Classifies totally geodesic submanifolds in symmetric spaces.
problem Classifying submanifolds in symmetric spaces.
method Classification of totally geodesic submanifolds in products of rank one symmetric spaces.
result Infinitely many examples of irreducible totally geodesic submanifolds in Hermitian symmetric spaces.
Classifies totally geodesic submanifolds in exceptional symmetric spaces.
problem Classifying totally geodesic submanifolds in exceptional symmetric spaces.
method Classification and introduction of an invariant (Dynkin index) for totally geodesic embeddings.
result Existence of a totally geodesic submanifold of minimal codimension with specific properties.
We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.
Study on Gauss images of specific minimal surfaces with finite curvature.
problem Characterizing Gauss images of minimal surfaces with finite total curvature.
method Analyzing the number and weight of omitted and totally ramified values of Gauss maps.
result Construction of new minimal surfaces with specific Gauss map properties.
A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spa…
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.
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
Study on totally real flat minimal surfaces in quaternionic projective space.
problem Characterizing the moduli space of totally real flat minimal immersions in HP^3.
method Analyzing the moduli space of linearly full totally real flat minimal immersions from C into HP^3.
result The moduli space has three components, each a 6-dimensional manifold.
The paper classifies submanifolds in pseudo-Riemannian space forms.
problem Characterizing and classifying totally umbilical submanifolds.
method Classification of congruent classes of totally umbilical submanifolds in non-flat pseudo-Riemannian space forms.
result Some moduli spaces of isometric immersions between space forms are non-Hausdorff.
Totally geodesic subvarieties in moduli spaces are studied.
problem Characterizing totally geodesic subvarieties in moduli spaces.
method Analyzing the Deligne-Mumford boundary and its strata.
result Boundary loci of totally geodesic subvarieties are themselves totally geodesic.
One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M− introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symm…
Totally geodesic submanifolds in hyperbolic space up to codimension two.
problem Characterizing minimal homogeneous submanifolds in hyperbolic spaces.
method Analyzing properties of minimal submanifolds in hyperbolic spaces up to codimension two.
result Minimal homogeneous submanifolds of hyperbolic space up to codimension two are totally geodesic.
We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of (κ,μ)-spaces with Boeckx invariant I≤−1. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.
Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.
problem Understanding the Steklov spectrum of covering and total spaces.
method Analyzing Dirichlet-to-Neumann maps on Riemannian manifolds with boundary and bounded geometry.
result Existence and properties of the bottom of the Dirichlet spectrum on covering and total spaces.
Study Kähler geometry on vector bundles over elliptic curves.
problem Characterize Kähler metrics on vector bundle total spaces.
method Analyzing function theory and Kähler geometry on vector bundles of degree zero.
result Biholomorphic total spaces correspond to isomorphic vector bundles.
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…
Paper proves curvature estimate for curved spaces.
problem Estimating total curvature of closed hypersurfaces.
method Used non-positively curved symmetric spaces.
result Proved total curvature estimate and isoperimetric inequality.
In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known construction, it is not always possible to totally geodesically embed a pos…
Characterizes curves in totally umbilical surfaces of space forms.
problem Identifying curves in totally umbilical surfaces of space forms.
method Analyzes curvature and torsion conditions for curves in $\h^3$ and $\s^3$.
result Necessary and sufficient conditions for curves in totally umbilical surfaces.
In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetr…
The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.
problem Semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.
method Intertwining results from dynamics, algebraic geometry, geometric group theory, and Teichmüller theory.
result Each component of the boundary is a product of simple factors, each behaving like a diagonal embedding.
A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a …
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic sp…
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.
Study classifies totally geodesic surfaces in nearly Kähler space.
problem Classifying totally geodesic surfaces in nearly Kähler space.
method Detailed description of nearly Kähler space and classification of surfaces.
result Classification of totally geodesic almost complex surfaces.
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
New condition ensures submanifolds are skew in small areas.
problem Ensuring submanifolds are skew in Euclidean space.
method Introduces a third-order differential condition.
result Constructs improved totally skew embeddings for Rn. 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.
We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller that each o…
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…
We show that there is a correspondence between totally umbilic null hypersurfaces in generalized Robertson-Walker spaces and twisted decompositions of the fibre. This allows us to prove that nullcones are the unique totally umbilic null hypersurfaces in the closed Friedmann Cosmological model. We also apply this kind o…
Characterizes totally umbilical hypersurfaces in product spaces.
problem Classifying totally umbilical hypersurfaces in product spaces.
method Characterization through local graphs built on isoparametric families.
result Extends classification results to SnimesR and HnimesR. Totally geodesic submanifolds in product spaces imply special curvature properties.
problem Characterizing manifolds based on the existence of totally geodesic submanifolds.
method Analyzing totally geodesic submanifolds in products of non-positively curved manifolds with transversality conditions.
result If infinitely or just a single dense such submanifolds exist, the manifold must be locally symmetric.
Study of f-biharmonic hypersurfaces in conformally flat spaces.
problem Characterize f-biharmonic hypersurfaces in conformally flat spaces. method Analyze f-biharmonicity of totally umbilical hypersurfaces in various contexts. result Properties of f-biharmonic hypersurfaces in nonpositively curved manifolds. The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.
problem Characterizing D'Atri spaces using geometric properties.
method Characterization of D'Atri spaces via total scalar curvature of geodesic hemispheres.
result A 3D Riemannian manifold is a D'Atri space if and only if the total scalar curvature of tubes about geodesic segments holds.
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 study finds minimal surfaces in complex space forms are often totally geodesic.
problem Characterizing minimal surfaces with specific geometric properties in complex space forms.
method Analyzing free-boundary minimal surfaces in geodesic balls of complex space forms.
result Minimal surfaces in certain complex space forms are either totally geodesic or superminimal.
New proofs given for space curves with totally positive torsion.
problem Description of convex hulls of space curves with totally positive torsion.
method New proofs of parametric representation, surface area, and volume formulas.
result Recovery of formulas for convex hull's surface area and volume.
The study classifies flows of finite curvature in 3D space.
problem Classifying flows of finite curvature in 3D space.
method Partial classification of eternal mean convex flows.
result Topologically nonplanar flows must exit a catenoid.
Paper extends rigidity and vanishing results for totally real submanifolds under Lp-integrable conditions.
problem Rigidity and vanishing properties of totally real submanifolds in complex space forms.
method Extends results by Cuong et al. to a broader range of p-integrable conditions. result Extends the range of p for rigidity and vanishing results. 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. The paper classifies totally geodesic Lagrangian submanifolds in a specific pseudo-nearly Kähler space.
problem Understanding Lagrangian submanifolds in pseudo-nearly Kähler spaces.
method Examining four classes of submanifolds based on their behavior with respect to an almost product structure, then classifying totally geodesic ones.
result A complete classification of totally geodesic Lagrangian submanifolds in the pseudo-nearly Kähler SL(2,R)imesSL(2,R). In this paper, we study totally real minimal surfaces in the quaternionic projective space HPn. We prove that the linearly full totally real flat minimal surfaces of isotropy order n in HPn are two surfaces in CPn, one of which is the Clifford solution, up to symplectic congruence.
In this paper we investigate what kind of manifolds arise as the total spaces of iterated S1-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated S1-bundle. We show that the total space of an iterated S1-bundle is homeomorphic to an infra-nilmanifold. A real Bo…
In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2, published by Chen and Nagano (B.…