arXiv research

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

The paper studies spectral convergence of connections on vector and principal bundles.

New equivalence found for flat vector bundles without extra conditions.

Quantum bundles use metrics to measure distances.

Study on Ricci solitons on tangent and unit tangent bundles.

We derive a bound on the $L^{\infty}$-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with sma…

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

Study on embedding sphere subbundles with prescribed mean curvature in Riemannian vector bundles.

Classifies sections of Riemannian bundles on Lie groups.

Researchers find hypersurfaces in a Riemannian vector bundle with specific curvature.

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometricians. Here we define a Riemannian or pseudo-Riemannian lift metric on TM, which is in some senses more…

Paper constructs $L^2$ estimates for flat vector bundles and generalizes Prékopa's theorem.

Study on $H$-contact structures on Riemannian manifolds.

Estimates small eigenvalues for geometrically finite manifolds.

The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately. The equivalence of the existence of Killing vector field on M and TM is proved. Ke…

The paper extends Laplacian spectra approximations to vector bundles.

The paper studies the geometry of Sasaki metric on vector bundles.

We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class.

This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.

The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.

Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.

Standard Laplace operator extends Hodge and Casimir operators to broader geometric contexts.

Study biharmonic vector fields and unit vector fields on Riemannian manifolds.

The paper introduces new metrics on Finsler manifolds and characterizes associated vector fields.

Constructs metrics for surfaces to show uniform positive scalar curvature.

New characterization of Riemannian metric positivity and $L^2$ estimates for $d$ operator.

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle $E$ of even rank over a closed compact orientable manifold $M$. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when $M$ is a Riemannian manifold and $E$ is the tangent bundle of $M$ endow…

Theory of covariant Schrödinger semigroups on Riemannian manifolds developed.

Researchers solve the Calderón problem for fractional Dirac operators.

Characterizes magnetic unit vector fields on Lie groups.

Study essential spectrum of differential operators on geometrically finite orbifolds.

The study finds conditions for embedding sphere subbundles with specific mean curvatures.

This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori.…

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on $TM$, are particular examples…

Functional-analytic method for stochastic parallel transport in bundles.

In the present paper we develop a framework in which questions of quantum ergodicity for operators acting on sections of hermitian vector bundles over Riemannian manifolds can be studied. We are particularly interested in the case of locally symmetric spaces. For locally symmetric spaces, we extend the recent construct…

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein …

The abstract discusses the equivalence of transnormal and isoparametric functions on compact manifolds.

Study of pseudo-bundles and pseudo-metrics on them.

The main purpose of the paper is to investigate Killing vector field on the tangent bundle T(M_{n}) of the Riemannian manifold with respect to the Levi-Civita connection of the metric II+III .

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact…

Establishes a connection between Kähler metrics and vector bundle sections.

Unified framework for Sobolev spaces on vector bundles, including explicit integration by parts.

Using screen distributions and lightlike transversal vector bundles we develop a theory of degenerate foliations of semi-Riemannian manifolds.

Develops global pseudo-differential calculus on homogeneous vector bundles.

Extends compactness theory to variable-coefficient pseudo-differential operators on manifolds.

The purpose of this Note is to prove that each of the following conditions is equivalent to that of the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the bundle of $r$-transverse jets is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on the slashed ${\cal J}_{0}^{r}$ is…