Geodesic Hypersurfaces Research Articles

CONVEXITY AND TWO PIECE PROPERTY IN S^n

A subset A of S n , n ≥ 2, has spherical two piece property if any hypersphere cuts A into at most two pieces. A subset A of S n has totally geodesic two piece property if any totally geodesic hypersurface in S n , n ≥ 2, cuts A into at most two pieces. In this article we study some geometric properties of subsets in S n -as a manifold with focal points and these types of two piece property in S n .

On the stability of minimal cones in warped products

In a seminal paper published in 1968, J. Simons proved that, for n ≤ 5, the Euclidean (minimal) cone CM, built on a closed, oriented, minimal and non totally geodesic hypersurface M n of $\mathbb{S}^{n + 1} $ is unstable. In this paper, we extend Simons’ analysis to warped (minimal) cones built over a closed, oriented, minimal hypersurface of a leaf of suitable warped product spaces. Then, we apply our general results to the particular case of the warped product model of the Euclidean sphere, and establish the unstability of CM, whenever 2 ≤ n ≤ 14 and M n is a closed, oriented, minimal and non totally geodesic hypersurface of $\mathbb{S}^{n + 1} $ .

The stability index of hypersurfaces with constant scalar curvature in spheres

The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.

Homotopy groups, focal points, and totally geodesic immersions

In this paper we consider on a complete Riemannian manifold $M$ an immersed totally geodesic hypersurface $\Si$ existing together with an immersed submanifold $N$ without focal points. No curvature condition is needed. We obtained several connectedness results relating the topologies of $M$ and $\Si$ which depend on the codimension of $N$.

Extrinsic hyperspheres in manifolds with special holonomy

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kähler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\), provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.

Existence and Topological Uniqueness of Compact CMC Hypersurfaces with Boundary in Hyperbolic Space

It is proved that if Γ is a compact, embedded hypersurface in a totally geodesic hypersurface ℍn of ℍn+1 satisfying the enclosing H-hypersphere condition with |H|<1, then there is one and only one (up to a reflection on ℍn) compact embedded constant mean curvature H hypersurface M such that ∂M=Γ. Moreover, M is diffeomorphic to a ball.

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.

Hyperplane sections in arithmetic hyperbolic manifolds

We prove that the fundamental groups of ‘standard’ arithmetic hyperbolic manifolds virtually retract onto their geometrically finite subgroups. In particular, this implies that the homology groups of immersed totally geodesic hypersurfaces of compact arithmetic hyperbolic manifolds virtually inject in the homology groups of the ambient manifold.

Hypersurfaces with null higher order mean curvature

A hypersurface M n immersed in a space form is r-minimal if its (r + 1) th -curvature (the (r + 1) th elementary symmetric function of its principal curvatures) vanishes identically. Let W be the set of points which are omitted by the totally geodesic hypersurfaces tangent to M. We will prove that if an orientable hypersurface M n is r-minimal and its r th -curvature is nonzero everywhere, and the set W is nonempty and open, then M n has relative nullity n − r. Also we will prove that if an orientable hypersurface M n is r-minimal and its r th -curvature is nonzero everywhere, and the ambient space is euclidean or hyperbolic and W is nonempty, then M n is r-stable.

New characterizations of totally geodesic hypersurfaces in anti-de Sitter space [formula omitted

By extending Yau’s technique, we obtain rigidity results to complete maximal spacelike hypersurfaces in anti-de Sitter space, imposing suitable conditions on both the norm of the second fundamental form and a certain height function naturally attached to the hypersurface. We also characterize complete maximal spacelike graphs satisfying a certain assumption on the gradient of the function which determines the graph.

Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|2 ≥ n. The natural generalization for this pinching is −(r + 2)Sr+2 ≥ (n − r)Sr > 0. Under this condition we shall extend such result for closed oriented hypersurface Σn of the Euclidean unit sphere \({\mathbb{S}^{n+1}}\) with null Sr+1 mean curvature by showing that the index of r-stability, \({Ind_{\Sigma^n}^{r}}\), also satisfies \({Ind_{\Sigma^n}^{r}\ge n+3}\). Instead of the previous hypothesis if we consider \({\frac{S_{r+2}}{{S_r}}}\) constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces \({\Sigma^{n}\subset \mathbb{S}^{n+1}}\) with Sr+1 = 0 and Sr+2 < 0 have index of r-stability greater than or equal to 2n + 5.

A cylinder theorem for a (1, 1)-geodesic affine hypersurface with non-flat induced connection

The purpose of this paper is to show a cylinder theorem for a (1, 1)-geodesic affine hypersurface to an affine space with non-flat induced connection.

