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Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the…

Differential Geometry · Mathematics 2016-01-20 Ryszard Deszcz , Małgorzata Głogowska , Jan Jełowicki , Georges Zafindratafa

We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators $\mathcal{A}^\varepsilon$ in divergence…

Spectral Theory · Mathematics 2023-12-15 Matteo Capoferri , Mikhail Cherdantsev , Igor Velčić

We study the covariant derivatives of an eigenfunction for the Laplace-Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant…

Differential Geometry · Mathematics 2022-08-30 Fei Qi

We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law.…

Combinatorics · Mathematics 2012-12-10 Jia-Yu Shao

In this paper, we define the $W_2$-curvature tensor on super Riemannian manifolds. And we compute the curvature tensor, the Ricci tensor and the $W_2$-curvature tensor on super twisted product spaces. Furthermore, we investigate the…

Differential Geometry · Mathematics 2024-02-13 Tong Wu , Yong Wang , Xue Wang

We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the…

Differential Geometry · Mathematics 2018-12-17 I. G. Shandra , S. E. Stepanov

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray-Singer…

Differential Geometry · Mathematics 2007-05-23 Dan Burghelea , Stefan Haller

A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally…

Metric Geometry · Mathematics 2025-04-24 Mohamed A. Mouamine , Fabian Mussnig

Through exploring the embedded transnormal systems of codimension 1, we show the existence of a transnormal function on a connected complete Riemannian manifold requires the underlying manifold to have a vector bundle structure or a linear…

Differential Geometry · Mathematics 2025-02-18 Minghao Li , Ling Yang

A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where…

Classical Analysis and ODEs · Mathematics 2022-12-27 Arian Nadjimzadah

We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics.…

Differential Geometry · Mathematics 2023-01-10 Alexander Pigazzini , Cenap Ozel , Saeid Jafari , Richard Pincak , Andrew DeBenedictis

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain are obtained. When zero Dirichlet, Robin or Neumann boundary conditions are specified on each factor, then the eigenfunctions on the product domain are…

Analysis of PDEs · Mathematics 2015-11-06 Giles Auchmuty , M. A. Rivas

In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class $[g_0]$ of the Riemannian…

Differential Geometry · Mathematics 2007-05-23 A. Raouf Chouikha

Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a…

Numerical Analysis · Mathematics 2019-10-08 Liqun Qi , Shenglong Hu , Xinzhen Zhang

A. Derdzinki [D] gave examples of Riemannian metrics with harmonic curvature and non parallel Ricci tensor on some compact manifolds $(M,g]$ . We examine their existence as well as their number wich naturally depends on the geometry of the…

Differential Geometry · Mathematics 2007-05-23 A. Raouf Chouikha

We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it…

Analysis of PDEs · Mathematics 2009-12-07 Jean Dolbeault , Maria J. Esteban , Gabriella Tarantello

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for…

Differential Geometry · Mathematics 2010-09-27 Joakim Arnlind , Jens Hoppe , Gerhard Huisken

We consider Laplacian eigenfunctions on a $d-$dimensional bounded domain $M$ (or a $d-$dimensional compact manifold $M$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(e_\ell)_{\ell \in \mathbb{N}}$.…

Analysis of PDEs · Mathematics 2018-11-27 Jianfeng Lu , Christopher D. Sogge , Stefan Steinerberger

Meta-conformal invariance is a novel class of dynamical symmetries, with dynamical exponent $z=1$, and distinct from the standard ortho-conformal invariance. The meta-conformal Ward identities can be directly read off from the Lie algebra…

Mathematical Physics · Physics 2022-11-14 Malte Henkel , Michal Dariusz Kuczynski , Stoimen Stoimenov

Osserman manifolds are a generalization of locally two-point homogeneous spaces. We introduce $k$-root manifolds in which the reduced Jacobi operator has exactly $k$ eigenvalues. We investigate one-root and two-root manifolds as another…

Differential Geometry · Mathematics 2023-01-27 Vladica Andrejić