1.

Suppose $\alpha$ is in generic position, meaning its evolute does not pass
through the origin. Then the finite critical points have Poincaré index $+1$
(which we shall call ‘centres’) or $-1$ (which we shall call ‘saddles’); the
saddles all lie above the line $\rho=1/k$, and the centres lie below it. There
are as many saddles as there are centres (so the sum of the Poincaré indices
of the finite critical points is zero), and there is at least one of each.

###### Proof.

If the evolute does not pass through the origin then the critical points are
non-degenerate and their Poincaré index is, from (1), the sign of $1-\rho k$
(see [22]); therefore saddles have $\rho>1/k$ and centres $\rho<1/k$. To
distinguish saddles and centres geometrically we recall the distance squared
function
$D^{2}:\mathbb{S}^{1}\to\mathbb{R}^{+}:s\to\mbox{\boldmath$\alpha$}.\mbox{\boldmath$\alpha$}$
which is stationary when
$\mbox{\boldmath$\alpha$}.\dot{\mbox{\boldmath$\alpha$}}=0$, i.e. when the
position vector at $s=s^{*}$ is perpendicular to the tangent line. Hence the
normal to $\alpha$ at $s^{*}$ passes through the origin, in fact
$\mbox{\boldmath$\alpha$}(s^{*})=-\rho^{*}\mbox{\boldmath$n$}(s^{*})$. Thus
critical points of the vector field correspond to stationary points of the
$D^{2}$ function. To classify the stationary points we note

$\frac{d^{2}(D^{2})}{ds^{2}}=\dot{\mbox{\boldmath$\alpha$}}.\dot{\mbox{\boldmath$\alpha$}}+\mbox{\boldmath$\alpha$}.\ddot{\mbox{\boldmath$\alpha$}}=1-\rho
k.$

Therefore critical points with Poincaré index $-1$ (saddles) correspond to
maxima of $D^{2}$, and those with Poincaré index $+1$ (centres) correspond to
minima. Since $D^{2}$ is defined on a circle there must be at least one
maximum and one minimum, and since $\chi(\mathbb{S}^{1})=0$ there must be as
many saddles as centres. ∎

An immediate advantage to the vector field approach is that we can tell at a
glance from the right of Figure 1 how many normals to $\alpha$ pass through
the origin; this is a locally constant function on
$\mathbb{R}^{2}/\mbox{\boldmath$\beta$}$. We can also quickly see what point
on $\alpha$ is closest to the origin, and how far it is. Suppose we were
interested in some point other than the origin, say $p$? We can either move
our coordinate system or move the curve so the origin is at $p$. As we do, we
may find the origin crosses the evolute and then the number of normals through
the origin will change by $\pm 2$; this is manifested in the vector field by
the creation or annihilation of a saddle-centre pair of critical points.

To sketch the details we suppose there is a critical point on the curve
$\rho=1/k$, and then we perturb it. The exposition is clearest if we suppose
we translate $\alpha$ in such as way that the origin passes though the evolute
perpendicularly, i.e. we let
$\mbox{\boldmath$\alpha$}\to\mbox{\boldmath$\alpha$}+\epsilon\mbox{\boldmath$t$}$.
Then, to second order in $\delta s,\delta\rho$, the normal map looks like

$F(s^{*}+\delta
s,\rho^{*}+\delta\rho)=\mbox{\boldmath$n$}[\delta\rho+O(3)]+\mbox{\boldmath$t$}[\epsilon-\tfrac{1}{2}\rho\dot{k}\delta
s^{2}-k\delta s\delta\rho+O(3)]$

(we have dropped the ∗ on the RHS). Where are the critical points? Since $t$
and $n$ are orthonormal we require $\delta\rho=0$ which means $\delta
s=\pm\sqrt{2\epsilon/\rho\dot{k}}$. Therefore if $2\epsilon/\rho\dot{k}>0$
there will be two critical points, a saddle and a centre on either side of the
curve $\rho=1/k$; otherwise none.

The following corollary is now as easy to prove as it is to state: suppose $p$
is some point in the plane not on $\alpha$ or $\beta$; emanating from $p$ are
an even number of line segments that reach $\alpha$ normally; precisely half
of these line segments are tangential to $\beta$ on the way to meeting
$\alpha$, the other half reach $\alpha$ first. As proof we note that, after
shifting the origin to $p$, there are as many saddles as there are centres and
the saddles all have $\rho>1/k$, which means the normals to $\alpha$ pass
through their focal point before reaching $p$; the centres have $\rho<1/k$ and
so the normals reach $p$ first.

