# Ruled surfaces in $3$-dimensional Riemannian manifolds

Marco Castrillón1 M. Eugenia Rosado María2 Alberto Soria3
1 Departamento de Álgebra, Geometría y Topología,
Facultad de Matemáticas, Universidad Complutense de Madrid,
Plaza de las Ciencias, 3, 28040 Madrid, Spain
<EMAIL_ADDRESS>
2 Departamento de Matemática Aplicada,
ETS de Arquitectura, Universidad Politécnica de Madrid,
Avenida de Juan de Herrera, 4, 28040 Madrid
<EMAIL_ADDRESS>
3 Departamento de Matemática Aplicada a las TIC,
ETSI de Telecomunicación, Universidad Politécnica de Madrid,
Avenida Complutense, 30, 28040 Madrid
<EMAIL_ADDRESS>
###### Abstract

In this work ruled surfaces in $3$-dimensional Riemannian manifolds are
studied. We determine the expression for the extrinsic and sectional curvature
of a parametrized ruled surface, where the former one is shown to be non-
positive. We also quantify the set of ruling vector fields along a given base
curve which allow to define a relevant reference frame along it and that we
refer to as _Sannia_. The fundamental Theorem of existence and equivalence of
Sannia ruled surfaces in terms of a system of invariants is given. The second
part of the article tackles the concept of striction curve, which is proven to
be the set of points where the so-called _Jacobi evolution function_ vanishes
on a ruled surface. This provides independent proofs for their existence and
uniqueness in space forms, and to disprove its existence or uniqueness in some
other cases.

_Mathematics Subject Classification 2020:_ Primary: 53B25; Secondary: 53B20,
53A55.

_Keywords and phrases:_ Ruled surface, Riemannian manifold, Jacobi field,
Sannia frame, differential invariant, striction curve.

##  1 Introduction

Ruled surfaces constitute a distinguished family of submanifolds in the area
of Differential Geometry with a long history and a rich collection of works in
the literature. Even though their original framework in the Euclidean space is
still an active field of research (just to mention one, the reader may have a
look at the study of non-developable ruled surfaces in [13]), ruled surfaces
are also a prominent object in other Riemannian manifolds. For example, in the
case of the Heisenberg group, where parts of planes, helicoids and hyperbolic
paraboloids are proven to be the only minimal ruled surfaces (see [20]), or
the minimal (resp. maximal) ruled surfaces in the Bianchi-Cartan-Vranceanu
spaces $\mathbb{E}(\kappa,\tau)$ (resp. in their Lorentzian counterparts), see
[1]. In [5] several results for ruled surfaces in $3$-dimensional Riemannian
manifolds are established, where formulas for the striction curve, the
distribution parameter and the first and second fundamental forms of ruled
surfaces in space forms are obtained. In addition, a necessary and sufficient
condition for the Riemann tensor of an extrinsically flat surface to be ruled
is also derived, together with a proof for the well-known fact that surfaces
in $3$-dimensional space forms with vanishing Gauss curvature are necessarily
ruled. However, this question is still open for more general contexts. Ruled
surfaces have also been studied in Lorentzian contexts as in [6], where all
Weingarten surfaces are characterized in the $3$-dimensional Minkowski
spacetime. Other results concerning ruled surfaces in the $3$-dimensional
Minkowski spacetime have been established in [4, 9, 10]. Lightlike ruled
hypersurfaces have also been of great utility to address the problem of some
geometric inequalities as in [14], where some cases of the so-called null
Penrose inequality for the Bondi energy of a spacetime are proven.

All Euclidean ruled surfaces except cylinders admit a unique special curve
called _striction_ , which is made up of the so-called central points of each
generator [18]. In the case of the cylinder the uniqueness property is not
fulfilled for every curve in it is a striction. On the other hand, striction
curves can degenerate to a single point, as occurs with the cone, where it is
made up exclusively of its vertex. A central point in a geodesic generator of
the surface furnish a critical point to the distance from points on that
generator to neighboring geodesics. The family of generators along the
striction curve is geodesically parallel along it. Striction curves are
important, among other reasons, because they contain the singular points of
the ruled surface in case they exist [8]. To the best of our knowledge there
are no results in the literature involving a method to determine the striction
curve of ruled surfaces in generic Riemannian manifolds. In this work we
present a new strategy to determine the existence of striction curves in such
context based on a function referred to as _Jacobi evolution function_ , which
vanishes at these type of curves in case they lie on the surface. Even though
the presence of striction curves is addressed in [5] for ruled surfaces in
$S^{3}(r)$ and $\mathbb{H}^{3}(r)$, no results are found in the literature
where either the existence or uniqueness of the striction curve is violated in
such contexts. In fact, in this work, the aforementioned Jacobi evolution
function is applied to prove that there exists at most one striction curve on
ruled surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^{3}$, with
examples of surfaces with no striction.

In case that the striction curve of a ruled surface in a $3$-dimensional
Riemannian manifold has a regular striction curve, it is possible to define a
specific reference frame called Sannia after the Italian mathematician who
first proposed it (see [18], [19]), and which is characterized for having the
generating vector of the rulings as the first element of the basis. The
evolution of the Sannia frame provide two Euclidean invariants associated to
the surface referred to as curvature and torsion, which together with the
(striction) angle enclosed, determine a complete system of Euclidean
invariants for the ruled surface. A fundamental problem in Riemannian Geometry
is that of equivalence of objects in a determined class, namely to provide a
criterion to know whether two given objects in this class are congruent under
isometries or not. In [3] the problem of curves with values in an arbitrary
dimensional Riemannian manifold is solved with respect to the Frenet
curvatures. However, spaces with non-constant curvature require additional
invariants to establish such result. The classical Sannia problem of
reconstructing suitable surfaces provided a set of associated invariant
functions is known is presented in Theorem 5.3.8 in [18]. In this work we
establish an analogous result of local existence and uniqueness of ruled
surfaces in generic $3$-dimensional Riemannian manifolds. In our main Theorem
we prove that four functions of suitable regularity and the Sannia frame at a
given point $p_{0}\in M$ determine uniquely a (local) ruled surface passing
through $p_{0}$. Moreover, we do not require the base to be a striction curve.
This is a great advantage since, as we shall see, certain ruled surfaces lack
this sort of curves.

The paper is structured as follows. In Section 2 we introduce the concept of
parametrized ruled surface, which is the most practical way to present ruled
surfaces for the purposes of this work. As a first result, we obtain the
expression for the sectional curvature of any ruled surfaces in a
$3$-dimensional Riemannian manifold in terms of the ambient geometry and its
second fundamental form. We also define the so called distribution parameter
function which reduces to the classical distribution parameter when is
evaluated on the so-called striction curve of any ruled surface in the
Euclidean space. At the end of the section we introduce the concept of Sannia
ruled surface, which we define as the ones admitting a ruling vector field
with linearly independent covariant derivative. In particular this means that
a Sannia frame can be defined along the base curve. An idea of the size of the
set of vectors satisfying such a property is put forward in Proposition 13.
Deriving the evolution equations for the vectors of a Sannia frame along a
given base curve gives rise to a set of two invariant functions associated to
the Sannia ruled surface which, together with two additional angle functions,
provide the fundamental theorem of ruled surfaces (see Theorem 20 below). The
existence tackled in the Theorem is completed with the uniqueness only in
space forms. In Section 3 we focus our interest on the existence and
uniqueness of striction curves. We define a new function on the ruled surface
that we refer to as Jacobi evolution function, which vanishes on the striction
curves in case of existence. This makes such a function a useful tool for
proving the presence of striction curves on ruled surfaces. In the case of
constant curvature $k$, we compute its expression explicitly in the three
possible cases of sign of $k$. In particular, for $k<0$, the striction curve
may not exist (in contrast to [5]). Furthermore, some extra examples of rules
surfaces in product manifolds are put forward to disprove the uniqueness of
the striction curve in such backgrounds. In the text, Einstein’s summation
convention will be assumed.

##  2 Ruled surfaces in $3$-dimensional Riemannian manifolds

###  2.1 Definitions and properties

Let $(M,g)$ be a $3$-dimensional Riemannian manifold and let
$\psi\colon\Sigma\rightarrow(M,g)$ be an immersed orientable surface in
$(M,g)$. We recall some basic concepts of the geometry of hypersurfaces (see
[5, Volume II, Chapter VII, Section 3]). We denote by $\nabla$ and
$\nabla^{\Sigma}$ the Levi-Civita connections of $(M,g)$ and
$(\Sigma,\psi^{*}g=g_{\Sigma})$ respectively. Locally, we can assume $\Sigma$
to be embedded in $(M,g)$. Let us choose a unit normal vector $\xi$ in a
neighborhood $U$ of a point $p\in\Sigma$. Recall that the second fundamental
form of $\Sigma$ is defined as

$\displaystyle\vec{h}\colon T_{p}\Sigma\times T_{p}\Sigma$
$\displaystyle\rightarrow T^{\perp}_{p}\Sigma$ $\displaystyle(U,V)$
$\displaystyle\mapsto(\nabla_{U}V)_{p}^{\perp}=h(U,V)\xi,$

where $T^{\perp}_{p}\Sigma$ is the subspace of normal vectors on $\Sigma$, and
$h\colon T_{p}\Sigma\times T_{p}\Sigma\rightarrow\mathbb{R}$ is the associated
second fundamental form tensor of $\Sigma$ defined as

$h(U,V)=-g(\nabla_{U}\xi,V)$

Given any vectors fields $X,Y,Z,T\in\mathfrak{X}(M)$, we consider the
curvature tensor of $(M,g)$ as

(1) $R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,$

and the Riemann tensor is $\mathrm{Riem}(X,Y,Z,T)=-g(R(X,Y)Z,T)$. The induced
curvature tensor and Riemann tensor on $\Sigma$ will be denoted by
$R^{\Sigma}$ and $\mathrm{Riem}^{\Sigma}$ respectively. Given any $p\in M$ and
two linearly independent vectors $X,Y\in T_{p}M$ generating a plane
$T_{p}\Sigma\subset T_{p}M$, the sectional curvature $K(T_{p}\Sigma)$ in
$(M,g)$ is defined by

$K(T_{p}\Sigma)=\frac{\mathrm{Riem}(X,Y,X,Y)}{g(X,X)g(Y,Y)-g(X,Y)^{2}},$

and the sectional curvature of $\Sigma$ will be denoted by $K^{\Sigma}$.

