# 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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### 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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### Positive Ricci curvature on biquotients

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 $...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $M$; ...

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### 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.
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### 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 ...

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### 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 ...

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### 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) = \...

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### 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 ...

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### 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 $U$ ...

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### 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 ...

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### 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 ...

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### 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 ...