Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
140 questions from the last 365 days
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Boundary of convex sets in Riemannian manifolds
Let $M$ be Riemannian d-manifold and $C\subset M$ a closed convex subset with smooth interior and non-smooth boundary. I am aware that the $(d-1)$-dimensional Hausdorff measure of the set $nd(\partial ...
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Verification request: three geometric steps in a parameter-free derivation of α⁻¹ [closed]
I am seeking verification of three specific mathematical claims arising from a geometric framework. I am not asking for evaluation of the broader physical interpretation — only whether these three ...
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Topology of compact manifolds admitting codimension-one foliations with dense leaves
Let $M$ be a compact manifold endowed with a codimension-one smooth foliation $\mathcal{F}$, defined as the kernel of a closed, nowhere-vanishing 1-form $\omega \in \Omega^1(M)$.
It is classical that ...
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Riemannian manifolds with maximal number of symmetries
This question is induced by what seems to be a rather large disconnect between "old" Riemannian geometry and modern treatments of it. For example Killing vector fields are extremely ...
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Formula for Riemann curvature tensor -- does it have a name?
Say you have $N$, an n-dimensional submanifold of a Euclidean space $\mathbb R^k$. We consider it to be a Riemann manifold with the pull-back metric. Locally near a point $p \in N$ you express $N$ ...
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Topology of metric balls in non-compact surfaces of non-negative Gauss curvature
Let $M$ be a complete non-compact Riemannian surface of non-negative Gauss curvature, and let $B(x,r)$ denote an open metric ball in $M$. What can be said about the topology of $B(x,r)$?
More ...
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Warner's elegant papers: conjugate locus, sprays, and fundamental groups
Motivation
I am working on the problem of uniqueness of Frechet mean for probability measures on a Riemannian manifold of non-negative curvature, for which I need to understand properties and ...
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Non-attainment of infimum energy among smooth competitors in Dirichlet problem
Let $M$ be a compact Riemannian manifold with boundary $\partial M \ne \emptyset$ and dimension at least $2$, and let $N$ be a compact Riemannian manifold with no boundary of dimension at least $2$ ...
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Connecting efficiently metrics on a surface with lower curvature bound
Let $S$ be a closed surface and $\mathcal M$ be the space of metrics on $S$ equipped with the Lipschitz distance $d$. (Say, up to isometry isotopic to the identity and $d$ is also defined with respect ...
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How to prove that a generic Riemannian metric does not admit nontrivial local isometries
Definition: Let $(M,g)$ be a Riemannian manifold. We say that two points $p,q\in M$ are locally isometric iff there exists an open set $U$ around $p$, an open set $V$ around $q$, and an isometry $\phi:...
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Rigidity of natural Riemannian metrics on convex hypersurfaces
Let $E$ be a finite dimensional real vector space.
(You can think of $E=\mathbb R^n$, but I will not be using the Euclidean metric.)
If $\Sigma\subset E$ is a compact, smooth, and strictly convex ...
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Metric convex hull and geodesic convex hull in a non-Euclidean space
Given a metric space $(X,d)$ that is a geodesic space, i.e., for any two points $x_1,x_2\in X$, there is a unique constant-speed geodesic $\omega:[0,1]\to X$ linking $x_1$ and $x_2$, such that
$\...
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Counterexamples in Simon-Smith min-max theory
I am reading about the Simon-Smith min-max method for constructing minimal surfaces in 3-manifolds, following this survey by Colding and De Lellis.
Here is what I understand:
Given a min-max sequence ...
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2
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1
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Ways in which minimal surfaces locally minimize area
Let $(M,g)$ be an $n$-dimensional Riemannian manifold.
For a $k$-dimensional submanifold $\Sigma\subseteq M$, say that it is minimal if it is a critical point of the $k$-dimensional area functional ...
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Pointwise bound of a quantity in almost Kähler geometry
Let $(M,g,J)$ is an almost Kähler manifold. It means $(M,J)$ is an almost Hermitian manifold with the symplectic form $\omega(u,v)=g(Ju ,v )$ being closed. Moreover $\omega$ has constant norm and is ...
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