Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
Rectifying submanifolds are characterized by their tangential position vector field component.
problem Characterizing rectifying submanifolds in Euclidean spaces.
method Introducing rectifying submanifolds and proving their properties.
result Rectifying submanifolds are identified by a specific tangential vector field property.
Study on Yamabe and quasi-Yamabe solitons on Euclidean submanifolds.
problem Characterizing solitons on Euclidean submanifolds.
method Investigation of solitons with tangential components of position vector fields.
result Classification of Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.
Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
Defines observer-invariant time derivatives on moving surfaces.
problem Deriving appropriate definitions for time derivatives on surfaces that move.
method Systematically derived from spacetime settings, considering observer-invariance and covariance principles.
result Formulations applicable for computations of tangential n-tensor fields on moving surfaces.
The study of Euclidean submanifolds with incompressible canonical vector fields.
problem Characterizing Euclidean submanifolds with incompressible canonical vector fields.
method Analyzing the canonical vector field properties and conditions for incompressibility.
result Necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible.
The study characterizes Euclidean submanifolds with a conformal canonical vector field.
problem Characterizing Euclidean submanifolds with a specific vector field property.
method Investigating properties of the canonical vector field and its conformal nature.
result Characterization of Euclidean submanifolds with conformal canonical vector fields.
Study properties of specific solitons on submanifolds with special vector fields.
problem Characterize almost η-Ricci and Yamabe solitons on submanifolds. method Analyze submanifolds isometrically immersed into Riemannian manifolds with specific potential vector fields.
result Necessary and sufficient conditions for hypersurfaces in the unit sphere to be solitons.
Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.
problem Existence of compact Clifford-Klein forms for tangential symmetric spaces.
method Analyzes tangential homogeneous spaces and provides necessary conditions for their existence.
result New tangential symmetric spaces are found that do not admit compact Clifford-Klein forms.
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field φ. For the normal case, we prove that a φ-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a φ-invariant submanifold N everyw…
New method for mesh denoising using TGV of normal vector field.
problem Improving mesh quality by removing noise.
method Proposes a novel TGV formulation for normal vector fields on triangular meshes.
result New method outperforms existing techniques in mesh denoising experiments.
The paper investigates Ricci almost solitons linked to conformal vector fields.
problem Investigating Ricci almost solitons on manifolds.
method Analyzing semi-Riemannian manifolds and conformal vector fields.
result Connected totally umbilic manifolds inherit Ricci almost soliton structures via conformal vector fields.
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…
A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean 3-space has tangentially biharmonic normal bundle if and only if it is either minimal…
This paper solves a Calderón problem for Beltrami fields on manifolds.
problem Reconstructing a 3D manifold from boundary measurements of Beltrami fields.
method Defined a normal-to-tangential map for Beltrami fields and used it to reconstruct the manifold.
result A real-analytic 3-manifold can be reconstructed from its normal-to-tangential map.
New method preserves topology in Hodge decomposition for scalar and vector fields.
problem Topology-preserving Hodge decomposition on manifolds with boundaries.
method Comprehensive 5-component decomposition in Eulerian representation.
result Effective numerical experiments validate the method's accuracy and orthogonality.
The study classifies contact metric manifolds based on Ricci-Yamabe solitons.
problem Classifying contact metric manifolds based on Ricci-Yamabe solitons.
method Analyzing specific types of solitons in contact metric manifolds.
result Contact metric manifolds are classified based on the properties of Ricci-Yamabe solitons.
The notion of Lagrangian H-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…
Study contact structures on projective spaces, proving infinite non-isotopic structures.
problem Classify contact structures on projective spaces with specific surgery numbers.
method Detailed analysis of Gompf's Γ-invariant and d_3-invariant of tangential 2-plane fields.
result Infinitely many non-isotopic contact structures on real projective 3-space.
For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…
Develops Hodge theory for foliations using perturbed Laplacians.
problem Creating a Hodge theory for foliations.
method Mimicking Witten's approach to Morse theory with perturbations of the Laplacian.
result Establishes a Hodge theory for tangential cohomology of foliations.
Partial boundary regularity for area-minimizing currents at tangential boundary points.
problem Boundary regularity of co-dimension one area-minimizing currents.
method Proof closely follows Hardt and Simon's boundary regularity result.
result Partial regularity of tangent cones, uniqueness of tangent cone.
