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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.

1,807 questions
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Special Riemannian metric on the product

Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then ...
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I am working with biquotients and positive curvatures and I was able to give a relatively simple proof for the following: Theorem: Let $G$ be a compact connected Lie group with a bi-invariant metric $... 0answers 69 views Inheriting quasiconvexity from convex function after re-parametrisation of space into the Stiefel manifold Consider a function$f(S): \mathcal{S} \to \mathbb{R}$, where$\mathcal{S}$is the convex cone of all positive semidefinite complex$M\times M$Hermitian matrices. The function$f(\cdot )$is concave ... 0answers 18 views Can we force the orthogonal of the boundary of a compact manifold to lie in a prescribed hypersurface? Let$M$be a compact manifold with boundary$\partial M$. Suppose that we are given a hypersurface$A$which is transversal to$\partial M$. Is there a way to construct a metric$g$on$M$such that ... 0answers 59 views Existence of nonparabolic ends Let$M$a nonparabolic Riemannian manifold. If exists only one nonparabolic end$E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ... 0answers 119 views Asymptotic bound on minimum epsilon cover of arbitrary manifolds Let$M \subset \mathbb{R}^d$be a compact smooth$k$-dimensional manifold embedded in$\mathbb{R}^d$. Let$\mathcal{N}(\varepsilon)$denote the minimal cardinal of an$\varepsilon$-cover$P$of$M$; ... 0answers 155 views Hermitian sectional curvature Let$N$be a Riemannian manifold, denote$R$its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is$K(X,Y) = R(X,Y,X,Y)$for an orthonormal pair. ... 0answers 82 views Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant Let$M \subset \mathbb{R}^d$be a compact smooth$k$-dimensional manifold embedded in$\mathbb{R}^d$. Let$\mathcal{N}(\epsilon)$denote the size of the minimum$\epsilon$cover$P$of$M$; that is ... 0answers 38 views Realizing curves as geodesics of positively curved metrics It is a well known fact (see here for instance), that under reasonable conditions any curve on a smooth manifold can be realized as a geodesic for some given connection. The natural construction when ... 1answer 88 views Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2],$$\mathcal{P}(n) = \... 1answer 142 views Question about interpretation of algebraic notation in differential geometry paper I am unable to understand the notation of equations (1.1) and (1.6) in page 2 of Kowalski and Belger's paper "Riemannian metric with the prescribed curvature tensor and all its covariant derivatives ... 2answers 480 views If a (distance) metric on a connected Riemannian manifold locally agrees with the Riemannian metric, is it equal to the induced metric? Let$(M,g)$be a connected Riemannian manifold. Let$d_g$be the induced distance metric of$g$. Now let$d$be some other metric on$M$. Suppose that for each$x \in M$, there is a neighborhood$U$... 0answers 56 views Ricci flow on Riemannian submersions Let$(P,g) \to (S^2,h)$be a Riemannian submersion. Let$g(t)$be the Ricci flow on$P$with initial condition$g$. Does the induced flow on$S^2$converges to the round metric on$S^2?$I could ... 1answer 125 views Does fractallity depend on the Riemannian metric? Edit: According to comment of Andre Henriques we revise the question: In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ... 2answers 463 views Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold Let$(M,g)$be a compact Riemannian manifold, and let$\Delta_g$be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of$\Delta_g\$ have finite multiplicity and tend to ...

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