arXiv research

The study constructs a Legendrian cycle for $F_nW^{2,n}$-sets and proves Reilly-type variational formulae.

Given a submanifold $S \subset \mathbb R^n$ of codimension at least three, we construct an asymptotically Euclidean Riemannian metric on $\mathbb R^n$ with nonnegative scalar curvature for which the outermost apparent horizon is diffeomorphic to the unit normal bundle of $S$.

Harmonic unit normal sections studied for Grassmannians induced by cross products.

The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.

We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a …

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…

We develop model-based methods for solving stochastic convex optimization problems, introducing the approximate-proximal point, or aProx, family, which includes stochastic subgradient, proximal point, and bundle methods. When the modeling approaches we propose are appropriately accurate, the methods enjoy stronger conv…

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve…

CFR-Pro enhances treatment effect estimation by incorporating local proximity.

The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.

We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ 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…

Let $(\mathbb{M}_{1}^{2},g)$ be a Minkowski surface and $(T_1\mathbb{M}_1^2, g_1)$ its unit tangent bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves which the inner product of normal vector and Reeb vector is zero and nonzero constan…

The paper reformulates Legendrian contact homology using string topology.

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

A new method reformulates Optimal Transport Conditional Flow Matching using proximal operators.

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 …

The study pinches rigidity theorems for minimal submanifolds in spheres.

A new method solves convex optimization on curved spaces.

We present the non-trivial example how to generate non-Euclidean geometries from associative unital algebras. We consider bundles of the sphere of the degenerate non-Eucleadian space and its two models. The first (conformal) model is obtained by the mapping S onto a plane pass through the origin. It is analogous to the…

Characterizes special curves on surface tangent bundles.

We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…

We use the octonionic multiplication $\cdot$ of $\mathbb{S}^{7}$ to associate, to each unit normal section $η$ of a submanifold $M$ of $\mathbb{S}^{7},$ an octonionic Gauss map $γ_η:M\rightarrow\mathbb{S}^{6},$ $γ_η(x)=x^{-1}\cdotη(x),$ $x\in M,$ where $\mathbb{S}^{6}$ is the unit sphere of $T_{1}\mathbb{S}^{7},$ $1$ i…

Study shows how certain foliations in unit tangent bundles behave.

Minimal normal curvature immersions in the unit ball studied.

New examples of austere submanifolds and hypersurfaces with specific curvature properties.

We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…

We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a compact surface.

The paper develops quantitative estimates for holomorphic sections over bounded domains.

We prove uniqueness, up to diffeomorphism, of symplectically aspherical fillings of certain unit cotangent bundles, including those of higher-dimensional tori.

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

Study on Klein bottle's cotangent bundle using contact homology.

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $M$, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle $TM$, called the horizontal distribution. Similarly to the finite-dimensional case, we ar…

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $η$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $η$-E…

Characterizes magnetic unit vector fields on Lie groups.

The paper proves Morse estimates for translated points on unit tangent bundles.

The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizonta…

We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…

Minimal vector fields on oscillator groups studied, with specific conditions for minimality.

Researchers create metrics on hyperbolic space's tangent bundle.

The paper studies vector fields on manifolds and their embeddings into tangent bundles.

The study identifies surfaces with Maslovian normal bundles.

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal…

We study Selberg zeta functions $Z(s,σ)$ associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete odd-dimensional hyperbolic manifold of finite volume. We assume a certain condition on the fundamental group of the manifold. A priori, the Selberg zeta functions are defined only for s in…

It is important in many applications to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of the paper is to provide an elementary proof of the existence and uniqueness of such an extension.

Paper generalizes Hamiltonian mechanics using line bundles.

Geodesics on modular surface yield arithmetic 3-manifolds.

It is showed that many examples of AMD submanifolds of higher dimensions come from SL normal bundles. A symmetry property of SL submanifolds and Björling type problem for SL normal bundles are also mentioned.

Unique symplectic fillings of odd spheres' cotangent bundles proven.