##  3 Global results

To understand the global properties of the vector field we need to look at
infinity. The cylinder has two points at infinity, both $\rho\to+\infty$ and
$\rho\to-\infty$. A two-point compactification of the cylinder could be the
Mercator projection: we orient the cylinder in $\mathbb{R}^{3}$ so
$\hat{\mbox{\boldmath$s$}}\times\hat{\mbox{\boldmath$\rho$}}$ is outward
pointing, and consider the sphere tangential to the cylinder at $\rho=0$. A
point on the cylinder $Q$ is mapped to the point on the sphere $Q^{\prime}$
which is the intersection of the line segment connecting $Q$ to $O$ with the
sphere. In what follows it may also be necessary to further vertically project
a hemisphere of the sphere onto its equatorial plane. To visualize the vector
field on the sphere we may find the integral curves on the cylinder and plot
their projections on the sphere; an example is in Figure 2. We can see clearly
the saddles and centres as before (the Poincaré index is preserved under the
projection), but the points of interest are now the north and south poles, $N$
and $S$, as these correspond to the points at infinity on the cylinder.
Clearly these points are not saddles or centres; we will derive their Poincaré
index shortly. First we prove the following lemma as it will be useful to make
rigorous the rest of this section.

Figure 2: The normal map vector field from Figure 1 projected onto the unit
sphere. The north and south poles have Poincaré index 3 and -1 respectively,
as predicted in Theorem 2.

######  Lemma 1.

Let $\mbox{\boldmath$u$},\mbox{\boldmath$v$},\mbox{\boldmath$w$}$ be unit
tangent vector fields on the cylinder, and let $\gamma$ be a simple closed
positively oriented but not contractible curve on the $\rho>0$ portion of the
cylinder. We will denote by
$deg(\mbox{\boldmath$u$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$})$ the
total anticlockwise rotation of $u$ with respect to $v$ as we perform one
circuit of $\gamma$. Let $(\phantom{a})^{\prime}$ denote an object after
projection from the cylinder to the sphere and then to the equatorial plane.
Then

  1. 1.

$deg(\mbox{\boldmath$u$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$})=deg(\mbox{\boldmath$u$},\mbox{\boldmath$w$};\mbox{\boldmath$\gamma$})+deg(\mbox{\boldmath$w$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$})$,

  2. 2.

$deg(\mbox{\boldmath$u$}^{\prime},\mbox{\boldmath$v$}^{\prime};\mbox{\boldmath$\gamma$}^{\prime})=deg(\mbox{\boldmath$u$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$})$.

###### Proof.

The first statement is clear but worth stating for what follows. For the
second statement, we note that the projections described above do not preserve
angle but they do preserve parallelism and orientation. Therefore
$\mbox{\boldmath$u$}^{\prime}=\mbox{\boldmath$v$}^{\prime}$ if and only if
$\mbox{\boldmath$u$}=\mbox{\boldmath$v$}$ and $\mbox{\boldmath$u$}^{\prime}$
will cross $\mbox{\boldmath$v$}^{\prime}$ in the same sense that $u$ crosses
$v$. Therefore in traversing $\mbox{\boldmath$\gamma$}^{\prime}$ we see
$\mbox{\boldmath$u$}^{\prime}$ will have completed as many anticlockwise
rotations about $\mbox{\boldmath$v$}^{\prime}$ as $u$ will complete about $v$
as we traverse $\gamma$. ∎

######  Theorem 2.

If $\alpha$ is a closed plane curve as described in Section 2 with rotation
index $r^{\alpha}$, then the normal map associated with $\alpha$ as a vector
field projected onto the sphere has Poincaré indices

$\displaystyle i_{N}=1+r^{\alpha},\qquad i_{S}=1-r^{\alpha}.$ (2)

###### Proof.

Let $\gamma$ be the curve $\rho=\rho_{0}$ on the cylinder, and let $u$ be the
normalized $F(s,\rho_{0})$. As $\rho\to+\infty$,
$\mbox{\boldmath$\gamma$}^{\prime}$ is a small positively oriented circle
enclosing $O$, the vertical projection of $N$ onto the equatorial plane. If
$k$ is some fixed direction in this plane, then
$i_{N}=deg(\mbox{\boldmath$u$}^{\prime},\mbox{\boldmath$k$};\mbox{\boldmath$\gamma$}^{\prime})$
but by the previous lemma

$\displaystyle
deg(\mbox{\boldmath$u$}^{\prime},\mbox{\boldmath$k$};\mbox{\boldmath$\gamma$}^{\prime})$
$\displaystyle=deg(\mbox{\boldmath$u$}^{\prime},\hat{\mbox{\boldmath$s$}}^{\prime};\mbox{\boldmath$\gamma$}^{\prime})+deg(\hat{\mbox{\boldmath$s$}}^{\prime},\mbox{\boldmath$k$};\mbox{\boldmath$\gamma$}^{\prime})$
$\displaystyle=deg(\mbox{\boldmath$u$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})+1.$

Since

$\lim_{\rho\to\infty}\mbox{\boldmath$u$}=\lim_{\rho\to\infty}\frac{F(s,\rho_{0})}{|F(s,\rho_{0})|}=\mbox{\boldmath$n$}$

then
$deg(\mbox{\boldmath$u$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})\to
deg(\mbox{\boldmath$n$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})=r^{\alpha}$
and hence $i_{N}=1+r^{\alpha}$. For $i_{S}$ we can repeat the construction for
$\rho\to-\infty$, or note that the sum of the Poincaré indices of all the
critical points on the sphere must sum to 2, whereas from Theorem 1 the sum of
the Poincaré indices at the finite critical points is zero. ∎

