Constant sectional curvature surfaces with a semi-symmetric non-metric connection

Friedmann and Schouten introduced in 1924 the notion of a semi-symmetric connection in a Riemannian manifold [4]. An affine connection $\widetilde{\nabla}$ in a Riemannian manifold $(\widetilde{M},\tilde{g})$ is said to be semi-symmetric connection if there is a non-zero vector field $\mathsf{C}\in\mathfrak{X}(\widetilde{M})$ such that its torsion $T$ satisfies the identity

If in addition $\widetilde{\nabla}\tilde{g}=0$, the connection $\widetilde{\nabla}$ is called a semi-symmetric metric connection [6]. Yano studied semi-symmetric metric connections with zero curvature and when the covariant derivative of the torsion tensor vanishes [14]. Submanifolds of Riemannian manifolds with semi-symmetric metric connections have been also investigated: without aiming a complete list, we refer to the readers to [7, 8, 9, 10, 12, 13].

If $\widetilde{\nabla}\tilde{g}\not=0$, the connection is called semi-symmetric non-metric connection (snm-connection to abbreviate) [1, 2]. In this case, there is a relation between $\widetilde{\nabla}$ and the Levi-Civita connection $\widetilde{\nabla}^{0}$ of $(\widetilde{M},\tilde{g})$, namely,

Such as it occurs for semi-symmetric metric connections, it is natural to study submanifolds of Riemannian manifolds endowed with a snm-connection $\widetilde{\nabla}$. Let $M$ be a submanifold of $\widetilde{M}$. Denote by $\nabla$ (resp. $\nabla^{0}$) the induced connection on $M$ by $\widetilde{\nabla}$ (resp. $\widetilde{\nabla}^{0}$). The Gauss formulas are given by

for all $X,Y\in\mathfrak{X}(M)$, where $h$ is a $(0,2)$-tensor field on $M$ and $h^{0}$ is the second fundamental form of $M$. It is known that $h=h^{0}$ [2]. Hence that problems of extrinsic nature are the same one that for the Levi-Civita connection.

We consider intrinsic geometry of submanifolds. One of the main concepts in intrinsic Riemannian geometry is that of sectional curvature. It is natural to carry this concept for snm-connections. However, the sectional curvature of $(\widetilde{M},\tilde{g})$ with respect to $\widetilde{\nabla}$ cannot be defined by the usual way as the Levi-Civita connection $\widetilde{\nabla}^{0}$. This is because if $\widetilde{R}$ is the curvature tensor of $\widetilde{\nabla}$, the quantity $\tilde{g}(\widetilde{R}(e_{1},e_{2})e_{2},e_{1})$, where $\{e_{1},e_{2}\}$ is an orthonormal basis of $\pi$, depends on the basis $\{e_{1},e_{2}\}$: see Sect. 2 for details. In contrast, the third author of this paper, jointly with I. Mihai, proved that $\tilde{g}(\widetilde{R}(e_{1},e_{2})e_{2},e_{1})+\tilde{g}(\widetilde{R}(e_{2},e_{1})e_{1},e_{2})$ is independent on the basis [11]. Then they introduced the following notion of sectional curvature for snm-connections.

Once we have the notion of sectional curvature, it is natural to ask for those submanifolds with constant sectional curvature. As for the Levi-Civita connection, this question is difficult to address in all its generality.

In this paper, we consider that the ambient space is the $3$-dimensional Euclidean space $\mathbb{R}^{3}$ endowed with the Euclidean metric $\langle,\rangle$. The amount of snm-connections of $\mathbb{R}^{3}$ is given by the vector fields $\mathsf{C}$ in the definition (1) of a semi-symmetric connection. One of the simplest choices of snm-connections of $\mathbb{R}^{3}$ is that $\mathsf{C}$ is a canonical vector field. To be precise, let $(x,y,z)$ be canonical coordinates of $\mathbb{R}^{3}$ and let $\{\partial_{x},\partial_{y},\partial_{z}\}$ be the corresponding basis of $\mathfrak{X}(\mathbb{R}^{3})$. In fact, if the vector field $\mathsf{C}$ is assumed to be canonical, namely $\mathsf{C}\in\{\partial_{x},\partial_{y},\partial_{z}\}$ then, after a change of coordinates of $\mathbb{R}^{3}$, $\mathsf{C}$ is a unit constant vector field.

From now on, unless otherwise specified, we denote by $\mathsf{C}$ a unit constant vector field on $\mathbb{R}^{3}$.