Ruled surfaces are a remarkable family of hypersurfaces in $3$-dimensional
Riemannian manifolds. They are generally defined as follows:

###### Definition 1

We say that the immersed hypersurface $\psi\colon\Sigma\rightarrow(M,g)$ in
$(M,g)$ is ruled if there exists a foliation of $\Sigma$ by complete curves
which are geodesics in the ambient space $(M,g)$.

However, in the following we are going to work with a notion, closer to the
classical constructions in the Euclidean space, of ruled surface defined by
base curves and ruling directions.

###### Definition 2

Let $(M,g)$ be a complete $3$-dimensional Riemannian manifold. Let
$\alpha\colon I\rightarrow(M,g)$ be a smooth regular curve and $Z$ a non-
vanishing smooth vector field along $\alpha$, i.e. a curve in $TM$ such that
$Z(u)\in T_{\alpha(u)}M$, $\forall u\in I$. The _parametrized ruled surface_
defined by $\alpha$ and $Z$ is the differentiable map

$\displaystyle\mathbf{X}\colon I\times\mathbb{R}$
$\displaystyle\rightarrow(M,g)$ (2) $\displaystyle(u,v)$
$\displaystyle\mapsto\gamma_{Z(u)}(v)=\exp_{\alpha(u)}(vZ(u)),$

where $\gamma_{Z(u)}\colon\mathbb{R}\rightarrow(M,g)$ is the geodesic
satisfying

$\displaystyle\gamma_{Z(u)}(0)$ $\displaystyle=\alpha(u),$
$\displaystyle\gamma_{Z(u)}^{\prime}(0)$ $\displaystyle=Z(u).$

###### Remark 3

Obviously, the image of the map $\mathbf{X}$ is not necessarily an embedded
surface in $(M,g)$. But if we require that rank of $d\mathbf{X}$ to be 2, it
will be an immersed surface. On the other hand, if $(M,g)$ is not complete,
the definition above is still valid by restricting the domain of $\mathbf{X}$.

Note that the partial derivative $\mathbf{X}_{v}=\partial\mathbf{X}/\partial
v$ is a geodesic vector field, that is

$\nabla_{\mathbf{X}_{v}}\mathbf{X}_{v}=0.$

In addition, $\mathbf{X}_{v}(u,0)=Z(u)$. The other partial derivative
$\mathbf{X}_{u}=\partial\mathbf{X}/\partial u$ is a Jacobi vector field along
the geodesic $\gamma_{Z(u)}$ since it is a geodesic variation. Hence it
satisfies the Jacobi equation

(3)
$\nabla_{\mathbf{X}_{v}}\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}+R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{u}=0.$

In the following result we derive the expression for the (intrinsic) curvature
$K_{p}^{\Sigma}$ of a ruled surface $\Sigma$ in $(M,g)$.

###### Proposition 4

Let $\Sigma$ be a parametrized ruled surface in $(M,g)$ defined by
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$, where $d\mathbf{X}$ is of rank
$2$. Then, we have

(4)
$K_{p}^{\Sigma}=\frac{-g(\nabla_{\mathbf{X}_{v}}\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{u})}{||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}}-\frac{\mathrm{vol}_{g}(\mathbf{X}_{u},\mathbf{X}_{v},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})^{2}}{(||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2})^{2}}.$

Proof. Taking into account the Gauss equation (see [11, Volume II, Chapter
VII, Proposition 4.1]), the sectional curvature for $T_{p}\Sigma$ is written
as follows

$\displaystyle K_{p}^{\Sigma}$
$\displaystyle=\frac{\mathrm{Riem}^{\Sigma}(\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v})}{||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}}$
$\displaystyle=\frac{\mathrm{Riem}(\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v})+g(\vec{h}(\mathbf{X}_{u},\mathbf{X}_{u}),\vec{h}(\mathbf{X}_{v},\mathbf{X}_{v}))-g(\vec{h}(\mathbf{X}_{u},\mathbf{X}_{v}),\vec{h}(\mathbf{X}_{u},\mathbf{X}_{v}))}{||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}}.$

Since $\mathbf{X}_{v}$ is geodesic and $[\mathbf{X}_{u},\mathbf{X}_{v}]=0$, we
have

$K_{p}^{\Sigma}=\frac{-g(\nabla_{\mathbf{X}_{v}}\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{u})-h(\mathbf{X}_{u},\mathbf{X}_{v})^{2}}{||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}},$

The proof is complete by taking into account

$|h(\mathbf{X}_{u},\mathbf{X}_{v})|=|g(\xi,\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})|=\frac{|\mathrm{vol}_{g}(\mathbf{X}_{u},\mathbf{X}_{v},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})|}{\sqrt{||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}}}.$

###### Remark 5

Given any immersed surface $\psi\colon\Sigma\rightarrow(M,g)$, the sectional
curvatures $K(T_{p}\Sigma)$ in $(M,g)$ and $K^{\Sigma}_{p}$ in
$(\Sigma,g_{\Sigma})$ defined by $T_{p}\Sigma$ respectively are related as

$K^{\Sigma}_{p}=K(T_{p}\Sigma)+K^{\Sigma}_{ext},$

where $K^{\Sigma}_{ext}$ is the extrinsic or Gauss curvature (the determinant
of the second fundamental form endomorphism). By virtue of (4), it follows
that

$\displaystyle K(T_{p}\Sigma)$ $\displaystyle=$
$\displaystyle\frac{-g(\nabla_{\mathbf{X}_{v}}\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{u})}{||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}},$
(5) $\displaystyle K_{\mathrm{ext}}^{\Sigma}$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{vol}_{g}(\mathbf{X}_{u},\mathbf{X}_{v},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})^{2}}{(||\mathbf{X}_{u}||^{2}||\mathbf{X}_{v}||^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2})^{2}}.$

Note that the extrinsic curvature $K^{\Sigma}_{ext}$ of a parametrized ruled
surface in a generic Riemannian background is non-positive, as (5) shows. In
particular, the (intrinsic) curvature of a ruled surface is always less or
equal than the ambient sectional curvature. Also notice that the extrinsic
curvature relation $K_{\mathrm{ext}}^{\Sigma}$ in (5) is still valid for any
other basis of $T_{p}\Sigma$ since it is a tensorial expression.

The expression of the extrinsic curvature that we have obtained above is
connected with a classical invariant in the theory of ruled surfaces, the
distribution parameter $\lambda$, which is defined on the so-called striction
curve of ruled surfaces in the Euclidean space (see [8], [18], [21] for more
details). In this work we devote Section 3 to study the main properties of
this curve in the current setting in case of existence. In the Euclidean space
ruled surfaces always admit a striction curve which can always be considered
the base curve of the parametrized ruled surface. For this reason the
classical distribution parameter is defined as

$\lambda(u)=\frac{\mathrm{vol}_{g}(\alpha^{\prime},Z,\nabla_{\alpha^{\prime}}Z)}{\|\nabla_{\alpha^{\prime}}Z\|^{2}},$

where $\alpha:I\rightarrow(\Sigma,g_{\Sigma})$ is the striction base curve and
$Z$ is the ruling vector field along $\alpha$. This function plays an
important role in many results for classical ruled surfaces in
$\mathbb{R}^{3}$. We next define a function defined on a parametrized ruled
surface $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ motivated by the
classical concept of distribution parameter:

######  Definition 6 (Extended distribution parameter)

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface as in (2) in a Riemmanian $3$-manifold $(M,g)$ with $d\mathbf{X}$ of
rank $2$. We define the _extended distribution parameter function_ of
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ as

(6)
$\lambda(u,v)=\frac{\mathrm{vol}_{g}(\mathbf{X}_{u},\mathbf{X}_{v},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})}{\|\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v}\|^{2}}.$

###### Remark 7

As we will illustrate in Section 3, for $v=0$ the above formula reduces to the
classical distribution parameter $\lambda(u,0)=\lambda(u)$ provided
$\alpha:I\rightarrow\mathbf{X}(I\times\mathbb{R})$ is an striction base curve.

