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 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.
We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds.
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.
Geodesic flow on submanifolds is shown to be Ck−1.
problem Regularity of geodesic flow on submanifolds.
method Analysis of Ck submanifolds with k≥2. result Geodesic flow and exponential map are Ck−1. 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.
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
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…
Study on CR-lightlike submanifolds in golden semi-Riemannian manifolds.
problem Exploring properties of CR-lightlike submanifolds in golden semi-Riemannian manifolds.
method Definition and investigation of properties of geodesic CR-submanifolds.
result Obtained results for totally geodesic and totally umbilical CR-submanifolds.
Totally geodesic Lagrangian submanifolds in nearly Kähler S³×S³.
problem Characterizing Lagrangian submanifolds in nearly Kähler manifolds.
method Analyzing H-umbilical properties and their implications for geodesicity. result In nearly Kähler S³×S³, H-umbilical Lagrangian submanifolds are totally geodesic. It is proved, that if M is a connected, complete submanifold of a complex space form N and each geodesic of M lies in an 1-dimensional totally geodesic complex submanifold of N, then M is totally geodesic in N and is a real space form or a complex space form.
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.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
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.
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.…
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 give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.
Study on cut locus of submanifolds in Finsler geometry.
problem Characterizing the cut locus of submanifolds in Finsler manifolds.
method Deformation and characterization of the cut locus, extending previous results.
result Generalization of Klingenberg's lemma for N-geodesic loops in reversible Finsler setting. 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.
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 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). Study on null submanifolds in indefinite complex contact geometry.
problem Geometry of null submanifolds in indefinite complex contact manifolds.
method Analysis of quaternion null submanifolds and distributions on screen submanifolds.
result Quaternion null submanifolds are always totally geodesic.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.
The paper explores totally geodesic submanifolds in SPD matrices and their properties.
problem Characterizing and understanding totally geodesic submanifolds in SPD matrices.
method Detailed geometric analysis and projection properties of SPD matrices.
result A non-linear projection on totally geodesic submanifolds has the minimizing property.
Proofs and descriptions of totally geodesic submanifolds in symmetric spaces.
problem Classifying totally geodesic submanifolds in symmetric spaces.
method Independent proof and descriptions using algebraic and geometric properties.
result Natural descriptions and classifications of totally geodesic submanifolds.
Study on lightlike submanifolds in statistical manifold geometry.
problem Characterizing contact CR and SCR-lightlike submanifolds.
method Developed characterization theorems on integrability and geodesicity.
result Obtained results on geometry of contact CR and SCR-lightlike submanifolds.
Extends flat submanifold properties from hyperbolic plane to symmetric spaces.
problem Spectral asymptotics for orbital integrals in symmetric spaces.
method Generalizes geodesic properties to maximal flat submanifolds.
result Establishes geometric properties of maximal flat submanifolds in symmetric spaces.
In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost co…
Planes are the only calibrated submanifolds with flat normal bundles.
problem Characterizing submanifolds with specific geometric properties.
method Using constant-coefficient differential forms and parallel calibrations.
result Calibrated submanifolds with flat normal bundles are planes.
We show that large classes of non-arithmetic hyperbolic n-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds o…
Constructs geodesics near intersection points of Lagrangian submanifolds.
problem Geodesics of positive Lagrangian submanifolds near intersection points.
method Cylindrical transform and C1,1 regularity proof. result First examples of C1,1 geodesics in arbitrary dimensions. 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.
In hyperbolic space Hn we set a geodesic ball of radius ρ. Consider a k dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic k dimensional submanifold passing through…
Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.
problem Characterize free boundary minimal submanifolds in geodesic balls of hyperbolic and spherical spaces.
method Define and analyze functionals related to critical metrics and spectral indices.
result Critical metrics of defined functionals arise from free boundary minimal immersions in geodesic balls of hyperbolic and spherical spaces.
Defines natural tensors for submanifolds of pseudo-Riemannian manifolds.
problem Characterizing tensors for submanifolds of pseudo-Riemannian manifolds.
method Constructs geodesic normal coordinates and expresses metric coefficients as polynomials in curvature and second fundamental form derivatives.
result Natural tensors are linear combinations of contractions of curvature and second fundamental form derivatives.
The article generalizes Clairaut's formula for geodesics on submanifolds.
problem Conditions for geodesics on specific submanifolds.
method Study of geodesics on submanifolds involving Euclidean distance.
result Generalization of Clairaut's formula for higher dimensions.
NR retraction approximates geodesics on submanifolds efficiently.
problem Efficiently approximating geodesics on submanifolds for practical algorithms.
method Introducing Newton retraction (NR) as a class of retractions on submanifolds induced by a foliation of the ambient manifold.
result NR is more stable and computationally cheaper than oblique projection, with superlinear convergence regions.
New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.
problem Determining conditions for positive simplicial volume in non-positively curved manifolds.
method Analyzing isolated, closed totally geodesic submanifolds of codimension one and their impact on simplicial volume.
result Positive simplicial volume for certain non-positively curved manifolds with specific submanifolds.
In the paper we investigate submanifolds in a tangent bundle endowed with g-natural metric G, defined by a vector field on a base manifold. We give a sufficient condition for a vector field on M to defined totally geodesic submanifold in (TM,G). The parallel vector field is discussed in more detail.
We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic…
Extends Polydisk Theorem to Cartan-Hartogs domains.
problem Characterizing geodesics in Cartan-Hartogs domains.
method Applying the Polydisk Theorem to Cartan-Hartogs domains.
result Cartan-Hartogs domains inherit geodesic submanifolds from bounded symmetric domains.
Random 3-manifolds have no totally geodesic submanifolds.
problem Existence of totally geodesic submanifolds in random 3-manifolds.
method Analysis of metrics on compact 3-manifolds in Cq-topology. result The set of such metrics contains an open and dense set in the Cq-topology for any q≥3. The L2-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N,g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…
Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.
problem Geodesic preservation in infinite-dimensional manifolds.
method Definition of spray-invariant sets and analysis of their properties.
result Different geometric properties of spray-invariant sets based on their regularity.
Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.
problem Uncertainty quantification in high-dimensional stochastic systems.
method Principal Geodesic Analysis on the Grassmann manifold, adaptive algorithm for local submanifolds, polynomial chaos expansion.
result Efficient surrogate modeling of system behavior across different parameter spaces.
We introduce the equation of n-dimensional totally geodesic submanifolds of a manifold E as a submanifold of the second order jet space of n-dimensional submanifolds of E. Next we study the geometry of n-Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n-dim…