We should note for the next section that there are two other ways of proving
this theorem: if $\alpha$ does indeed have rotation index $r^{\alpha}$, then
$n$ is regular homotopic to $(\cos(2\pi r^{\alpha}s/l),\sin(2\pi
r^{\alpha}s/l))$ where $l$ is the length of $\alpha$. Now we can either
explicitly project this vector field onto the sphere and then the plane and
calculate $i_{N}$ directly, or we can use the methods of Firby and Gardiner
[7] and note that each anticlockwise rotation of $n$ on the cylinder generates
an elliptic sector in the plane. We have instead used the approach of the
lemma as it leads nicely to the following theorems. First we define a ‘normal
segment’ to be the normal to $\mbox{\boldmath$\alpha$}(s)$ with
$-\infty<\rho<1/k(s)$.

######  Theorem 3.

Let $\alpha$ be a closed plane curve as before, and $\beta$ be its evolute.
Let $n_{p}$ be the number of normal segments to $\alpha$ through a point in
the plane $p$ (assuming $p$ is not on $\alpha$ or $\beta$). Then

$\displaystyle w^{\beta}_{p}=r^{\alpha}-n_{p}.$ (3)

###### Proof.

A translation of coordinates puts the origin at $p$. Let $\gamma$ be the curve
$\rho=1/k$ on the cylinder, and let $u$ be the normal map restricted to this
curve and normalized. Then
$deg(\mbox{\boldmath$u$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})=w^{\beta}_{p}$,
but also if $v$ is the tangent vector to $\gamma$ then

$w^{\beta}_{p}=deg(\mbox{\boldmath$u$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})=deg(\mbox{\boldmath$u$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$})+deg(\mbox{\boldmath$v$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})=deg(\mbox{\boldmath$u$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$}).$

Now projecting to the sphere and hence to the plane,
$\mbox{\boldmath$\gamma$}^{\prime}$ is a simple closed positively oriented
curve which contains $n_{p}$ saddle points and the projection of $N$;
therefore from Theorems 1 and 2 its Poincaré index is

$n_{p}(-1)+(1+r^{\alpha}).$

But this is the total rotation of $\mbox{\boldmath$u$}^{\prime}$ with respect
to $k$ as we traverse $\mbox{\boldmath$\gamma$}^{\prime}$, and

$\displaystyle
deg(\mbox{\boldmath$u$}^{\prime},\mbox{\boldmath$k$};\mbox{\boldmath$\gamma$}^{\prime})$
$\displaystyle=deg(\mbox{\boldmath$u$}^{\prime},\mbox{\boldmath$v$}^{\prime};\mbox{\boldmath$\gamma$}^{\prime})+deg(\mbox{\boldmath$v$}^{\prime},\mbox{\boldmath$k$};\mbox{\boldmath$\gamma$}^{\prime})$
$\displaystyle=deg(\mbox{\boldmath$u$},\mbox{\boldmath$v$};\mbox{\boldmath$\gamma$})+1$

which means

$n_{p}(-1)+(1+r^{\alpha})=w^{\beta}_{p}+1$

and the result follows. ∎

We can rephrase the last result in terms of simply the number of normals to
$\alpha$ through $p$. If we denote this by $N_{p}$ then (since there are as
many saddles as there are centres) $N_{p}=2n_{p}$ and

$\displaystyle w^{\beta}_{p}=r^{\alpha}-N_{p}/2,$ (4)

which extends the result of [18] to non-simple curves.

Now suppose we instead consider separately the portions of the normal segments
with $\rho<0$ and $0<\rho<1/k$. Then we have:

######  Theorem 4.

With the definitions as before,

$\displaystyle w^{\alpha}_{p}=r^{\alpha}-m_{p}$ (5)

where $m_{p}$ is the number of normal segments with $\rho<0$ through $p$.

###### Proof.

As in the previous theorem, only now we let $\gamma$ be the curve $\rho=0$ on
the cylinder. There will be $m_{p}$ centres below this line, and if $u$ is
$F(s,0)$ normalized then
$deg(\mbox{\boldmath$u$},\hat{\mbox{\boldmath$s$}};\mbox{\boldmath$\gamma$})=w^{\alpha}_{p}$.
∎

Looking at Figure 1 left and right we see examples of both the previous
theorems in action: the point $A$ has two normal segments through it, the red
line corresponding to the critical point above the horizontal axis but below
the line $\rho=1/k$, and the green line the critical point below the axis;
therefore $n_{A}=2$ and $m_{A}=1$ and we see $w^{\beta}_{A}=2-2=0$ and
$w^{\alpha}_{A}=2-1=1$. For the point $B$ we observe $n_{B}=1,m_{B}=0$ and
$w^{\beta}_{B}=2-1=1$ and $w^{\alpha}_{B}=2-0=2$. Finally we observe
$n_{C}=3,m_{C}=1$ and $w^{\beta}_{C}=2-3=-1$ and $w^{\alpha}_{C}=2-1=1$.