The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two diffe…
We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The lin…
The paper classifies hypersurfaces in product spaces with specific curvature properties and finds that only rotational ones admit almost Ricci solitons.
problem Characterizing hypersurfaces in product spaces with specific curvature properties.
method Local classification and necessary/sufficient conditions for almost Ricci soliton structures.
result Only rotational hypersurfaces in the studied product spaces admit almost Ricci solitons.
Gradient flows for surface energies with tensor fields are derived and analyzed.
problem Deriving consistent gradient flows for surface energies involving tensor fields.
method Introducing different gauges of surface independence and demonstrating their effects on energy decrease.
result Consistent choice of gauge and time derivative is necessary for energy decrease.
Characterizes convergence of orbits of holomorphic semigroups in the unit disc.
problem Understanding the convergence behavior of orbits of holomorphic semigroups.
method Analyzes the shape of the starlike domain image of the Koenigs function and uses properties of quasi-geodesics.
result Characterizes convergence types (non-tangential and tangential) in terms of the domain's shape and quasi-geodesic properties.
Reconstructing 3D manifolds from boundary electromagnetic data.
problem Reconstructing a compact, connected, real-analytic Riemannian 3-manifold from tangential electric and magnetic fields on its boundary.
method Factorizing Maxwell's equations and using an isometric transform to reconstruct the metric.
result The electromagnetic Dirichlet-to-Neumann map uniquely determines all derivatives of electromagnetic parameters on the boundary.
Study on affine surfaces with specific algebraic properties.
problem Characterize homogeneous affine surfaces with Hessian rank 2.
method Investigate algebra of differential invariants under affine transformation group.
result Organize homogeneous models into inequivalent branches.
The study characterizes rectifying curves in n-dimensional space.
problem Understanding rectifying curves in arbitrary dimensions.
method Characterization through various conditions and constructions.
result Different ways to characterize rectifying curves in n-dimensional Euclidean space.
We study the local differential geometry of varieties Xn⊂CPn+a with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…
Study space-like surfaces in Robertson-Walker spacetimes with specific geometric conditions.
problem Characterize space-like surfaces in Robertson-Walker spacetimes with given geometric conditions.
method Investigate surfaces satisfying specific conditions on tangential and normal parts of the unit vector field, using shape operators and minimal surfaces.
result Classification theorem and parametrizations of space-like class A surfaces in L14(f,0). The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.
problem Analyzing boundary conditions and thin-shell limits for viscous operators on arbitrary smooth hypersurfaces.
method Decomposing the ambient Bochner Laplacian into intrinsic and radial pieces, proving results for stress-free and Hodge boundary conditions.
result Universal thin-shell limits for viscous operators on arbitrary smooth hypersurfaces, including stress-free and Hodge boundary conditions.
Algorithm finds real-analytic Legendrian representatives for every link type.
problem Finding explicit expressions for Legendrian representatives and Bateman fields.
method Algorithm based on trigonometric polynomials and solving linear equations.
result No compact subset of R^3 can contain an electromagnetic knot indefinitely.
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of C1,α submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…
New insights into biharmonic and biconservative hypersurfaces in Euclidean spaces.
problem Characterizing biharmonic and biconservative hypersurfaces in Euclidean spaces.
method Analyzing biharmonic and biconservative equations, studying holonomic properties.
result Holonomic biharmonic hypersurfaces are minimal.
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …
Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…
We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its comput…
We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tange…
The continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regulari…
Study on properties of tangential hypersurfaces in product-like manifolds.
problem Investigating properties of tangential hypersurfaces in product-like manifolds.
method Analyzing basic properties and computing curvature tensor relations.
result Computed relations involving the Riemannian curvature tensor of tangential hypersurfaces.
Study of GCR hypersurfaces in Minkowski spaces, including classification and examples.
problem Classifying and understanding GCR hypersurfaces in Minkowski spaces.
method Geometrical analysis and classification of GCR hypersurfaces in Minkowski spaces.
result Complete classification of GCR surfaces in the Minkowski 3-space.
New results affirmatively answer the stable converse soul question for many curved spaces.
problem Determining if vector bundles over curved spaces admit metrics with non-negative curvature.
method Topological K-theory and homotopy equivalence.
result The stable converse soul question has an affirmative answer for many curved spaces, except possibly for one specific space.
An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space PN(C). By means of the foca…
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
Study examines tangential real hypersurfaces on Hermite-like manifolds.
problem Characterizing real hypersurfaces on Hermite-like manifolds.
method Introduced tangential real hypersurfaces and derived main identities.
result Discussed contact metric structures in K-contact and cosymplectic cases.
Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base. This is done via imposing certain conditions on the Lie derivative…