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.
6 questions from the last 30 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 $B(x,r)$?
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