We can see now the advantage of viewing the normal map as a vector field: we
can see at a glance the number of centres $n_{p}$, the number of centres below
$\rho=0$ (this is $m_{p}$), and the index at the north pole $i_{N}$. From
these we can calculate $r^{\alpha},\ w^{\alpha}_{p}$ and $w^{\beta}_{p}$.
What’s more, if we look at the curve $\rho=1/k$ we may count the number of
turning points of this curve, $\nu$; these correspond to vertices of $\alpha$
which lead to cusps of $\beta$. From another work of the first author [28] we
can therefore find $r^{\beta}$ via $r^{\beta}=(\nu+2r^{\alpha})/2$. In summary

$\begin{pmatrix}r^{\alpha}\\\ r^{\beta}\\\ w^{\alpha}_{p}\\\
w^{\beta}_{p}\end{pmatrix}=\begin{pmatrix}1&0&0&0\\\ 1&\tfrac{1}{2}&0&0\\\
1&0&-1&0\\\ 1&0&0&-1\end{pmatrix}\begin{pmatrix}i_{N}\\\ \nu\\\ m_{p}\\\
n_{p}\end{pmatrix}-\begin{pmatrix}1\\\ 1\\\ 1\\\ 1\end{pmatrix},$

and all the elements of $(i_{N},\nu,m_{p},n_{p})$ can be quickly seen by
looking at the normal map as a vector field.

##  4 Normal map for surfaces

In this section we extend the dicussion to consider the normal map of surfaces
in $\mathbb{R}^{3}$. To fix ideas we at first suppose
$\mathcal{S}:\mathbb{S}^{2}\to\mathbb{R}^{3}$ is a regular, smooth, convex
topological sphere, but later we consider the more general case of
$\mathcal{S}$ having self-intersections. If
$\mbox{\boldmath$\sigma$}=\mbox{\boldmath$\sigma$}(u,v)$ is a coordinate patch
then we fix the orientation as follows: $n$ is inward pointing so the
principal curvatures are positive, and $u,v$ are ordered so
$(\mbox{\boldmath$\sigma$}_{,u},\mbox{\boldmath$\sigma$}_{,v},\mbox{\boldmath$n$})$
is right handed; then
$F(u,v,\rho)=\mbox{\boldmath$\sigma$}+\rho\mbox{\boldmath$n$}$ is the normal
map as before. As $DF$ is now $3\times 3$ there are more possibilities: for a
given $u,v$ there will be two values of $\rho$ where the normal map has corank
1: $\rho=1/k_{1}$ and $\rho=1/k_{2}$ where $k_{1}$ and $k_{2}$ are the
principal curvatures at $u,v$ (we label $k_{2}\geq k_{1}>0$). The corank may
be 2 at a point where $k_{1}=k_{2}$, known as an umbilic point . In this
context the images under $F$ of the singular sets tend to be known as ‘focal
sets’ rather than evolutes (we will label them $\mbox{\boldmath$\beta$}_{1}$
and $\mbox{\boldmath$\beta$}_{2}$ respectively), and thus the surface
$\mathcal{S}$ will have two focal sets in $\mathbb{R}^{3}$ which will
intersect each other in the large but also at umbilic points. There is much in
the literature to read: Porteous [26] and Izumiya et al [13] classify the
different types of focal point; Thielhelm [27] provides in-depth visualization
of these focal sets; Gutiérrez et al [10] describe the neighbourhood of
umbilic points and the necessity of at least two umbilic points on spheres is
the Carathéodory conjecture (see Guilfoyle [9]). Interestingly the global
picture of the focal sets is largely missing; an archetypal example to dwell
on would be the focal sets associated with the triaxial ellipsoid, and even
this picture is largely absent from the literature (but see Joets and Ribotta
[15] for a rare example). While the triaxial ellipsoid is illuminating, we
alert the reader to the fact that even for convex spheres, such as the
spherical harmonic surfaces described in [29], the focal sets are exceedingly
complex and difficult to visualize in $\mathbb{R}^{3}$; we think the vector
field approach provides some clarity.

The normal map for $\mathcal{S}$ is
$\mathbb{S}^{2}\times\mathbb{R}\to\mathbb{R}^{3}$ but vector fields in
$\mathbb{R}^{3}$ are very cluttered to visualize and so we will not provide a
plot here; see a representative sketch instead in Figure 3. As before there
are critical points which represent normals to $\mathcal{S}$ which pass
through the point we have chosen to be the origin. We classify them as
follows: suppose $(u^{*},v^{*},\rho^{*})$ corresponds to a normal through the
origin. Then $(u^{*},v^{*})$ is a stationary point of the function
$D^{2}:\mathbb{S}^{2}\to\mathbb{R}^{+}:(u,v)\to\mbox{\boldmath$\sigma$}.\mbox{\boldmath$\sigma$}$,
and a critical point of the corresponding gradient vector field on
$\mathcal{S}$, with Poincaré index

$\displaystyle\text{sgn}[(1-\rho k_{1})(1-\rho k_{2})].$ (6)

Similarly $(u^{*},v^{*},\rho^{*})$ is a critical point of the normal map
vector field and its Poincaré index is also (6). As $\mathcal{S}$ is a sphere
and $\chi(\mathbb{S}^{2})=2$ this means the sum of the Poincaré indices of the
finite critical points of the normal map vector field is 2.

