Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.
problem Characterizing minimal hypersurfaces in S^5 with certain curvature conditions.
method Analyzing hypersurfaces with constant scalar curvature and zero Gauss curvature.
result Minimal hypersurfaces in S^5 with these curvature properties are totally geodesic.
Classifies special hypersurfaces in Gödel spacetimes.
problem Characterizing hypersurfaces in Gödel spacetimes.
method Classification of parallel and totally geodesic hypersurfaces.
result Identified specific types of hypersurfaces in Gödel spacetimes.
Classifies special hypersurfaces in specific types of manifolds.
problem Characterizing geometric structures in non-reductive homogeneous spaces.
method Classification of parallel and totally geodesic hypersurfaces.
result Found a complete classification of these hypersurfaces.
Totally geodesic null hypersurfaces found in Lorentzian manifolds.
problem Characterizing null hypersurfaces in Lorentzian manifolds.
method Analyzing light-like geodesically complete Lorentzian manifolds with null energy condition.
result Null hypersurfaces are totally geodesic.
Classifies and describes hypersurfaces in Siklos spacetimes.
problem Characterizing hypersurfaces in Siklos spacetimes.
method Classification and description of totally geodesic and parallel hypersurfaces.
result A large class of minimal hypersurfaces is described.
Totally geodesic minimal hypersurfaces in H5 with specific curvature properties.
problem Characterizing minimal hypersurfaces in hyperbolic space with certain curvature conditions.
method Analyzing properties of minimal hypersurfaces in H5 with constant scalar curvature and zero Gauss-Kronecker curvature. result Any complete minimal hypersurface in H5 with constant scalar curvature and zero Gauss-Kronecker curvature is totally geodesic. Totally geodesic hypersurfaces in a sphere have small total curvature.
problem Characterizing hypersurfaces with constant scalar curvature in a sphere.
method Analyzing the total curvature of locally conformally flat hypersurfaces.
result Hypersurfaces with small total curvature are totally geodesic.
Study shows mean curvature flows converge to a geodesic graph over a totally geodesic hypersurface.
problem Mean curvature flows in warped product manifolds with closed hypersurfaces.
method Investigation of mean curvature flows in specific warped product manifolds with conditions on warping function and Ricci curvature.
result Existence and convergence of mean curvature flows for certain initial hypersurfaces.
The paper classifies hypersurfaces in space forms with a totally geodesic foliation.
problem Characterizing hypersurfaces in space forms with a specific foliation.
method Complete local parametric classification for hypersurfaces of dimension at least three.
result There exists exactly one further class of local examples in Euclidean space with rank two.
Classifies totally geodesic submanifolds in specific spaces and finds Einstein hypersurfaces.
problem Classifying totally geodesic submanifolds in Damek-Ricci spaces and Cayley projective plane.
method Analyzing properties of harmonic manifolds, using curvature conditions, and applying classification techniques.
result Characterizes totally geodesic submanifolds and Einstein hypersurfaces in specific spaces.
Characterizes curves on geodesic spheres and totally geodesic hypersurfaces in hyperbolic and spherical spaces.
problem Characterizing curves in curved spaces.
method Rotation minimizing frames and exponential maps.
result Characterizes geodesic spherical curves in hyperbolic and spherical spaces through linear equations.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
Totally biharmonic hypersurfaces in space forms and 3D BCV spaces classified.
problem Characterizing totally biharmonic hypersurfaces in space forms and 3D BCV spaces.
method Analyzing geodesics and isoparametric properties to classify hypersurfaces.
result Classification of totally biharmonic hypersurfaces in space forms and 3D BCV spaces.
Classifies hypersurfaces in Minkowski space with a specific foliation.
problem Characterizing hypersurfaces with a totally geodesic foliation in Minkowski space.
method Classification based on properties of the foliation and hypersurface structure.
result Hypersurfaces are ruled, partial tubes over curves, or contain strips.
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. Totally geodesic hypersurfaces in hyperbolic manifolds are rigid under certain conditions.
problem Conditions under which totally geodesic hypersurfaces in hyperbolic manifolds are rigid.
method Study of homotopy equivalence and sectional curvature properties.
result Conditions for rigidity of totally geodesic hypersurfaces in hyperbolic manifolds.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.
problem Analyzing the Morse index and eigenvalues of free-boundary CMC hypersurfaces in the upper hemisphere.
method Proved results using the norm squared of the second fundamental form and eigenvalue estimates.
result Proved bounds on Morse index and eigenvalues for free-boundary CMC hypersurfaces.
We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also con…
Paper characterizes a special hypersurface in 5D sphere.
problem Characterizing minimal hypersurfaces in S5. method Analyzes hypersurfaces satisfying a specific curvature condition.
result Closed minimal hypersurfaces in S5 satisfying a certain curvature condition are either totally geodesic or congruent to the Cartan minimal hypersurface. The paper examines stable capillary hypersurfaces in hyperbolic space.
problem Stability of capillary hypersurfaces with free boundary on a horosphere.
method Analysis of umbilical and totally geodesic hypersurfaces using stability criteria.
result Umbilical and totally geodesic hypersurfaces are the only stable capillary hypersurfaces with boundary on a horosphere.
Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.
problem Analyzing free boundary minimal hypersurfaces in the Riemannian Schwarzschild space.
method Variational methods and geometric analysis.
result Zero Morse index for certain free boundary rotationally symmetric totally geodesic hypersurfaces in the Riemannian Schwarzschild space.
The study examines spacelike foliations on Lorentz manifolds under specific conditions.
problem Investigating geometric properties of spacelike foliations on Lorentz manifolds.
method Analyzing conditions for stability, total geodesy, and total umbilicity of foliation leaves.
result Conditions for the stability, total geodesy, and total umbilicity of spacelike foliation leaves are established.
Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.
problem Properties of Lorentzian manifolds influenced by totally geodesic null hypersurfaces.
method Coupling rigging technique with null foliation existence to prove Riemann flow structure.
result Proves curvature conditions restrict causal structure of spacetime.
Study geometric properties of spacelike foliations on Lorentz manifolds.
problem Investigate conditions for spacelike foliations to be totally umbilical or geodesic.
method Develop an equation relating foliation to ambient manifold, apply Maximum principle.
result Obtain an obstruction for totally geodesic foliations in spacetimes with positive Ricci curvature.
Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.
problem Classifying hypersurfaces in the Sol4_0 geometry.
method Analyzing hypersurfaces with Codazzi tensors and parallel second fundamental forms.
result Full classification of hypersurfaces in Sol4_0, including parallel and totally umbilical types.
Study disproves a conjecture about constant curvature hypersurfaces.
problem Whether every hypersurface with constant principal curvatures is isoparametric.
method Constructed an explicit example of a conformally flat Riemannian manifold.
result Disproved the conjecture by providing a counterexample.
New tensor introduced for complex quadric hypersurfaces, no Hopf hypersurfaces found.
problem Characterizing real hypersurfaces in complex quadric using star-Ricci tensors.
method Introducing and analyzing star-Ricci tensors in real hypersurfaces of complex quadric.
result No Hopf hypersurfaces exist in Qm,m≥3, with certain star-Ricci tensors. In the year 1984 Shibata investigated the theory of a change which is called a β-change of a Finsler metric. On the other hand in 1985 a systematic study of geometry of hypersurfaces in Finsler spaces was given by Matsumoto. In the present paper is to devoted to the study of a condition for a Randers conformal chang…
New proof shows minimal submanifolds of sphere are totally geodesic.
problem Characterize minimal submanifolds of spheres.
method Develops a new proof strategy.
result Obtains analogous result for codimension 2 minimal submanifolds.
Study finds new minimal surfaces in Schwarzschild space.
problem Existence of non-totally geodesic minimal surfaces in Schwarzschild space.
method Family of properly embedded free boundary minimal hypersurfaces of revolution.
result Existence of new minimal surfaces with circular boundaries in Schwarzschild space.
We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product o…
The paper classifies A-principal Hopf hypersurfaces in complex quadrics.
problem Characterizing A-principal Hopf hypersurfaces in complex quadrics. method Analyzing the singularities of the unit normal vector field and using geometric properties.
result A A-principal Hopf hypersurface in Qm is an open part of a tube around a totally geodesic Qm+1 in Qm. Degenerate submanifolds of pseudo-Riemannian manifolds are quite difficult to study because there is no prefered connection when the submanifold is not totally geodesic. For the particular case of degenerate totally umbilical hypersurfaces, we show that there are "Weyl" connections adapted to the induced structure on t…
We prove that a Riemannian product of type M x R (where R denotes the Euclidean line) admits totally umbilical hypersurfaces if and only if M has locally the structure of a warped product and we give a complete description of the totally umbilical hypersurfaces in this case. Moreover, we give a necessary and sufficient…
We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non semisimple Lie group with an hypersurface orbit.
Study geometric properties and spectral estimates on warped products.
problem Investigate Ricci curvature and spectral estimates in warped products.
method Establish integral inequalities and sufficient conditions for geometric properties.
result Sufficient conditions for intersection of warped products with totally geodesic hypersurfaces.
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
Classification of hypersurfaces in homogeneous spaces with specific properties.
problem Classifying hypersurfaces in Riemannian homogeneous spaces with additional assumptions.
method Analyzing hypersurfaces under various conditions in homogeneous spaces CP3. result All extrinsically homogeneous hypersurfaces are classified in all homogeneous CP3 spaces. 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 th…
The paper studies geodesic hypersurfaces in hyperbolic manifolds and their fundamental groups.
problem Understanding the fundamental groups of hyperbolic manifolds through geodesic hypersurfaces.
method Analyzing sequences of asymptotically geodesic hypersurfaces and their properties.
result If a closed hyperbolic manifold contains a sequence of asymptotically geodesic hypersurfaces, its fundamental group is virtually special and linear over integers.
The study defines conditions for mean curvature of umbilical leaves in hyperbolic space.
problem Understanding mean curvature of umbilical leaves in hyperbolic space.
method Formulated necessary and sufficient conditions for a function along geodesics and hypercycles in hyperbolic space.
result Conditions for mean curvature of umbilical leaves are established for geodesics and hypercycles.
Study shows only hyperplanes in Heisenberg groups have zero curvature.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.
A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater tha…
Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.
problem Behavior of minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.
method Analysis of free boundary minimal hypersurfaces and totally geodesic hyperplanes in Schwarzschild n-manifolds. result A free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region.