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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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