###### Definition 8

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ a parametrized ruled
surface in $(M,g)$ as in (2). The function $\sigma\colon
I\rightarrow\mathbb{R}$ satisfying

$\cos\sigma_{\alpha(u)}=\frac{g(\alpha^{\prime}(u),Z(u))}{\|\alpha^{\prime}(u)\|},$

is called base angle along the curve $\alpha$.

In the following result we prove that the inner product of the vectors
$\mathbf{X}_{u}$ and $\mathbf{X}_{v}$ remains constant along the rulings of a
parametrized ruled surface.

###### Proposition 9

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ a parametrized ruled
surface in $(M,g)$ as in (2) by a unitary vector field $Z$. Then
$g(\mathbf{X}_{u},\mathbf{X}_{v})$ is constant along its rulings. Moreover,
the angle $\sigma$ between $\mathbf{X}_{u}$ and $\mathbf{X}_{v}$ at any point
$q$ of the ruling containing $p=\alpha(u)$ is given by

(7)
$\cos{\sigma}_{q}=\frac{\|\alpha^{\prime}(u)\|\cos{\sigma_{p}}}{\|\mathbf{X}_{u}\|_{q}}.$

Proof. Differentiating the function $g(\mathbf{X}_{u},\mathbf{X}_{v})$ along
the vector field $\mathbf{X}_{v}$ gives

$\mathbf{X}_{v}(g(\mathbf{X}_{u},\mathbf{X}_{v}))=g(\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{v})+g(\mathbf{X}_{u},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{v})=g(\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v},\mathbf{X}_{v})=0,$

where we have taken into account that $[\mathbf{X}_{u},\mathbf{X}_{v}]=0$ and
$\mathbf{X}_{v}$ is geodesic. This means that
$g(\mathbf{X}_{u},\mathbf{X}_{v})=\|\mathbf{X}_{u}\|\cos{\sigma}$ is constant
along the rulings of $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$. This
value can be obtained by evaluating it at the point $p=\alpha(u)$. Indeed,
given any $q$ of the ruling at $p=\alpha(u)$,

$g(\mathbf{X}_{u},\mathbf{X}_{v})|_{q}=\|\mathbf{X}_{u}\|_{q}\cos{\sigma_{q}}=\|\mathbf{X}_{u}\|_{p}\cos{\sigma}_{p}=\|\alpha^{\prime}(u)\|\cos{\sigma_{p}},$

from where relation (7) follows.

###### Remark 10

Relation (7) shows that whenever the vectors
$\mathbf{X}_{u}(u,0)=\alpha^{\prime}(u)$ and $Z_{p}=\mathbf{X}_{v}(u,0)$ are
orthogonal at the base curve, they remain orthogonal along the ruling
$\gamma_{Z(u)}(v)=\mathbf{X}(u,v)$. Nevertheless, the coordinate basis
$(\mathbf{X}_{u},\mathbf{X}_{v})$ associated to the parametrization (2) with
$d\mathbf{X}$ of rank $2$ is not necessarily orthogonal.

As already mentioned, $\mathbf{X}_{u}$ is a Jacobi vector field on every
parametrized ruled surface, not necessarily orthogonal to its rulings.
However, it is sometimes useful to consider the orthogonal component to the
surface generators, which turns out to be a Jacobi field too. In the following
Proposition we compute the decomposition of $\mathbf{X}_{u}$ into its tangent
and normal components to the rulings.

###### Proposition 11

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface in $(M,g)$ as in (2). Then the decomposition of the Jacobi field
$\mathbf{X}_{u}$ into its tangential and normal part with respect to the
ruling is

(8)
$\mathbf{X}_{u}=\|\alpha^{\prime}(u)\|(\cos{\sigma_{p}})\mathbf{X}_{v}+\mathbf{X}_{u}^{\bot},$

where $\mathbf{X}_{u}^{\bot}$ is a Jacobi field along the ruling
$\gamma_{Z(u)}$ and orthogonal to it, and $\sigma_{p}$ is the base angle at
$p=\alpha(u)$.

Proof. It is a well known fact that any Jacobi vector field along a geodesic
curve $\gamma(v)$ parametrized by its arc length decomposes as

$J(v)=(a+bv)\gamma^{\prime}(v)+J^{\bot}(v),$

where $a=g(J(0),\gamma^{\prime}(0))$,
$b=g((\nabla_{\gamma^{\prime}}J)_{p},\gamma^{\prime}(0))$ and $J^{\bot}$ is an
orthogonal Jacobi field along $\gamma$. In this background, the Jacobi field
$\mathbf{X}_{u}$ evaluated at $p$ reads
$\mathbf{X}_{u}|_{p}=\alpha^{\prime}(u)$, so

$a=g(\alpha^{\prime}(u),Z_{p})=\|\alpha^{\prime}(u)\|\cos{\sigma_{p}},$

and

$b=g((\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})|_{p},Z_{p})=\frac{1}{2}\mathbf{X}_{u}(g(\mathbf{X}_{v},\mathbf{X}_{v}))|_{p}=0,$

since $\|\mathbf{X}_{v}\|=1$. Decomposition (8) follows from such values.

###  2.2 Sannia invariants and the Fundamental Theorem of Ruled Surfaces

Orthonormal frames along curves are often considered in Geometry and Physics
to address problems in relation to the geometry of manifolds and submanifolds.
In this work we consider frames whose first vector is determined by the
geodesics defining the ruled surface. Under suitable regularity hypothesis, a
possible way of constructing the rest of the vectors in the frame is by
considering successive derivatives of the first one along the tangent
direction to the curve (see for example [3]). However, it may occur that some
of the derivatives are linearly dependent with some other vector in the frame,
which would prevent such a set of vectors to become a basis. With the
following results we first intend to give an idea of the size of the set of
ruled surfaces admitting such a relevant frame, and then we prove the main
result of this work, the _fundamental theorem of ruled surfaces_ , in which we
state that certain ruled surfaces are uniquely determined from certain
invariants.

###### Definition 12

A vector field $Z\in\mathfrak{X}(\alpha)$ along a smooth curve $\alpha\colon
I\rightarrow M$ taking values into a manifold $M$ endowed with a linear
connection $\nabla$ is said to be in _general position_ at $u_{0}\in I$ if the
vector fields $Z$ and $\nabla_{\alpha^{\prime}}Z$ along $\alpha$ are linearly
independent at $u_{0}$. The vector field $Z\in\mathfrak{X}(\alpha)$ is in
general position up to order $2$ if it is in general position up to this order
for every $u\in(a,b)$.

The following result quantifies the set of vectors in general position along a
curve on a manifold endowed with a linear connection. To this purpose it will
be necessary to make use of the so-called jet bundles. We refer the reader to
[12, section 41] for more details on this topic.

###### Proposition 13

Let $M$ be a $3$-dimensional manifold endowed with a linear connection
$\nabla$ and let $\alpha\colon I\rightarrow M$ be a smooth curve on $M$. The
set of vector fields $Z\in\mathfrak{X}(\alpha)$ along $\alpha$ that are in
general position is a dense set on $\mathfrak{X}(\alpha)$ with respect to the
strong topology.

Proof. The sections of the bundle $E=\alpha^{\ast}TM\rightarrow I$ are vector
fields along $\alpha$. Let consider the $1$-jet bundle of $E$. The morphism on
$J^{1}E$ given by

$\displaystyle\varrho\colon J^{1}E$ $\displaystyle\longrightarrow E\oplus E$
$\displaystyle j_{u}^{1}Z$
$\displaystyle\mapsto\left(Z(u),(\nabla_{\alpha^{\prime}}Z)(u)\right)$

is an isomorphism. We define the singular set

$S=\\{(Z_{1},Z_{2})\in E\oplus E:Z_{1}\wedge Z_{2}=0\\},$

which can written as the (non-disjoint) union $S=S_{1}\cup S_{2}$, with

$\displaystyle S_{1}$ $\displaystyle=\\{(Z_{1},Z_{2})\in E\oplus
E:Z_{2}=fZ_{1}\\}\simeq E\times\mathbb{R},$ $\displaystyle S_{2}$
$\displaystyle=\\{(Z_{1},0)\in E\oplus E\\}\simeq E.$

Both $S_{1}$ and $S_{2}$ are closed submanifolds of $E\oplus E$, of dimensions
$3+1$ (resp. codimension $2-1$) and $3$ (resp. codimension $3$) respectively.
The same properties apply to $T_{1}=\varrho^{-1}(S_{1})$ and
$T_{2}=\varrho^{-1}(S_{2})$. According to Thom’s transversality Theorem (_cf._
[22, VII, Théorème 4.2]), the set of curves $Z(u)\in\Gamma(E)$ such that
$j^{1}Z$ is transversal to both $T_{1}$ and $T_{2}$ is open and dense in
$\Gamma(E)$ with the strong topology. For these curves $Z(u)$, the codimension
of $(j^{1}Z)^{-1}(T_{1})$ is $2$ and the codimension of $(j^{1}Z)^{-1}(T_{2})$
is $3$. Since these are set in $I\subset\mathbb{R}$, they must be empty.
Therefore, for this dense set of curves $Z(u)$, we have that $j^{1}Z\cap
S=\varnothing$, and the proof is complete.