A Poincaré index of $+1$ or $-1$ in $\mathbb{R}^{3}$ is somewhat less
informative than in $\mathbb{R}^{2}$, in that there are 8 non-equivalent cases
(4 from each) to consider (rather than 5) [17], however we can arrange the
phase space as follows: there are two sheets $\rho=1/k_{1}$ and $\rho=1/k_{2}$
which touch at umbilic points. There are at least two critical points, with
those above $\rho=1/k_{1}$ and below $\rho=1/k_{2}$ having Poincaré index $+1$
(maxima and minima of $D^{2}$) and those between the sheets having index $-1$
(saddles of $D^{2}$); see Figure 3 for the case of the triaxial ellipsoid. If
we now perform a two-point compactification of
$\mathbb{S}^{2}\times\mathbb{R}$ onto $\mathbb{S}^{3}$, we again may consider
the Poincaré index at the poles. Since $\chi(\mathbb{S}^{3})=0$ we know these
indices must sum to $-2$; indeed direct calculations along the lines described
after Theorem 2 shows the indices at the poles are both $-1$ (recall we are
considering $\mathcal{S}$ to be a convex sphere).

As we move the origin of the coordinate system (or move $\mathcal{S}$ around
in $\mathbb{R}^{3}$) we may have the origin cross one of the focal sets; in
the vector field this means a pair of critical points are created or
annihilated via a saddle-centre bifurcation on either $\rho=1/k_{1}$ or
$1/k_{2}$ depending on which focal set was crossed, but notice one critical
point is a maximum or minimum of $D^{2}$ depending on whether we cross the
$1/k_{1}$ or $1/k_{2}$ focal set respectively (the other critical point will
be a saddle). Since the winding index of a focal set changes by $\pm 1$ when
we cross it, and since when the point $p$ goes to infinity there will be
precisely 2 normals through it, we can extend (4) to
$N_{p}=2(w^{\beta_{1}}_{p}+w^{\beta_{2}}_{p})+2$, but this does not
distinguish $w^{\beta_{1}}_{p}$ and $w^{\beta_{2}}_{p}$. Taking a hint from
the comment above, instead of counting the number of normals through $p$ we
count the number of normals which correspond to maxima, minima and saddles of
$D^{2}$; let’s label them $n^{+}_{p},n^{-}_{p}$ and $n^{0}_{p}$ respectively.
Then we have

$w^{\beta_{1}}_{p}=n^{+}_{p}-1,\quad w^{\beta_{2}}_{p}=n^{-}_{p}-1,$

which is consistent with $n^{+}+n^{-}-n^{0}=2$.

More generally, $\mathcal{S}$ may be the image of $\mathbb{S}^{2}$ under some
map that might have self-intersections - imagine the limaçon rotated about its
axis of symmetry. In this context rather than the term ‘rotation index’ the
term ‘degree of the Gauss map’ tends to be used [22]; let’s say the degree of
the Gauss map of $\mathcal{S}$ is $d$. The local structure at the critical
points is as described previously, but now since the Euler characteristic is
twice the degree of the Gauss map (see for example [21]) the sum of the
Poincaré indices of the finite critical points should be $2d$. When looking at
the projection onto $\mathbb{S}^{3}$ calculations suggest the Poincaré index
at the poles is $-d$ and $-d$, which agrees with $\chi(\mathbb{S}^{3})=0$.
Indeed we might conjecture an extension to (4) as
$w^{\beta_{1}}_{p}=n^{+}_{p}-d,w^{\beta_{2}}_{p}=n^{-}_{p}-d$. Rigorous proofs
of the statements in this paragraph are more involved and might require an
analogue of Lemma 1; as such we leave further consideration to future work.

Figure 3: As described in the text, the two sheets $\rho=1/k_{1}$ and
$\rho=1/k_{2}$ for the triaxial ellipsoid with semi-axes $(2,1.5,1)$. The
surface $\mathcal{S}$ has been opened into the lozenge shape by choosing two
points as poles, cutting along the line joining those poles, and flattening
the surface (hence the pointed end visible is one pole and the two curved
adges are identified). The vector field for this triaxial ellipsiod, with
origin of coordinates at the centre, would have two maximum/saddle/minimum
critical points above/between/below the sheets; the linear structure for each
case is shown (above $\rho=1/k_{1}$ both critical points have the same
eigenvalues).

