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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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      63 views

      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 ...
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      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 ...
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      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 ...
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      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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      70 views

      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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      116 views

      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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      114 views

      Lipschitz constant of exponential map

      I asked before this question on MSE but I was not able to work out the details on my own. Suppose $M$ is a smooth compact Riemannian manifold, take $p \in M$ and consider the map $$ T_pM \ni v \...
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      Gradient of squared riemannian distance on complete manifold

      Let $\theta: M \times M \to \mathbb{R}$ the squared distance function $\theta(x,y)=d(x,y)^{2}$ on complete Riemannian manifold $M$. I would like to calcule the gradient of $d^{2}$, where $d^{2}_{y}(x)=...
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      125 views

      Embed the hyperbolic plane into Euclidean spaces

      Can the complete simply-connected surface with constant Gauss curvature -1 be embedded smoothly in the 5-dimensional Euclidean space? Can the complete simply-connected surface with constant Gauss ...
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      1answer
      68 views

      Compact group actions with uniformly bounded derivatives

      Suppose we have a smooth action of a compact Lie group $G$ on a non-compact smooth manifold $M$, denoted by $$\phi:G\times M\rightarrow M.$$ Differentiating $\phi$ at a point $x\in M$ gives a map that ...
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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 ...
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      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 ...
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      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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