Definitively, the problem that we study is the classification of surfaces with constant sectional curvature for a given canonical snm-connection $\widetilde{\nabla}$ of $\mathbb{R}^{3}$. A way to tackle this problem is to impose a certain geometric condition on the surface. A natural condition is that the surface is invariant by a one-parameter group of rigid motions. Denote $K$ by the sectional curvature with respect to the induced connection on the surface from $\widetilde{\nabla}$. Assuming a certain invariance of the surface, it allows us to expect that the equation $K=c$ can be expressed as an ordinary differential equation, where, under mild conditions, the existence is assured. For example, we can assume that the surface is invariant by a group of translations or that the surface is invariant by a group of rotations. In the first case, the surface is called cylindrical and in the second one, rotational surface, or surface of revolution.

The organization of this paper is according to both types of surfaces. In Sect. 2 we prove an useful formula for computing the sectional curvature $K$ of a surface in terms of that of $\mathbb{R}^{3}$ and the Gaussian and mean curvatures of the surface. We will show some explicit examples of computations of sectional curvatures.

Section 3 is devoted to cylindrical surfaces. A cylindrical surface can be parametrized by $\psi(s,t)=\gamma(s)+t\vec{w}$, $s\in I\subset\mathbb{R}$, $t\in\mathbb{R}$, where $\vec{w}\in\mathbb{R}^{3}$ is a unitary vector and $\gamma\colon I\to\mathbb{R}^{3}$ is a curve contained in a plane orthogonal to $\vec{w}$. The surface is invariant by the group of translations generated by $\vec{w}$. After computing the sectional curvature $K$ in Thm. 3.1, in Cor. 3.2, we prove that any cylindrical surface whose rulings are parallel to $\mathsf{C}$ has constant sectional curvature $K$, being $K=\frac{1}{2}$. Another interesting case of cylindrical surfaces is that the rulings are orthogonal to $\mathsf{C}$. We obtain a full classification of these cylindrical surfaces with $K$ constant depending on the sign of $K$ (Cor. 3.3). For the particular values $K=1/2$ and $K=-1/2$, in Cor. 3.4 we obtain explicit parametrizations of the surfaces.

Rotational surfaces are invariant by rotations about an axis $L$ of $\mathbb{R}^{3}$ and such surfaces with $K$ constant will be studied in Sect. 4. It is worth to point out that there is not a priori relation between the axis $L$ and the vector field $\mathsf{C}$ that defines the canonical snm-connection. However, we prove in Thm. 4.1 that $L$ and $\mathsf{C}$ must be parallel. In Thm. 4.3, we classify all conical rotational surfaces with $K$ constant proving that these surfaces are planes or circular cylinders. As a last observation, when $K=\frac{1}{2}$, in Thm. 4.5, the existence of rotational surfaces orthogonally intersecting the rotation axis is also obtained.

Let $(\widetilde{M},\tilde{g})$ be a Riemannian manifold of dimension $\geq 2$ and let $\widetilde{\nabla}$ be an affine connection on $\widetilde{M}$. The torsion and curvature of $\widetilde{\nabla}$ are respectively a $(1,2)$-tensor field $\widetilde{T}$ and a $(1,3)$-tensor field $\widetilde{R}$ defined by

for $X,Y,Z\in\mathfrak{X}(\widetilde{M})$. Let $\widetilde{\nabla}$ be a snm-connection on $(\widetilde{M},\tilde{g})$ determined by a vector field $\mathsf{C}\in\mathfrak{X}(\widetilde{M})$. Using (2), there is also a relation between $\widetilde{R}$ and the Riemannian curvature tensor $\widetilde{R}^{0}$ of $\widetilde{\nabla}^{0}$ ([1, 11]). Indeed, for orthonormal vectors $e_{1},e_{2}\in T_{p}\widetilde{M}$, $p\in\widetilde{M}$, we have

Although the first term at the right hand-side is the sectional curvature of the plane section $\pi=\text{span}\{e_{1},e_{2}\}$, the term at the left hand-side depends on the choice of the basis of $\pi$. Therefore, the value $\tilde{g}(\widetilde{R}(e_{1},e_{2})e_{2},e_{1})$ does not stand for a sectional curvature. The quantity (3) was proposed in [11] as the definition of sectional curvature of $\pi$ with respect to $\widetilde{\nabla}$ because it is independent on the basis in $T_{p}\widetilde{M}$. In case that $\{e_{1},e_{2}\}$ is an arbitrary basis of $\pi$, it is immediate to see

From now on, suppose that $\widetilde{M}$ is the Euclidean space $\mathbb{R}^{3}$. We compute the sectional curvature of a plane of $\mathbb{R}^{3}$.