##  5 Conclusions

We have taken a novel approach in this paper by viewing the normal map as a
vector field, and by studying the critical points of this vector field and
making the connection with the distance function we have derived some simple
counting formulae. In this paper we have only considered the case of positive
curvature. If we allowed the curvature to change sign, then where $k$ passes
through 0 the curve $1/k$ in Figure 1 would diverge, and then when we map to
the sphere this curve would lead to loops passing through the poles. The
results from Sections 2 and 3 would need to be adapted as follows: Theorem 1
holds if we replace “the saddles all lie above the line $\rho=1/k$ and the
centres below it” with “the saddles lie above/below the line $\rho=1/k$ in the
regions where $k$ is positive/ negative respectively, with the centres
below/above it”. Theorem 2 holds with no change, however the problem with
Theorem 3 is how to meaningfully define $w_{p}^{\beta}$ if $\beta$ is
unbounded and disconnected (one option would be to ‘complete’ the evolute by
adding the normal lines to $\alpha$ where $k=0$, oriented by the asymptotic
portions of the evolute). Theorem 4 would become

$w_{p}^{\alpha}=r^{\alpha}-(m_{p}^{+}-m_{p}^{-})$

where $m_{p}^{+}/m_{p}^{-}$ are the numbers of normal segments with $\rho<0$
through $p$ which correspond to minima/maxima of the distance function
respectively.

For surfaces, even when we restrict our attention to convex surfaces, the
focal sets can be very complex. An interesting question would be to consider
the generalisation of the formula given at the end of Section 3, for the
rotation index of the focal sets. Now instead of cusps there are ribs
([26],[13]), and there may be a formula for the rotation index in terms of the
number of ribs, or (in analogue with the four cusp theorem) a ‘three ribs’
theorem. In summary, the caustic of normal lines to curves and surfaces has
been much studied over the years but nonetheless still provides interesting
and complex structure warranting further research.

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# A finite dimensional algebra with infinite delooping level

Luke Kershaw  School of Mathematics, University of Bristol, Bristol BS8 1UG,
UK<EMAIL_ADDRESS>and  Jeremy Rickard  School of Mathematics,
University of Bristol, Bristol BS8 1UG, UK<EMAIL_ADDRESS>
###### Abstract.

We give an example of a finite dimensional algebra with infinite delooping
level, based on an example of a semi-Gorenstein-projective module due to
Ringel and Zhang.

###### Key words and phrases:

delooping level, semi-Gorenstein-projective modules, finitistic dimension
conjecture

###### 2010 Mathematics Subject Classification:

16E10,16G10

##  1\. Introduction

One of the most celebrated open problems in the representation theory of
finite dimensional algebras is the finitistic dimension conjecture, publicized
by Bass [Bas60] in 1960, and subsequently proven to imply a host of other
homological conjectures.

In recent years other, potentially stronger, conditions that would imply the
finitistic dimension conjecture have been considered. One is the notion of
“injective generation” [Ric19], that we consider briefly in Section 5. But our
main focus is the finiteness of delooping level, introduced by Gélinas
[Gél22]. He works in greater generality, but since the main example in this
paper is a finite dimensional algebra, we shall describe the delooping level
in that context.

Given a finite dimensional algebra $A$ and a finitely generated right
$A$-module $M$, the delooping level of $M$ is the smallest nonnegative integer
$n$ such that $\Omega^{n}M$ is a direct summand (in the stable module
category) of $\Omega^{n+1}N$ for some other finitely generated module $N$, or
is infinite if no such $n$ exists. The delooping level is denoted by
$\operatorname{dell}M$.

Then the delooping level $\operatorname{dell}A$ is defined to be the largest
delooping level of a simple $A$-module.

One of the main properties of the delooping level is that
$\operatorname{dell}A$ is an upper bound for the (big) finitistic dimension of
the opposite algebra $A^{op}$ and so if $\operatorname{dell}A<\infty$ then the
big finitistic dimension conjecture holds for $A^{op}$ [Gél22, Proposition
1.3].

In this paper we shall give what we believe to be the first known example of a
finite dimensional algebra with infinite delooping level. This does not give a
counterexample to the finitistic dimension conjecture.

Although the delooping level of an algebra $A$ is defined in terms of the
delooping levels of _simple_ modules, we show in Section 2 that if there is
any finitely generated $A$-module $M$ with $\operatorname{dell}M=\infty$, then
there is another finite dimensional algebra $B$ with
$\operatorname{dell}B=\infty$.

Our main result, Proposition 4.1, is that a particular “semi-Gorenstein-
projective” module studied by Ringel and Zhang [RZ20] has infinite delooping
level. Gorenstein projective modules are easily seen to have delooping level
equal to zero, but the same argument does not apply to semi-Gorenstein-
projective modules. What inspired us to look at this example is that Gélinas
showed in [Gél22, Theorem 1.10] that the module $N$ in the definition of
delooping level can always be taken to be $\Sigma^{n+1}\Omega^{n}M$ (where
$\Sigma$ denotes the left adjoint of $\Omega$), which in the case of a semi-
Gorenstein-projective module simplifies to $\Sigma M$, so that proving
$\operatorname{dell}M=\infty$ reduces to proving that $\Omega^{n}M$ is never a
summand of $\Omega^{n}(\Omega\Sigma)M$.

There are other examples known of modules that are semi-Gorenstein-projective,
but not Gorenstein projective. The first was given by Jorgensen and Şega, and
Marczinzik, who calls such modules “stable”, gave others [JŞ06, Mar17]. It
might be interesting to study the delooping level of these examples.