Minimality of totally geodesic submanifolds in Finsler geometry

Using the symplectic definition of the Holmes-Thompson volume we prove that totally geodesic submanifolds of a Finsler manifold are minimal for this volume. Thanks to well suited technics the minimality of totally geodesic hypersurfaces (see Alvarez Paiva and Berck in Adv Math 204(2):647–663, 2006) and 2-dimensional totally geodesic surfaces (see Alvarez Paiva and Berck in Adv Math 204(2):647–663, 2006, Ivanov in Algebra i Analiz 13(1)26–38, 2001) had already been proved. However the corresponding statement for the Hausdorff measure is known to be wrong even in the simplest case of totally geodesic 2-dimensional surfaces in a 3-dimensional Finsler manifold (see Alvarez Paiva and Berck in Adv Math 204(2):647–663, 2006).

On Hypersurfaces with Zero r-Mean Curvature

We consider hypersurfaces of simply connected space forms, with zero r-mean curvature, associated to a totally geodesic hypersurface, by Ribaucour transformations. We characterize such a hypersurface in terms of solutions of a nonlinear partial differential equation. In particular, we obtain the differential equations whose solutions produce hypersurfaces of the Euclidean space Rn+1, with zero r-mean curvature. We characterize the hypersurfaces corresponding to special solutions of these differential equations. Such solutions provide cylinders, explicit hypersurfaces with zero (n−1)-mean curvature, rotational hypersurfaces, with zero r-mean curvature, and also hypersurfaces generated by the action of the groups O(s) × O(n − s) × {1} or O(n − 1) × I2 on 2-dimensional surfaces.

Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities

We prove that any totally geodesic hypersurface $N^5$ of a 6-dimensional nearly K\"ahler manifold $M^6$ is a Sasaki-Einstein manifold, and so it has a hypo structure in the sense of \cite{ConS}. We show that any Sasaki-Einstein 5-manifold defines a nearly K\"ahler structure on the sin-cone $N^5\times\mathbb R$, and a compact nearly K\"ahler structure with conical singularities on $N^5\times [0,\pi]$ when $N^5$ is compact thus providing a link between Calabi-Yau structure on the cone $N^5\times [0,\pi]$ and the nearly K\"ahler structure on the sin-cone $N^5\times [0,\pi]$. We define the notion of {\it nearly hypo} structure that leads to a general construction of nearly K\"ahler structure on $N^5\times\mathbb R$. We determine {\it double hypo} structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5-dimensional Lie algebras with non-zero first Betti number. An extension of the concept of nearly K\"ahler structure is introduced, which we refer to as {\it nearly half flat} SU(3)-structure, that leads us to generalize the construction of nearly parallel $G_2$-structures on $M^6\times\mathbb R$ given in \cite{BM}. For $N^5=S^5\subset S^6$ and for $N^5=S^2 \times S^3\subset S^3 \times S^3$, we describe explicitly a Sasaki-Einstein hypo structure as well as the corresponding nearly K\"ahler structures on $N^5\times\mathbb R$ and $N^5\times [0,\pi]$, and the nearly parallel $G_2$-structures on $N^5\times\mathbb R^2$ and $(N^5\times [0,\pi])\times [0,\pi]$.

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let M M be an n n -dimensional compact hypersurface with constant scalar curvature n ( n − 1 ) r n(n-1)r , r &gt; 1 r&gt; 1 , in a unit sphere S n + 1 ( 1 ) S^{n+1}(1) . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral ∫ M H d M \int _MHdM of the mean curvature H H . In this paper, we first study the eigenvalue of the Jacobi operator J s J_s of M M . We derive an optimal upper bound for the first eigenvalue of J s J_s , and this bound is attained if and only if M M is a totally umbilical and non-totally geodesic hypersurface or M M is a Riemannian product S m ( c ) × S n − m ( 1 − c 2 ) S^m(c)\times S^{n-m}(\sqrt {1-c^2}) , 1 ≤ m ≤ n − 1 1\leq m\leq n-1 .

HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS

In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let φ : (Mn, g) → (Nm, h) be a horizontally homothetic harmonic morphism from a Riemannian manifold into a Riemannian manifold of non-positive sectional curvature and let T be the tensor measuring minimality or totally geodesics of fibers of φ. We prove that if T is parallel and the horizontal distribution is integrable, then for any totally geodesic submanifold P in N , the inverse set, φ−1(P ), is volume-stable in M . In case that P is a totally geodesic hypersurface, the condition on the curvature can be weakened to Ricci curvature.

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface Mt meets such tgh orthogonally, we prove that: (a) the flow exists while Mt does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of Mt remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set (if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.

What is wrong with the Hausdorff measure in Finsler spaces

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily minimal, filling results such as those of Ivanov [On two-dimensional minimal fillings, St. Petersburg Math. J. 13 (2002) 17–25] do not hold, and integral-geometric formulas do not exist. On the other hand, using the Holmes–Thompson definition of volume, we prove a general Crofton formula for Finsler spaces and give an easy proof that their totally geodesic hypersurfaces are minimal.