We conclude this section establishing a relation between the sectional curvatures $K$ and $\widetilde{K}$ of a surface in $\mathbb{R}^{3}$ in terms of the Gaussian and the mean curvatures of the surface. Let $M$ be an oriented surface immersed in $\mathbb{R}^{3}$ and $N$ its unit normal vector field. Let also $\widetilde{\nabla}$ be a snm-connection on $\mathbb{R}^{3}$ determined by an arbitrary vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^{3})$. We have the decomposition of $\mathsf{C}$ in its tangential and normal components with respect to $M$,

If $X,Y,Z,U\in\mathfrak{X}(M)$, then the Gauss equation with respect to $\widetilde{\nabla}$ is ([2]):

Simple consequences of the relation (6) appear in the following result.

Thanks to this corollary we see that a plane and a cylindrical cylinder satisfy the equality $K=\widetilde{K}$. In general, a surface satisfies $K=\widetilde{K}$ if and only if $G=\langle\mathsf{C},N\rangle H$. In case that $\mathsf{C}$ is a canonical vector field, we construct such a surface as follows.

Let $M$ be a cylindrical surface in $\mathbb{R}^{3}$ whose rulings are parallel to $\vec{w}$, where $\vec{w}\in\mathbb{R}^{3}$, $|\vec{w}|=1$. If $\gamma=\gamma(s)$ is the generating curve of $M$ contained in a plane orthogonal to $\vec{w}$, then a parametrization of $M$ is

Without loss of generality, we suppose that $\gamma$ is parametrized by arc-length. Let ${\bf n}$ be the unit normal vector of $\gamma$ and let $\kappa$ be the Frenet curvature of $\gamma$ with $\gamma^{\prime\prime}=\kappa{\bf n}$. Since $\gamma$ is contained in a plane orthogonal to $\vec{w}$, consider the orientation on $\gamma$ such that $(\gamma^{\prime},\vec{w},{\bf n})=1$, where $(\vec{a},\vec{b},\vec{c})$ stands for the determinant of the matrix formed by three vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ of $\mathbb{R}^{3}$.

We distinguish two particular cases, when the rulings are parallel or orthogonal to the constant vector field $\mathsf{C}$.

Suppose that the rulings are orthogonal to $\mathsf{C}$. In the next result we are going to obtain explicit parametrizations of cylindrical surfaces with constant sectional curvature. Without loss of generality, we suppose that $\mathsf{C}=\partial_{z}$ and $\vec{w}=(0,1,0)$. Then $\gamma$ is contained in the $xz$-plane, say $\gamma(s)=(x(s),0,z(s))$, for smooth functions $x,z\colon I\to\mathbb{R}$. The case that $M$ is a plane is particular. Any plane of $\mathbb{R}^{3}$ perpendicular to $\partial_{z}$ can be viewed as a cylindrical surface with rulings orthogonal to $\partial_{z}$. By Prop. 2.1, we know that its curvature $K$ is constant with $0\leq K\leq\frac{1}{2}$. We discard this case.

In Fig. 2 we depict some graphics of the generating curves for different values of $K$. Notice that the domain of $\gamma$ is not $\mathbb{R}$ in general because the root that appears in the integrals that define the $x$-coordinate of $\gamma$. For example, if $K=0$, then $s\in[1,\infty)$.

It is worth to consider the cases $K=1/2$ and $K=-1/2$. In such a case, the integrals of Cor. 3.3 can be explicitly solved.

For $K=1/2$, the curve $\gamma$ in (1) is called grim reaper. The usual parametrization of the grim reaper is $y(x)=-\log(\cos(x))$ in the $(x,y)$-plane $\mathbb{R}^{2}$. This is deduced immediately by letting $x=\tan^{-1}(\sinh(s))$. The grim reaper is a remarkable curve in the theory of curve-shortening flow [5].

In this section we study rotational surfaces with constant sectional curvature. A first problem is the relation between the axis $L$ of the surface and the vector field $\mathsf{C}$ that defines the canonical snm-connection. As we said in the Introduction, there is no a priori a relation between both. However, we prove that they must be parallel.