###### Definition 14

Let $(M,g)$ be a Riemannian $3$-manifold. A parametrized ruled surface
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ is said to be a _Sannia ruled
surface_ if $Z\in\mathfrak{X}(\alpha)$ is in general position with respect to
the Levi-Civita connection of $g$.

###### Proposition 15

Let $(M,g)$ be an oriented $3$-dimensional Riemannian manifold and let
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a Sannia ruled surface
defined by a vector field $Z$ along a smooth curve $\alpha\colon
I\rightarrow(M,g)$. Then, there exist unique vector fields $X_{i}$, $1\leq
i\leq 3$, defined along $\alpha$ and smooth functions $\kappa_{i}\colon
I\rightarrow\mathbb{R}$, $\,0\leq i\leq 2$, with $\kappa_{0}>0$ and
$\kappa_{1}>0$, such that

  1. 1.

$(X_{1}(u),X_{2}(u),X_{3}(u))$ is a positively oriented orthonormal linear
frame of $T_{\alpha(u)}M$ for $u\in I$.

  2. 2.

The following formulas hold:

    1. (a)

$Z=\kappa_{0}X_{1}$,

    2. (b)

$\nabla_{\alpha^{\prime}}X_{1}=\kappa_{1}X_{2},$

    3. (c)

$\nabla_{\alpha^{\prime}}X_{2}=-\kappa_{1}X_{1}+\kappa_{2}X_{3},$

    4. (d)

$\nabla_{\alpha^{\prime}}X_{3}=-\kappa_{2}X_{2}.$

Proof. We define

(9) $\displaystyle X_{1}$ $\displaystyle=(\kappa_{0})^{-1}Z,$ (10)
$\displaystyle X_{2}$
$\displaystyle=(\kappa_{1})^{-1}\nabla_{\alpha^{\prime}}X_{1},$

where $\kappa_{0}=\left\|Z\right\|$,
$\kappa_{1}=\left\|\nabla_{\alpha^{\prime}}X_{1}\right\|$. For $X_{3}$ we
consider the unique vector field defining a positive orthonormal basis with
$X_{1}$ and $X_{2}$. Differentiating $g(X_{i},X_{j})$ with respect to $u$ we
get the formulas (2a)-(2d) in the statement.

###### Definition 16

The frame $(X_{1},X_{2},X_{3})$ along $\alpha$ determined in the above
Proposition is called _Sannia frame along_ $\alpha$, and the functions
$\kappa_{0},\kappa_{1},\kappa_{2}$ are the _Sannia invariants_ of the ruled
surface $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$.

Let $\theta$ and $\varphi$ denote the spherical angles of $\alpha^{\prime}$
with respect to $(X_{1},X_{2},X_{3})$; that is,

(11)
$\frac{\alpha^{\prime}}{\|\alpha^{\prime}\|}=\cos{\varphi}\cos{\theta}\,X_{1}+\sin{\varphi}\,X_{2}+\cos{\varphi}\sin{\theta}\,X_{3}.$

If $\varphi=0$, then $\theta$ is the base angle (see Definition 8). We will
show in Section 3 that this case corresponds to the base curve being a
striction one.

###### Remark 17

It would be more accurate to collect the angle invariants $\theta$ and
$\varphi$ into a single smooth function $\varsigma\colon I\to
S^{2}\subset\mathbb{R}^{3}$ assigning the coordinates of $\alpha^{\prime}$
with respect to the Sannia basis. However, in order to be close to the
classical results in the Euclidean space, we will keep track of the angles
$\theta$ and $\varphi$ instead of $\varsigma$.

In Definition 6 we have presented a function that extends the traditional
Euclidean distribution parameter to the whole surface in generic
$3$-dimensional Riemannian backgrounds. With a view to recovering the
distribution parameter in the classical sense, we next give the value of this
function in terms of the invariants $\kappa_{0}$, $\kappa_{1}$, $\kappa_{2}$,
$\theta$ and $\varphi$ of the Sannia frame associated to a base curve which
does not have to be necessarily a striction curve.

###### Proposition 18

If $(X_{1},X_{2},X_{3})$ is the Sannia frame of a Sannia ruled surface
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$, then the value of the
distribution parameter function $\lambda$ on the base curve $\alpha$ is

$\lambda(u,0)=\frac{\|\alpha^{\prime}\|\kappa_{0}^{2}\kappa_{1}\cos{\varphi}\sin{\theta}}{\left(\frac{d\kappa_{0}}{du}\right)^{2}+\kappa_{0}^{2}\kappa_{1}^{2}}.$

Proof. The result follows directly when (11) and relations (2a)-(2d) in
Proposition 15 are inserted into expression (6), evaluated at $v=0$.

###### Remark 19

Note that it is always possible to consider a unit ruling $Z$ along an arc-
length parametrized curve $\alpha$, i.e $\kappa_{0}=1$ and
$\|\alpha^{\prime}\|=1$. In such case the value of the distribution parameter
function along $\alpha$ reduces to

(12) $\lambda(u,0)=\frac{\cos{\varphi}\sin{\theta}}{\kappa_{1}}.$

We next state and prove the local existence and uniqueness theorem of ruled
surfaces in general $3$-Riemannian manifolds:

######  Theorem 20 (Fundamental theorem of ruled surfaces)

Let $(M,g)$ be a $3$-dimensional oriented Riemannian manifold and let
$(e_{1},e_{2},e_{3})$ be a positively oriented orthonormal basis of
$T_{p_{0}}M$, $p_{0}\in M$. Given smooth functions
$\overline{\kappa}_{i}\colon(u_{0}-\varepsilon,u_{0}+\varepsilon)\rightarrow\mathbb{R}$,
$0\leq i\leq 2$,
$\overline{\theta}\colon(u_{0}-\varepsilon,u_{0}+\varepsilon)\to\mathbb{R}$,
$\overline{\varphi}\colon(u_{0}-\varepsilon,u_{0}+\varepsilon)\to\mathbb{R}$
with $\overline{\kappa}_{0},\overline{\kappa}_{1}>0$, there exists
$\delta<\varepsilon$, a curve
$\alpha\colon(u_{0}-\delta,u_{0}+\delta)\rightarrow(M,g)$, parametrized by its
arc length, and a vector field $Z\in\mathfrak{X}(\alpha)$ such that the Sannia
ruled surface $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$,
$\mathbf{X}(u,v)=\exp_{\alpha(u)}(vZ(u))$ satisfies:

  1. 1.

$\alpha(u_{0})=p_{0}$,

  2. 2.

$X_{i}(u_{0})=e_{i}$ for $1\leq i\leq 3$,

  3. 3.

$\kappa_{i}=\overline{\kappa}_{i}$ for $0\leq i\leq 2$,
$\overline{\theta}=\theta$, $\overline{\varphi}=\varphi$,

where $\kappa_{i}$, $0\leq i\leq 2$, $\theta$ and $\varphi$ are the invariant
of Definition 16 and $X_{i}(u_{0})$ for $1\leq i\leq 3$ is the Sannia frame.
In addition, given any two points $p_{0},p_{0}^{\prime}\in M$ and two oriented
orthonormal bases $(e_{i})_{i=1}^{3}$, $(e^{\prime}_{i})_{i=1}^{3}$ of
$T_{p_{0}}M$ and $T_{p_{0}^{\prime}}M$ respectively, there always exists a
local isometry around $p_{0}$ and $p_{0}^{\prime}$ sending one ruled surface
to the other if and only if $(M,g)$ is of constant curvature.

Proof. We consider the vector bundle $p_{M}\colon\oplus^{3}TM\rightarrow M$
and a normal coordinate system $(U,(x_{i})_{i=1}^{3})$ centred at $p_{0}$
associated to the orthonormal basis $(e_{i})_{i=1}^{3}$. We define the natural
coordinate system $(x^{i},y_{k}^{j})$, $i,j,k=1,2,3$ in $p_{M}^{-1}(U)$ such
that

$w_{j}=y_{j}^{i}(w)\left.\frac{\partial}{\partial
x^{i}}\right|_{p},\qquad\forall
w=\left(w_{1},w_{2},w_{3}\right)\in\oplus^{3}T_{x}M,\quad p\in U.$

Let $X\colon I\rightarrow\oplus^{3}TM$, $a<u_{0}<b$, be the curve given by

$X(u)=\left(X_{1}(u),X_{2}(u),X_{3}(u)\right).$

Without loss of generality, the ruling can be considered to be unit, i.e.
$\overline{\kappa}_{0}=1$ in Proposition 15. This condition together with the
formulas (2a)-(2d) can be expressed as follows:

