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# 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.

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### Geodesics (Local vs Global)

Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
80 views

### Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
99 views

### “Inflating” a closed, defined metric, manifold

Consider a two-dimensional Riemannian manifold homeomorphic to the sphere, with a defined metric. Since we do not suppose that manifold to have a positive curvature, we are not in the hypotheses of ...
66 views

### Convergence of non-collapsing sequences of Riemannian manifolds with biliterally bounded sectional curvature

EDIT: I heard that there is the following result: Given a sequence $\{(M_i^n,g_i)\}$ of compact smooth $n$-dimensional Riemannian manifolds with uniformly bounded absolute value of sectional ...
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### Possible isometry groups of open manifolds

Consider a non-compact manifold $M$. Does there always exist a Riemannian metric on $M$ such that the isometry group is non-compact?
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### Gaussian null coordinates

I find it hard to find information on the so-called "Gaussian null coordinates", which Wikipedia says is used to describe "near horizon geometries". Can someone provide a reference where I can read ...
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### Norm of a Riemannian metric

I'm writing a thesis and I need to be able to say when two Riemannian metrics are close. I was reading a paper and there the definition was assumed to be known, so I guessed it had to be something ...
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### Volume form on unit normal bundle via moving frames

Let $M$ be an $m$-dimensional Riemannian submanifold of $\mathbb{R}^{m+n}$. Let $B_1$ denote the unit normal bundle of $M$, whose fiber at $p \in M$ is the $(n-1)$-sphere $\mathbb{S}^{n-1}$ in the the ...
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### On the Variable Coefficient Laplacian

This the copy of the question that I had asked in math stackexchange I read about Laplace Operator here. As given in the link, given the metric, we can find the expression for Laplace operator. I am ...
26 views

### Cylindrical coordinates in quotient of symmetric space

I am interested in the following situation. Suppose $G/K$ is a symmetric space of non-compact type and $\alpha$ is the axis of a hyperbolic isometry. I am interested in computing the Hessian of the ...
194 views

### Is the injectivity radius (semi) continuous on a non-complete Riemannian manifolds?

Let $\mathcal{M}$ be a Riemannian manifold, and let $\mathrm{inj} \colon \cal M \to (0, \infty]$ be its injectivity radius function. It is known that if $\cal M$ is connected and complete, then \$\...

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