Acknowledgments. The first author was supported by Engineering and Physical
Sciences Research Council Doctoral Training Partnership award EP/R513179/1.

##  2\. Delooping levels of one point extensions

######  Proposition 2.1.

If $M$ is a finitely generated module with infinite delooping level for a
finite dimensional $k$-algebra $A$, then there is a finite dimensional
$k$-algebra $B$ with infinite delooping level.

###### Proof.

Let $B$ be the one point extension algebra

$B=A[M]=\begin{pmatrix}k&M\\\ 0&A\end{pmatrix},$

and let $S$ be the simple $B$-module
$\begin{pmatrix}k&M\end{pmatrix}/\begin{pmatrix}0&M\end{pmatrix}$

Then for $n\geq 1$, $\Omega^{n}S=\begin{pmatrix}0&\Omega^{n-1}M\end{pmatrix}$.
If $S$ has finite delooping level, then for all sufficiently large $n$,
$\Omega^{n}S$ is a summand of $\Omega^{n+1}X$ for some $B$-module $X$, and
$\Omega X=\begin{pmatrix}0&N\end{pmatrix}$ for some $A$-module $N$, and
$\Omega^{n+1}X=\begin{pmatrix}0&\Omega^{n}N\end{pmatrix}$.

Therefore $\Omega^{n-1}M$ is a summand of $\Omega^{n}N$, and so $M$ has finite
delooping level. ∎

##  3\. Ringel and Zhang’s example

In [RZ20], Ringel and Zhang exhibited an example of a three-dimensional module
$M$ for a six-dimensional local finite dimensional algebra $\Lambda$ that is
semi-Gorenstein-projective (meaning that $\operatorname{Ext}^{i}(M,\Lambda)=0$
for all $i\geq 1$), but is not torsionless (i.e., is not a submodule of a
projective $\Lambda$-module), and hence not Gorenstein-projective.

In this section, we gather the information that we will need about this
example. All of the results in this section are taken from Ringel and Zhang’s
paper, except for the remarks about the subalgebra $\Gamma$. Note that they
work with left modules, whereas we are using right modules, so the algebra we
define is the opposite of theirs.

###  3.1. The algebra $\Lambda=\Lambda(q)$ and the subalgebra $\Gamma$

[RZ20, §6.1]

Let $k$ be a field, and $q\in k$ an element with infinite multiplicative
order. Then $\Lambda=\Lambda(q)$ is the algebra

$k\langle x,y,z\rangle/(x^{2},y^{2},z^{2},zy,yx+qxy,zx-xz,yz-xz),$

which is easily seen to be six-dimensional, with basis $1,x,y,z,yx,zx$.

Let $\Gamma$ be the two-dimensional subalgebra of $\Lambda$ generated by $x$.
Note that $\Lambda$ is a free right $\Gamma$-module.

###  3.2. The modules $M(\alpha)$

[RZ20, §6.1,§6.3]

For each $\alpha\in k$, $M(\alpha)$ is the three-dimensional $\Lambda$-module
with basis $v,v^{\prime},v^{\prime\prime}$, where $vx=\alpha v^{\prime}$,
$vy=v^{\prime}$, $vz=v^{\prime\prime}$, with $v^{\prime}$ and
$v^{\prime\prime}$ annihilated by $x$, $y$ and $z$.

Next, we gather the results that we need regarding the modules $M(\alpha)$. We
shall state them in terms of the effect of $\Omega$ and $\Sigma$ on the
modules, but apart from the brief remark about the subalgebra $\Gamma$,
everything in the following proposition is a restatement of results of Ringel
and Zhang [RZ20].

######  Proposition 3.1.

Let $\alpha\in k$.

  1. (1)

If $\alpha\neq 1$ then $\Omega M(\alpha)=M(q\alpha)$.

  2. (2)

If $\alpha\neq q$ then $\Sigma M(\alpha)=M(q^{-1}\alpha)$.

  3. (3)

$\Omega\Sigma M(q)$ is a two-dimensional module that is free as a
$\Gamma$-module.

###### Proof.

Both (1) and (2) follow immediately from [RZ20, Lemma 6.4 and Lemma 3.2]. Note
that Ringel and Zhang use the notation $\mho M(\alpha)$ where we use $\Sigma
M(\alpha)$.

For any module $M$, $\Omega\Sigma M$ is the maximal torsionless quotient of
$M$. For $M=M(q)$, this is shown in [RZ20, Lemma 6.2] to be $M(q)/M(q)z$,
which is freely generated as a $\Gamma$-module by $v$. ∎

##  4\. The delooping level of $M(q)$

######  Proposition 4.1.

The $\Lambda$-module $M(q)$ has infinite delooping level.

###### Proof.

Gélinas showed that if a module $M$ has delooping level $n<\infty$, then
$\Omega^{n}M$ is a stable direct summand of
$\Omega^{n+1}\Sigma^{n+1}\Omega^{n}M$ [Gél22, Theorem 1.10]. We shall show
that this is not the case for $M=M(q)$.

By Proposition 3.1(1), $\Omega^{n}M(q)\cong M(q^{n+1})$ is three-dimensional.