(13)
$\left\\{\begin{array}[c]{l}\dfrac{d\left(x^{i}\circ\alpha\right)}{du}\\!=\\!\cos{\overline{\varphi}}\cos{\overline{\theta}}\,(y_{1}^{i}\circ
X)+\sin{\overline{\varphi}}\,(y_{2}^{i}\circ
X)+\cos{\overline{\varphi}}\sin{\overline{\theta}}\,(y_{3}^{i}\circ X),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \dfrac{d\left(y_{1}^{i}\circ
X\right)}{du}\\!=\\!\overline{\kappa}_{1}\left(y_{2}^{i}\circ
X\right)\\!-\\!\Gamma_{jk}^{i}\dfrac{d\left(x^{j}\circ\alpha\right)}{du}\left(y_{1}^{k}\circ
X\right),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \dfrac{d\left(y_{2}^{i}\circ
X\right)}{du}\\!=\\!-\overline{\kappa}_{1}\left(y_{1}^{i}\circ
X\right)\\!+\\!\overline{\kappa}_{2}\left(y_{3}^{i}\circ
X\right)\\!-\\!\Gamma_{jk}^{i}\dfrac{d\left(x^{j}\circ\alpha\right)}{du}\left(y_{2}^{i}\circ
X\right),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \dfrac{d\left(y_{3}^{i}\circ
X\right)}{du}\\!=\\!-\overline{\kappa}_{2}\left(y_{2}^{i}\circ
X\right)\\!-\\!\Gamma_{jk}^{i}\dfrac{d\left(x^{j}\circ\alpha\right)}{du}\left(y_{3}^{i}\circ
X\right),\vskip 3.0pt plus 1.0pt minus 1.0pt\end{array}\right.$

with $1\leq i\leq 3$ and where $\Gamma_{jk}^{i}$ are the components of the
Levi-Civita connection $\nabla$ of $g$ with respect to the coordinate system
$(x_{i})_{i=1}^{3}$. From the general theory of ODE’s, the system (13) has
unique solutions $x^{i}\circ\alpha,y_{k}^{j}\circ X$, $i,j,k=1,2,3$,
satisfying the initial conditions
$\left(x^{i}\circ\alpha\right)(u_{0})=x^{i}(x_{0})$, $\left(y_{k}^{j}\circ
X\right)(u_{0})=\hat{\delta}_{k}^{j}$, $i,j,k=1,2,3$, where $\hat{\delta}$ is
the Kronecker delta. We consider the ruled surface
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$,
$\mathbf{X}(u,v)=\exp_{\alpha(u)}(vZ(u))$ with

$\alpha(u)=(x^{1}(u),x^{2}(u),x^{3}(u)),\qquad
Z(u)=y_{1}^{i}(u)\left(\frac{\partial}{\partial x^{i}}\right)_{\alpha(u)}.$

We first prove that

$X_{k}(u)=y_{k}^{i}(u)\left(\frac{\partial}{\partial
x^{i}}\right)_{\alpha(u)},\qquad 1\leq k\leq 3,$

define an orthonormal basis. To this end, consider the functions

$\phi_{ij}(u)=g_{\alpha(u)}(X_{i}(u),X_{j}(u)),\qquad 1\leq i\leq j\leq 3.$

As a consequence of the system (13), it is straightforward to check that
$\phi_{ij}$ satisfy the following equations,

$\left\\{\begin{array}[c]{l}\dfrac{d\phi_{11}}{du}=2\overline{\kappa}_{1}\phi_{12},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dfrac{d\phi_{12}}{du}=\overline{\kappa}_{1}\phi_{22}-\overline{\kappa}_{1}\phi_{11}+\overline{\kappa}_{2}\phi_{13},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dfrac{d\phi_{13}}{du}=\overline{\kappa}_{1}\phi_{23}-\overline{\kappa}_{2}\phi_{12},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dfrac{d\phi_{22}}{du}=-2\overline{\kappa}_{1}\phi_{12}+2\overline{\kappa}_{2}\phi_{23},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dfrac{d\phi_{23}}{du}=-\overline{\kappa}_{1}\phi_{13}+\overline{\kappa}_{2}\phi_{33}-\overline{\kappa}_{2}\phi_{22},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\dfrac{d\phi_{33}}{du}=-2\overline{\kappa}_{2}\phi_{23},\end{array}\right.$

with initial conditions $\phi_{ij}(u_{0})=\hat{\delta}_{j}^{i}$. But the
constant function $\hat{\phi}_{ij}(u)=\hat{\delta}_{j}^{i}$ also satisfy these
system of differential equations with these initial conditions. By virtue of
the unicity theorem of EDOs, it follows that
$\phi_{ij}(u)=\hat{\delta}_{j}^{i}$, for all $u$. In addition, since
$(X_{1}(u_{0}),X_{2}(u_{0}),X_{3}(u_{0}))$ is an oriented positive basis, by
continuity, so it is $(X_{1}(u),X_{2}(u),X_{3}(u))$ for any $u$. The frame
determined by $X$ verifies that $X_{1}=Z$ and
$\nabla_{\alpha^{\prime}}X_{1}=\overline{\kappa}_{1}X_{2}$, so since
$(X_{1},X_{2},X_{3})$ is positive, it is the Sannia basis of the ruled surface
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$. Finally, again from (13), it
follows that $\overline{\kappa}_{1}$ and $\overline{\kappa}_{2}$ are the
Sannia invariants.

We now prove the uniqueness of the ruled surface up to isometry. First, if
$(M,g)$ is of constant curvature, given $p_{0},p_{0}^{\prime}\in M$ and
$(e_{i})_{i=1}^{3}$, $(e^{\prime}_{i})_{i=1}^{3}$ oriented orthonormal bases
at $T_{p_{0}}M$ and $T_{p_{0}^{\prime}}M$ respectively, we consider a local
isometry $\phi$ sending $p_{0}$ to $p_{0}^{\prime}$ and $(e_{i})_{i=1}^{3}$ to
$(e^{\prime}_{i})_{i=1}^{3}$. Since $\phi$ preserves the Levi Civita
connection, the image $\phi\circ\mathbf{X}$ of the ruled surface defined by
$p_{0}$ and $(e_{i})_{i=1}^{3}$ is a ruled surface with the same parameters
$\overline{\kappa}_{1}$, $\overline{\kappa}_{2}$. By the uniqueness of the
system of differential equations (13), we have:
$\mathbf{X}^{\prime}=\phi\circ\mathbf{X}$. Conversely, if given $p_{0}$,
$p_{0}^{\prime}\in M$ and orthonormal bases $(e_{i})_{i=1}^{3}$,
$(e^{\prime}_{i})_{i=1}^{3}$ at $T_{p_{0}}M$ and $T_{p_{0}^{\prime}}M$
respectively, there is a local isometry sending one to the other, the space is
locally isotropic, and hence of contant curvature (_cf._ [11]).

##  3 Central points and striction curves

Euclidean ruled surfaces contain a special curve called _striction_ which is
the locus of the points whose distance with respect to neighbouring geodesic
is extremal. In particular striction curves are divided into _expanding-
contracting_ and _contracting-expanding_ curves depending on whether this
distance corresponds to a maximum or a minimum value respectively. In general,
the striction curve of a family of Euclidean curves is the locus for the
points where the geodesic curvature of the corresponding orthogonal set of
curves vanishes (read [15, 16] for more details). The striction curve of any
Euclidean ruled surface is also characterised by the fact that the generators
along it are parallel in the Levi-Civita sense. The following definition
extends such a property to general ambient Riemannian manifolds:

###### Definition 21

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface. A curve $s\colon I\to\mathbf{X}(I\times\mathbb{R})$ is said to be a
_striction curve_ if

(14) $g(s^{\prime},\nabla_{s^{\prime}}\mathbf{X}_{v})=0.$

As already mentioned, whenever the striction curve $s\colon
I\to\mathbf{X}(I\times\mathbb{R})$ exists, it can be taken as the base curve
of the ruled surface so that it can be reparametrized as
$\mathbf{X}(u,v)=\mathrm{exp}_{s(u)}(vZ(u))$.

###### Proposition 22

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a Sannia ruled surface
in a $3$-dimensional Riemannian manifold $\left(M,g\right)$ admitting a
striction curve $s\colon I\to\mathbf{X}(I\times\mathbb{R})$ and chosen to be
its base curve. For any $p=\mathbf{X}(u,0)$, the vectors $\\{X_{1},X_{3}\\}$
of the associated Sannia frame constitute an orthonormal basis of
$T_{p}\,(\mathbf{X}(I\times\mathbb{R}))$ and the angle $\varphi$ defined by
(11) vanishes identically. Therefore

$s^{\prime}(u)=\cos\sigma_{s(u)}\,X_{1}+\sin\sigma_{s(u)}\,X_{3}\quad\text{for
all $u\in I$}.$

Proof. The above relation holds since the tangent plane to
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ along $s$ is spanned by
$s^{\prime}$ and $Z$, and the second vector
$X_{2}=\nabla_{s^{\prime}}Z/||\nabla_{s^{\prime}}Z||$ of the corresponding
Sannia frame is perpendicular to both of them.

As mentioned above, in the Euclidean setting the ruling vector field of a
ruled surface is parallel along the striction with respect to its induced
Levi-Civita connection. This property also holds in a general Riemannian
background as shown in the following result.

###### Corollary 23

Under the hypotheses of Proposition 22, the ruling vector field $Z$ is
parallel along the striction curve with respect to the induced connection on
the Sannia ruled surface.

Proof. The Gauss identity on the striction reads

$\nabla_{s^{\prime}}Z=\nabla^{\Sigma}_{s^{\prime}}Z+\vec{h}(s^{\prime},Z),$

where $\nabla^{\Sigma}$ stands for the induced connection on the ruled surface
$\Sigma\equiv\mathbf{X}(I\times\mathbb{R})$. By Proposition 22,
$\nabla_{s^{\prime}}Z$ is orthogonal to $\Sigma$, which means that
$\nabla^{\Sigma}_{s^{\prime}}Z=0$.

