Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0505160

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

Miguel Angel Javaloyes·Paolo Piccione
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Miguel Angel Javaloyes, Paolo Piccione. 2005-05-09. Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds. https://arxiv.org/abs/math/0505160

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ancient mean curvature flow asymptotic to a minimal quadratic cone

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n +m \geq 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\textbf{singular minimal cone as the asymptotic model}$.

math.DG↗
arXiv2026

Log-Concavity of First Dirichlet Eigenfunctions on $\mathbb{CP}^2$

We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $Ω\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u) > \max\left\{ψ(s),\frac85\right\}g, \text{ where } ψ(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(1+s), \] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $λ_2-λ_1>46/5$.

math.DG↗
arXiv2026

Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications

We prove a Pólya--Szegö inequality for functions defined on an $n$-dimensional submanifold $Σ$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $Σ$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

math.DG↗