Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
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.
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.
Totally geodesic submanifolds in spheres have restricted curvature properties.
problem Characterizing submanifolds in spheres based on curvature conditions.
method Analyzing normal curvature, scalar curvature, and second fundamental form conditions.
result Compact pseudo-umbilical submanifolds in spheres are totally geodesic under specific curvature conditions.
We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
Study extends geodesic curvature formula to higher dimensions.
problem Extending curvature formula to higher-dimensional spheres.
method Using new integral-geometric formulas for Euclidean and geodesic total curvature.
result Explicit formula for geodesic total curvature on higher-dimensional spheres.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the techni…
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.
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.
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.
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Rieman…
We present a new equation with respect to a unit vector field on Riemannian manifold Mn such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and pro…
Paper calculates second variation of Graham-Witten energy for spheres.
problem Calculating the second variation of Graham-Witten energy.
method Explicit formula for second variation at minimal submanifolds in Einstein manifolds.
result Totally geodesic spheres in unit sphere are critical points with non-negative second variation.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic …
The study finds infinitely many twist knot complements with totally geodesic surfaces.
problem Finding infinitely many twist knot complements with a specific number of totally geodesic surfaces.
method Using a family of twist knot complements and their dihedral covers, the authors construct examples of hyperbolic 3-manifolds with totally geodesic surfaces.
result The construction of infinitely many non-commensurable hyperbolic 3-manifolds with exactly k totally geodesic surfaces for any positive integer k.
The paper studies Lagrangian submanifolds in a specific sphere and derives a Simons' type inequality.
problem Characterizing Lagrangian submanifolds in a homogeneous nearly Kähler 6-dimensional sphere. method Derives a Simons' type integral inequality for compact Lagrangian submanifolds and shows equality conditions.
result Equality in the inequality occurs only for totally geodesic S3(1) or Dillen-Verstraelen-Vrancken's Berger sphere. The study proves that geodesic spherical curves characterize manifolds of constant curvature.
problem Characterizing manifolds of constant curvature using spherical curves.
method Proving the converse of the known linear equation for RM frames, and providing two additional characterizations.
result Geodesic spherical curves on a manifold characterize constant sectional curvature.
We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…
We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field …
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvatu…
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
Let M be the image of a smooth CR embedding of a strictly pseudoconvex CR real hypersurface into a sphere. If the CR second fundamental form of M vanishes, we show that M is a totally geodesic submanifold.
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.
Develops PCA for symmetric spaces using geodesic submanifolds.
problem PCA for data on Riemannian symmetric spaces.
method Totally geodesic submanifolds for approximation.
result Illustrated on n-spheres, Grassmannians, n-tori and polyspheres.
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. Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese…
Study on flag manifolds using Hopf fibration to find geometrically special 2-spheres.
problem Computing and characterizing 2-spheres in flag manifolds.
method Hopf fibration, invariant geometry, Weyl group action.
result Generators of second homotopy group have invariant geometry and are classified.
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.
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.
Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.
The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M, proved in Math. Ann. 165 (1966), 309--317.
Slim curves on 3-sphere help spherical CR uniformizations.
problem Understanding curves on 3-sphere for CR uniformizations.
method Defining slimness, analyzing foliations, and applying to quasi-Fuchsian groups.
result Slim curves lead to spherical CR uniformizations of certain 3-manifolds.
The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
problem Proving nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
method Using quasi-Fuchsian surfaces and totally geodesic surfaces, the study proves filling properties with rigorous mathematical proofs.
result Proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.
problem Proving rigidity for minimal submanifolds in spheres with flat normal bundle.
method Explicit second-gap rigidity theorem for the squared norm of the second fundamental form.
result The theorem provides evidence for Chern's conjecture in higher codimension.
New isoparametric hypersurfaces found in Damek-Ricci spaces.
problem Characterizing new isoparametric hypersurfaces in Damek-Ricci spaces.
method Defining and studying 'sphere-like' hypersurfaces formed by extending horospheres.
result Found a new family of isoparametric hypersurfaces connecting geodesic spheres to previously known ones.
The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(…
The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N o…
In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere…
The paper studies Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics.
problem Investigating Finsler spheres with specific curvature properties and geodesic orbits.
method Analyzing the action of isometries and loops on Finsler spheres, focusing on finite orbits of prime closed geodesics.
result The existence of geometrically distinct orbits of prime closed geodesics and their properties.
Study on minimal hypersurfaces in a unit sphere, proving specific isometries.
problem Characterizing minimal hypersurfaces with constant scalar curvature.
method Analyzing n-dimensional complete minimal hypersurfaces in a unit sphere with constant scalar curvature. result Proves isometry to totally geodesic sphere or Clifford torus under certain conditions.
The paper studies how submanifolds of a sphere evolve over time.
problem Evolution of pinched submanifolds in the sphere.
method High codimension mean curvature flow with pinching conditions.
result Convergence to a round point or totally geodesic sphere under pinching conditions.
New characterization of Calabi torus in unit sphere found.
problem Rigidity of closed minimally immersed Legendrian submanifolds in unit sphere.
method Maximum principle and Simons' type integral inequality.
result New characterization of Calabi torus in unit sphere.
This paper deals with the study of some properties of immersed curves in the conformal sphere $\mathds{Q}_n$, viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. K…