###### Remark 24

In the classical statement of the fundamental theorem or ruled surfaces in the
Euclidean space $(\mathbb{R}^{3},\delta)$ an striction curve can always be
taken as the base curve in the parametrization for it always exists. By virtue
of the above proposition, in such a context $\varphi=0$ and $\theta=\sigma$,
i.e. the base angle on the striction. In particular, the distribution
parameter formula (12) on a general base curve reduces to

$\lambda(u,0)=\frac{\sin{\sigma_{s(u)}}}{\kappa_{1}},$

which is the classical expression for the Euclidean distribution parameter (on
a striction curve) [18, Lemma 5.3.7].

In the following result we obtain a condition for the striction curve in terms
of the coordinate basis $(\mathbf{X}_{u},\mathbf{X}_{v})$.

###### Proposition 25

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ with $\alpha\colon
I\to\mathbf{X}(I\times\mathbb{R})$ an associated base curve, and with a
unitary ruling vector field $Z\in\mathfrak{X}(\alpha)$. A curve $s\colon
I\to\mathbf{X}(I\times\mathbb{R})$ on such a surface is a striction curve if
and only if the squared norm of the vector field $\mathbf{X}_{u}$ along each
generator $\gamma_{Z(u)}(v)$ is critical at the corresponding point $s(u)$,
i.e.

(15) $\mathbf{X}_{v}(\|\mathbf{X}_{u}\|^{2})|_{p}=0,\quad\text{for all
$p=s(u)$}.$

Proof. Given any curve of the form
$s:I\rightarrow\mathbf{X}(I\times\mathbb{R})$, $s(u)=\mathbf{X}(u,v(u))$,
where $v:I\rightarrow\mathbb{R}$ is smooth, we have
$s^{\prime}=\mathbf{X}_{u}+v^{\prime}\mathbf{X}_{v}$. Then

$\displaystyle g(s^{\prime},\nabla_{s^{\prime}}\mathbf{X}_{v})$
$\displaystyle=$ $\displaystyle
g(\mathbf{X}_{u}+v^{\prime}\mathbf{X}_{v},\nabla_{\mathbf{X}_{u}+v^{\prime}\mathbf{X}_{v}}\mathbf{X}_{v})=g(\mathbf{X}_{u}+v^{\prime}\mathbf{X}_{v},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})$
$\displaystyle=$ $\displaystyle
g(\mathbf{X}_{u},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})=g(\mathbf{X}_{u},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})=\tfrac{1}{2}\mathbf{X}_{v}(\|\mathbf{X}_{u}\|^{2}),$

where we have taken into account that $g(\mathbf{X}_{v},\mathbf{X}_{v})$ is
constantly $1$, and that that
$\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v}-\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}=[\mathbf{X}_{u},\mathbf{X}_{v}]=0$.
The proof is complete by the definition of striction curve.

###### Remark 26

Since the length of the Jacobi vector field $\mathbf{X}_{u}$ gives a local
idea of the deviation of a congruence of geodesics, relation (15) is in
accordance with the definition of striction line established in a Euclidean
sense, where any point in these curves has a critical distance with respect to
neighbouring geodesics, as mentioned before.

The previous result motivates the following definition.

###### Definition 27

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface in a $3$-dimensional Riemannian manifold $(M,g)$. The function that
describes the derivative of the squared norm of the Jacobi field

(16) $F\colon\mathbf{X}(I\times\mathbb{R})\rightarrow\mathbb{R},\quad
F(p)=\tfrac{1}{2}(\mathbf{X}_{v}\|\mathbf{X}_{u}\|^{2})|_{p}$

will be referred to as the _Jacobi evolution function_ of the ruled surface.
The points of the parametrized ruled surface where $F$ vanishes are called
_central points_.

###### Remark 28

Therefore, striction curves lie on the vanishing set of the function $F$.
Furthermore, the function $F$ describes the evolution of the norm of the
Jacobi variational field $\mathbf{X}_{u}$ along each ruling. The study of the
sign of $F$ provides relevant information with regard to the behavior of the
rulings in a local manner. If $F$ is strictly positive at a point, the norm of
the Jacobi is increasing and hence the rulings are locally spreading apart.
Just the opposite happens in case that $F$ is strictly negative at some point.
Since $(u,v)$ is a system of coordinates associated to the parametrization (2)
of the ruled surface, we will simply refer to $(F\circ\mathbf{X})(u,v)$ as
$F(u,v)$ for the sake of simplicity.

We next explore the behaviour of the first and second derivatives of the
function $F$.

###### Theorem 29

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface in a Riemannian $3$-manifold $(M,g)$. The first and second derivative
of the Jacobi evolution function $F$ are

(17) $\displaystyle\frac{\partial F}{\partial
v}=\mathrm{Riem}(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u})+\|\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}\|^{2},$
(18)
$\displaystyle\frac{\partial^{2}F}{\partial\,v^{2}}=-g\left(\left(\nabla_{\mathbf{X}_{v}}R\right)(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v}),\mathbf{X}_{u}\right)+4\mathrm{Riem}(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}).$

Proof. If we differentiate $F$ along the $\mathbf{X}_{v}$ direction, we obtain

$\frac{\partial F}{\partial
v}=\nabla_{\mathbf{X}_{v}}g(\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{u})=g(\nabla_{\mathbf{X}_{v}}\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{u})+g(\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}).$

As $\mathbf{X}_{u}$ is a Jacobi vector field along the rulings, it satisfies
(3) so that

$\displaystyle\frac{\partial F}{\partial v}$ $\displaystyle=$
$\displaystyle-g(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v},\mathbf{X}_{u})+\|\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}\|^{2}$
$\displaystyle=$
$\displaystyle\mathrm{Riem}(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u})+\|\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}\|^{2},$

and

$\displaystyle\frac{\partial^{2}F}{\partial\,v^{2}}$ $\displaystyle=$
$\displaystyle-g\left(\nabla_{\mathbf{X}_{v}}(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v}),\mathbf{X}_{u}\right)-g(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v}),\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})$
$\displaystyle+2g(\nabla_{\mathbf{X}_{v}}\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})$
$\displaystyle=$
$\displaystyle-g\left(\nabla_{\mathbf{X}_{v}}(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v}),\mathbf{X}_{u}\right)-3g(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}).$

On the other hand, taking into account the symmetries of the Riemann tensor
and the fact that $\mathbf{X}_{v}$ is a geodesic vector field we obtain

(20) $\displaystyle
g(\nabla_{\mathbf{X}_{v}}(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v}),\mathbf{X}_{u})$
$\displaystyle=$ $\displaystyle
g((\nabla_{\mathbf{X}_{v}}R)(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v}),\mathbf{X}_{u})$
$\displaystyle+g(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}).$

Plugging (20) into (3) finally gives (18), which finally concludes the proof
of the theorem.

As a consequence, in the special case where the ambient manifold has constant
curvature the above derivatives of $F$ read as follows.

###### Corollary 30

Let $(M(k),g)$ be $3$-Riemannian manifold of constant sectional curvature $k$
and $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface with $Z$ a unitary vector field. Then the first and second derivatives
of the Jacobi evolution function $F$ read

(21) $\displaystyle\frac{\partial F}{\partial v}$ $\displaystyle=$
$\displaystyle-k(\|\mathbf{X}_{u}\|^{2}-\|\alpha^{\prime}(u)\|^{2}\cos^{2}\sigma_{p})+\|\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}\|^{2}$
(22) $\displaystyle\frac{\partial^{2}F}{\partial\,v^{2}}$ $\displaystyle=$
$\displaystyle(-2k)\mathbf{X}_{v}(\|\mathbf{X}_{u}\|^{2})=-4kF(u,v)$

where $p=\alpha(u)=\mathbf{X}(u,0)$ and $\sigma_{p}$ is the base angle between
$\alpha^{\prime}(u)$ and $Z$ at $p$.

Proof. By the sectional curvature relation for a space form, we have

(23)
$\mathrm{Riem}(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u})=-k\left(\|\mathbf{X}_{u}\|^{2}-g(\mathbf{X}_{u},\mathbf{X}_{v})^{2}\right),$

since $\mathbf{X}_{v}$ is a unitary vector field. On the other hand, we know
by Proposition 9 that
$g(\mathbf{X}_{u},\mathbf{X}_{v})=\|\alpha^{\prime}(u)\|\cos{\sigma_{p}}$ is
constant along the associated ruling. Hence, (23) can be rewritten as

(24)
$\mathrm{Riem}(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v},\mathbf{X}_{u})=-k\left(\|\mathbf{X}_{u}\|^{2}-\|\alpha^{\prime}(u)\|^{2}\cos^{2}\sigma_{p}\right),$

so (17) becomes (21) when relation (24) is inserted. Let us derive now
relation (22). By virtue of formula (18), the second derivative of the Jacobi
evolution function $F$ in a space form becomes

$\frac{\partial^{2}F}{\partial\,v^{2}}=4\,\mathrm{Riem}(\mathbf{X}_{v},\mathbf{X}_{u},\mathbf{X}_{v},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})=-4\,g(R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}),$

for $\nabla R=0$ holds. Since

$R(\mathbf{X}_{v},\mathbf{X}_{u})\mathbf{X}_{v}=R(\mathbf{X}_{v},\mathbf{X}_{u}^{\bot})\mathbf{X}_{v}=k\mathbf{X}_{u}^{\bot},$

where the last equality is fulfilled as a consequence of the ambient manifold
being a space form, we have

(25) $\frac{\partial^{2}F(u,v)}{\partial
v^{2}}=(-4k)g(\mathbf{X}_{u}^{\bot},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u}).$

Using the splitting (8) given in Proposition 11 for the Jacobi field
$\mathbf{X}_{u}$, we obtain

$\displaystyle g(\mathbf{X}_{u}^{\bot},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})$
$\displaystyle=$ $\displaystyle
g(\mathbf{X}_{u},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})-\|\alpha^{\prime}(u)\|(\cos{\sigma_{p}})g(\mathbf{X}_{v},\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{X}_{v}(\|\mathbf{X}_{u}\|^{2}),$

with $\sigma_{p}$ the base angle at $p=\alpha(u)$, where
$[\mathbf{X}_{u},\mathbf{X}_{v}]=0$ has been applied in combination with the
fact that $\mathbf{X}_{v}$ is geodesic. Inserting (3) into (25) finally gives
(22).