By Proposition 3.1(2), $\Sigma^{n}\Omega^{n}M(q)\cong M(q)$, so

$\Omega^{n+1}\Sigma^{n+1}\Omega^{n}M\cong\Omega^{n+1}\Sigma M(q).$

But since $\Omega\Sigma M(q)$ and $\Lambda$ are both free as $\Gamma$-modules,

$\Omega^{n+1}\Sigma M(q)=\Omega^{n}\left(\Omega\Sigma M(q)\right)$

is also free as a $\Gamma$-module and therefore every direct summand has even
dimension. ∎

######  Corollary 4.2.

The one-point extension algebra $\Lambda[M(q)]$ has infinite delooping level.

###### Proof.

Proposition 3.1 and Proposition 4.1. ∎

##  5\. Other properties of $\Lambda[M(q)]$

In this final section, we consider some other related properties of the
algebra $\Lambda[M(q)]$.

One of the main reasons that the delooping level was introduced was its
connection with the finitistic dimension conjecture: for Artinian rings $A$,
Gélinas proved that the delooping level $\operatorname{dell}A$ of $A$ is an
upper bound for the big finitistic dimension $\operatorname{Findim}A^{op}$ of
its opposite algebra, so that finiteness of the delooping level of $A$ implies
the big finitistic dimension conjecture for $A^{op}$ [Gél22, Proposition 1.3].

It is therefore natural to wonder whether $\Lambda[M(q)]^{op}$ might have
infinite finitistic dimension. This is not the case.

######  Proposition 5.1.

$\operatorname{Findim}\Lambda[M(q)]^{op}=1$

###### Proof.

Since

$\Lambda[M(q)]^{op}=\begin{pmatrix}k&0\\\ M(q)&\Lambda^{op}\end{pmatrix}$

there is an obvious module with projective dimension equal to one – namely,
$\begin{pmatrix}0&\Lambda^{op}\end{pmatrix}=\begin{pmatrix}M(q)&\Lambda^{op}\end{pmatrix}/\begin{pmatrix}M(q)&0\end{pmatrix},$
and so $1\leq\operatorname{Findim}\Lambda[M(q)]^{op}$.

But since $\Lambda[M(q)]^{op}$ is a triangular matrix ring, it is well-known
[FGR75, Corollary 4.21] that

$\operatorname{Findim}\Lambda[M(q)]^{op}\leq
1+\operatorname{Findim}k+\operatorname{Findim}\Lambda^{op},$

and since $\Lambda^{op}$ is a local finite dimensional algebra,
$\operatorname{Findim}\Lambda^{op}=0$. So
$\operatorname{Findim}\Lambda[M(q)]^{op}\leq 1$. ∎

Another property that has recently been shown to imply the big finitistic
conjecture for a finite dimensional algebra $A$ is “injective generation”. If
the unbounded derived category $\mathcal{D}(A)$ is equal to
$\operatorname{Loc}(\operatorname{Inj}A)$, the localizing subcategory of
$\mathcal{D}(A)$ generated by injective modules, then the big finitistic
dimension conjecture holds for $A$ [Ric19, Theorem 4.3].

We shall end by showing that the algebra $\Lambda[M(q)]^{op}$ is not a
counterexample to injective generation.

######  Proposition 5.2.

Injectives generate for $\Lambda[M(q)]^{op}$.

###### Proof.

Since $\Lambda[M(q)]^{op}$ is a triangular matrix ring, by a result of
Cummings it is sufficient to show that injectives generate for $\Lambda^{op}$
[Cum23, Example 6.11].

Since $M(0)=\Omega M(0)$, $M(0)$ has finite syzygy type as a $\Lambda$-module,
and so, taking the vector space dual, $DM(0)$ has finite cosyzygy type as a
$\Lambda^{op}$-module, which implies that $DM(0)$ is in
$\operatorname{Loc}(\operatorname{Inj}\Lambda^{op})$ [Ric19, Proposition 7.2].

Since $DM(0)$ has radical length two, there is a short exact sequence

$0\to S_{1}\to DM(0)\to S_{2}\to 0$

where $S_{1}$ and $S_{2}$ are nonzero semisimple $\Lambda^{op}$-modules.
Taking the coproduct of infinitely many copies of this sequence, we get a
short exact sequence

$0\to S\to\bigoplus DM(0)\to S\to 0$

where $S$ is an infinitely generated semisimple module. Splicing together
copies of this sequence gives a resolution

$\cdots\to\bigoplus DM(0)\to\bigoplus DM(0)\to S\to 0$

by coproducts of copies of $DM(0)$, and so $S$ is in
$\operatorname{Loc}(\operatorname{Inj}\Lambda^{op})$ [Ric19, Proposition
2.1(h)]. Therefore the unique simple $\Lambda^{op}$-module is in
$\operatorname{Loc}(\operatorname{Inj}\Lambda^{op})$ and so
$\mathcal{D}(\Lambda^{op})=\operatorname{Loc}(\operatorname{Inj}\Lambda^{op})$
[Ric19, Proposition 2.1(e) and Lemma 6.1]. ∎

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