In the next result we integrate the differential equation (22) in the
different models for positive, zero or negative $k$.

###### Theorem 31

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a parametrized ruled
surface in a complete manifold $(M,g)$ of constant sectional curvature $k$,
with $Z$ a unitary ruling vector field. Then the expression of the Jacobi
evolution function $F$ is

(27) $\displaystyle F(u,v)$ $\displaystyle=$ $\displaystyle
C_{1}\cos{(\sqrt{4k}v)}+C_{2}\sin{(\sqrt{4k}v)},\quad\text{if $k>0$},$ (28)
$\displaystyle F(u,v)$ $\displaystyle=$ $\displaystyle
C_{1}+C_{2}v,\quad\text{if $k=0$}$ (29) $\displaystyle F(u,v)$
$\displaystyle=$ $\displaystyle
C_{1}\cosh{(\sqrt{-4k}v)}+C_{2}\sinh{(\sqrt{-4k}v)},\quad\text{if $k<0$}$

where

(30) $C_{1}=g_{p}(\nabla_{\alpha^{\prime}}Z,\alpha^{\prime}),$

and

(31) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{|4k|}}\left(-k\|\alpha^{\prime}\|_{p}^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|_{p}^{2}\right),\quad\text{if
$k>0$ or $k<0$},$ (32) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p},\quad\text{if $k=0$}.$

Proof. Evaluating the Jacobi evolution function $F(u,v)$ at $v=0$ gives

$F(u,0)=\tfrac{1}{2}\mathbf{X}_{v}(\|\mathbf{X}_{u}\|^{2})|_{p}=g_{p}(\nabla_{\mathbf{X}_{v}}\mathbf{X}_{u},\mathbf{X}_{u})=g_{p}(\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v},\mathbf{X}_{u})=g_{p}(\nabla_{\alpha^{\prime}}Z,\alpha^{\prime}),$

which implies (30), after evaluating (27), (28) and (29), at $v=0$. To compute
$C_{2}$, we use the Jacobi first derivative relation (21), which becomes

(33) $\frac{\partial F}{\partial
v}(u,0)=-k\|\alpha^{\prime}(u)\|_{p}^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p}$

since $\mathbf{X}_{u}(u,0)=\alpha^{\prime}(u)$. Differenciating (27) at $v=0$
from (33) we obtain

$C_{2}\sqrt{4k}=-k\|\alpha^{\prime}(u)\|^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p},$

which is none other than $C_{2}$ in (31) for $k>0$. The respective value of
$C_{2}$ for $k<0$ is obtained in an analogous way differenciating (29) at
$v=0$. Likewise, in the Euclidean context $k=0$, $(\partial F/\partial
v)(0)=\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p}$, which implies the value of
$C_{2}$ in (32) for the Jacobi evolution function is $F(u,v)=C_{1}+C_{2}v$ in
the flat case.

###### Theorem 32

Let $\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ be a Sannia ruled surface
$\mathbf{X}:I\times\mathbb{R}\rightarrow(M,g)$ in a complete manifold $(M,g)$
of constant sectional curvature $k$, where $\alpha\colon
I\to\mathbf{X}(I\times\mathbb{R})$ is an associated base curve, and with a
unitary ruling vector field $Z\in\mathfrak{X}(\alpha)$. If $s\colon
I\to\mathbf{X}(I\times\mathbb{R})$, $s(u)=\mathbf{X}(u,v(u))$ is an striction
curve, then

(34) $\displaystyle v(u)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4k}}\arctan{\left(\frac{-\sqrt{4k}\,g_{p}(\nabla_{\alpha^{\prime}}Z,\alpha^{\prime})}{-k\|\alpha^{\prime}\|_{p}^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p}}\right)}\quad\text{if
$k>0$},$ (35) $\displaystyle v(u)$ $\displaystyle=$
$\displaystyle-\frac{g_{p}(\nabla_{\alpha^{\prime}}Z,\alpha^{\prime})}{\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p}}\quad\text{if
$k=0$},$ (36) $\displaystyle v(u)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-4k}}\mathrm{arctanh}{\left(\frac{-\sqrt{-4k}\,g_{p}(\nabla_{\alpha^{\prime}}Z,\alpha^{\prime})}{-k\|\alpha^{\prime}\|_{p}^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p}}\right)}\quad\text{if
$k<0$},$

where $\sigma_{p}$ is the angle betwen the vectors $\alpha^{\prime}$ and $Z$
at $p=\alpha(u)$. In addition, if $M$ is simply connected (that is, $M=S^{3}$
for $k>0$, $M=\mathbb{R}^{3}$ for $k=0$ or $M=\mathbb{H}^{3}$ for $k<0$ with
their standard metrics scaled by $1/\sqrt{|k|}$ when $k\neq 0$), then for
$k\geq 0$, every Sannia surface has a unique striction curve. For $k<0$, every
Sannia surface has at most one striction curve.

Proof. As already noted in Remark 28, a necessary condition for a point $p$ to
lie on the striction curve is that the Jacobi evolution function verifies
$F(p)=0$. Then the expressions for $v(u)$ are directly obtained from (27),
(28) and (29) respectively, taking into account (31) or (32). For $k>0$, the
expression must be understood to provide $v(u)=\pm\pi/2$ if
$-k\|\alpha^{\prime}\|_{p}^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|^{2}_{p}=0$.
In any case, the value of $v(u)$ is periodic with period $2\pi/\sqrt{k}$,
which is exactly the period of the geodesics in $S^{3}$, so that the striction
curve is geometrically unique. The uniqueness holds trivially true for $k=0$.
For $k<0$, the injectivity of the function $\mathrm{arctanh}$ implies that
there is at most a solution for $v(u)$.

###### Remark 33

Formula (35) is the classical formula for the striction in the Eucliden space.
Formulas (34) in $S^{3}$ and (36) in $\mathbb{H}^{3}$ were given in [5], where
the uniqueness of striction curves is also addressed. However, they obtain the
results by working with the sphere or the hyperbolic space as submanifolds of
$\mathbb{R}^{4}$ with the standard Euclidean or Lorentzian metric
respectively, that is, in a less intrinsic fashion. Our approach involves a
differential equation that, in principle, could be analysed in arbitrary
Riemannian manifolds. Note that in such a case, the equation will involve the
curvature of the ambient space. Finally, the authors also claim in [5] that
the striction curve always exists for $k<0$, a fact that is wrong as the
following example illustrates.

Example 1: A ruled surface in the hyperbolic space without striction curve. We
consider in $(\mathbb{H}^{3},g=\frac{1}{z}(dx^{2}+dy^{2}+dz^{2}))$ the ruled
surface with base curve $\alpha\colon[0,2\pi)\rightarrow(\mathbb{H}^{3},g)$
determined by $\alpha(u)=(\cos{u},\sin{u},1)$ and unitary ruling vector
$Z(u)=(\cos{u},\sin{u},0)$ defined along $\alpha$. A straightforward
computation shows that

$\nabla_{\alpha^{\prime}}Z=-\sin{u}\,\partial_{x}+\cos{u}\,\partial_{y}.$

The form for the Jacobi evolution function $F$ is described by relation (29)
for negative constant sectional curvature. The coefficients $C_{1}$ and
$C_{2}$ read this time
$C_{1}=g_{p}(\alpha^{\prime},\nabla_{\alpha^{\prime}}Z)=1$ and
$C_{2}=\frac{1}{2}\left(\|\alpha^{\prime}\|_{p}^{2}\sin^{2}{\sigma_{p}}+\|\nabla_{\alpha^{\prime}}Z\|_{p}^{2}\right)=1$,
since $\sigma_{p}=\pi/2$. Therefore, the Jacobi evolution function associated
to such ruled surfaces is

$F(u,v)=\cosh{2v}+\sinh{2v}.$

Condition $F=0$ holds on the striction line in case of existence, which is
equivalent to

$1+\tanh{2v}=0,$

which clearly does not have any solution.

On the other hand, when the ambient manifold has not constant curvature, the
uniqueness of the striction curve cannot be guaranteed. This is shown by the
following examples.

Example 2: A ruled surface in a product manifold equipped with different
striction curves. We consider the surface of revolution
$\psi\colon\Sigma\rightarrow(\mathbb{R}^{3},\delta)$ in
$(\mathbb{R}^{3},\delta)$ generated by the rotation of the curve
$f(x)=2+\sin{x}$ in the $xy$-plane about the $x$-axis. Such a surface is
defined by the following parametrization:

$\displaystyle\mathbf{X}\colon[0,2\pi)\times\mathbb{R}\rightarrow(\mathbb{R}^{3},\delta)$
$\displaystyle\mathbf{X}(u,v)=(v,(2+\sin{v})\cos{u},(2+\sin{v})\sin{u}).$

The expression for the induced metric $g_{\Sigma}$ on $\Sigma$ reads

$g_{\Sigma}=(2+\sin{v})^{2}du^{2}+(1+\cos^{2}{v})dv^{2}$

in local coordinates associated to the parametrization (3). It is a well known
fact that the generating curves of a surface of revolution in
$(\mathbb{R}^{3},\delta)$ are geodesics of the surface. Consider the product
manifold $\mathbb{R}\times\Sigma$ endowed with the product metric
$g=dt^{2}+g_{\Sigma}$. Since $\Sigma\subset(\mathbb{R}\times\Sigma,g)$ is a
totally geodesic slice of this Riemannian product, its Gauss curvature is zero
and the generating geodesic curves of $(\Sigma,g_{\Sigma})$ are also geodesics
in the ambient product manifold. Let us choose $\alpha(u)=\mathbf{X}(u,0)$ as
base curve of $(\Sigma,g_{\Sigma})$. When we consider the embedded ruled
surface $\psi\colon\Sigma\rightarrow(\mathbb{R}\times\Sigma,g)$,
$\alpha^{\prime}(u)=\mathbf{X}_{u}|_{\alpha}$ is the initial $v=0$ value of
the Jacobi vector field $\mathbf{X}_{u}$ along the rulings, and
$Z=(1/\sqrt{1+\cos^{2}{v}})\mathbf{X}_{v}$ is a unit geodesic vector tangent
to them. Let us compute the Jacobi evolution function $F$ in this context. A
straightforward calculation gives

$\nabla_{\alpha^{\prime}}Z=\frac{\cos{v}}{(1+\cos^{2}{v})(2+\sin{v})}\mathbf{X}_{u}.$

As a consequence

$F(u,v)=g(\mathbf{X}_{u},\nabla_{\mathbf{X}_{u}}Z)=\frac{\cos{v}(2+\sin{v})}{1+\cos^{2}{v}}.$

The striction curves are given by the solution of $F=0$, and this happens if
and only if $v=\pi/2+k\pi$, where $k\in\mathbb{Z}$. This means that each curve
$\alpha_{v_{k}}(u)=(u,\pi/2+k\pi)$ with $k\in\mathbb{Z}$ is a striction curve
of $\psi\colon\Sigma\rightarrow(\mathbb{R}\times\Sigma,g)$.

Example 3: A ruled surface in a warped product manifold with an arbitrary
number of striction curves. Given an open interval $I\subset\mathbb{R}$ and a
$2$-dimensional Riemannian manifold $(F,g_{F})$, consider the product
$3$-manifold $I\times F$ endowed with the metric $g=dt^{2}+f^{2}(t)g_{F}$,
where $f\colon I\rightarrow\mathbb{R}$ is a smooth positive function. We will
refer to the warped product manifold $(I\times F,g)$ as $I\times_{f}F$.
Consider a closed unit curve $\alpha^{F}\colon[a,b)\rightarrow(F,g_{F})$ in
the fiber of $I\times_{f}F$. Given any $v\in I$, $\alpha^{F}$ can be lifted in
a natural way to the slice $\\{t=v\\}$ as the curve
$\alpha_{v}\colon[a,b)\rightarrow I\times_{f}F$ defined by
$\alpha_{v}(u)=(v,\alpha^{F}(u))$, with $Z=\partial_{t}$ as ruling unit vector
field along $\alpha_{v}$. As usual, the initial value of the Jacobi field
$\mathbf{X}_{u}$ on $\alpha_{v}$ is
$\mathbf{X}_{u}|_{\alpha_{v}}=\alpha^{\prime}_{v}$. Since
$Z=\partial_{t}|_{\alpha_{v}}$ is orthogonal to the base curve,
$\mathbf{X}_{u}$ will remain orthogonal to the ruling by virtue of Proposition
9. Since $\mathbf{X}_{u}$ is tangent to the fiber in $I\times_{f}F$ and
$\mathbf{X}_{v}=\partial_{t}$ is orthogonal to it. It holds

$\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v}=(\partial_{t}\log f)\mathbf{X}_{u},$

so the corresponding Jacobi evolution function reads

$F(u,v)=g(\mathbf{X}_{u},\nabla_{\mathbf{X}_{u}}\mathbf{X}_{v})=(\partial_{t}\log
f)\|\mathbf{X}_{u}\|^{2}.$

The solution to the equation $F=0$ which determines the striction curves is
given in this case by $f^{\prime}(v)=0$. Hence, there will be as many
striction curves as there are values at which the function $f^{\prime}$
vanishes in $I$. For instance, if we consider $I=\mathbb{R}$ and $(F,g_{F})$
isometric to the two-dimensional Euclidean space $(\mathbb{R}^{2},\delta)$,
for the choice $f(t)=\sin{t}$, every curve of the form
$\alpha_{v_{k}}(u)=(\pi/2+k\pi,\cos{u},\sin{u})$ with $k\in\mathbb{Z}$ is a
striction line in the ruled surface determined by it and
$\mathbf{X}_{v}=\partial_{t}$.

## References

  * [1] A.L. Albujer, F.R. dos Santos, On the geometry of non-degenerate surfaces in Lorentzian homogeneous $3$-manifolds, arXiv preprint arXiv:2108.06823 (2021).
  * [2] G. Calvaruso, M. Castrillón, _Pseudo-Riemannian homogeneous structures_ , Springer Nature Switzerland, 2019.
  * [3] M. Castrillón, V. Fernández Mateos, J. Muñoz, The equivalence problem of curves in a Riemannian manifold, Annali di Matematica Pura ed Applicata (1923-) 194(2) (2015), 343–367.
  * [4] S.M. Choi, On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space, Tsukuba journal of mathematics 19.2 (1995), 285-304.
  * [5] L.C.B da Silva, J.D. da Silva, Characterization of manifolds of constant curvature by ruled surfaces, São Paulo Journal of Mathematical Sciences (2022), 1–25.
  * [6] F. Dillen, W. Kühnel, Ruled Weingarten surfaces in Minkowski $3$-space, Manuscripta. Math. 98 (1999), 307–320.
  * [7] M.P. Do Carmo, Riemannian Geometry, Birkhauser (1992).
  * [8] M.P. Do Carmo, Differential geometry of curves and surfaces: revised and updated second edition, Courier Dover Publications (2016).
  * [9] Y.H. Kim, D.W. Yoon, Ruled surfaces with pointwise $1$-type Gauss map, Journal of Geometry and Physics. 34.3-4 (2000), 191-205.
  * [10] Y.H. Kim, D.W. Yoon, Classification of ruled surfaces in Minkowski $3$-spaces, Journal of Geometry and Physics. 49.1 (2004), 89–100.
  * [11] S. Kobayashi and K. Nomizu, _Foundations of Differential Geometry_ , Volumes I, II, Interscience Publishers, John Wiley and Sons, Inc., New York, London, 1963, 1969.
  * [12] A. Kriegl and P.W. Michor, _The convenient setting of global analysis_ , Vol.53 American Mathematical Soc., 1997.
  * [13] H. Liu, Y. Yu, S.D. Jung Invariants of non-developable ruled surfaces in Euclidean 3-space, Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry. 55.1 (2014), 189–199.
  * [14] M. Mars, A. Soria On the Penrose inequality along null hypersurfaces, Classical and Quantum Gravity. 33.11 (2016): 115019.
  * [15] H.R. Müller, _Über die Striktionslinien von Kurvenscharen_ , 50, Monatsh. für Math. und Phys., 1941, 101–110.
  * [16] H.R. Müller, _Über Striktionslinien von Kurven- und Geradenscharen im elliptischen Raum_ , 52, Monatsh. für Math., 1948, 138–161.
  * [17] B. O’Neill, Semi-Riemannian geometry with applications to relativity, Academic Press (1983).
  * [18] H. Pottmann, J. Wallner, _Computational Line Geometry_ , Springer-Verlag Berlin Heidelberg, 2001.
  * [19] G. Sannia, Una rappresentazione intrinseca delle rigate, Giorn. di Mat. 63 (1925), 31-47.
  * [20] H. Shin, Y.W. Kim, S.-E. Koh, H.Y. Lee and S.-D. Yang, Ruled minimal surfaces in the three-dimensional Heisenberg group, Pacific Journal of Mathematics 261.2 (2013), 477–496.
  * [21] Dirk J. Struik, _Lectures on Classical Differential Geometry: Second Edition_ , Dover Publications 1988.
  * [22] Tougeron, J.-C. _Idéaux de fonctions différentiables_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. Springer, Berlin (1972).