Questions tagged [riemannian-geometry]

Question 1

I am reading the paper "Liouville Theorems, Volume growth, and Volume comparison for Ricci Shrinkers" by Li Ma. In the proof of Theorem 3, there is a claim: Let $M$ be a complete noncompact ...

Question 2

An irreducible Hyperkähler manifold is an Riemannian manifold of complex dimension $2n$, whose holonomy group with respect to a Kähler metric is $Sp(n)$, the symplectic group. A Riemannian manifold is ...

Question 3

Show that if $\tilde g=e^{2f}g$ for some function $f,$ then: $$\tilde R^l_{ijk}= R^l_{ijk}-a^l_i g_{jk}-a_{jk}\delta^l_i+a_{ik}\delta^l_j+a^l_jg_{ik}$$ where $a_{ij}$ is given to be: $$a_{ij}=\nabla_i\...

Question 4

In Petersen's Riemannian Geometry, we have the following passage. The Ricci tensor: For now this is simply an abstract $(1,1)$-tensor: $\mathrm{Ric}(E_i) = \mathrm{Ric}\,_i^j E_j$; thus $$\mathrm{Ric} ...

Question 5

Let us consider a Lagrangian system for which the equations of motion come from Hamilton's principle and are such that the control variable $\tau$ equals the equations of motion of a free system, ...

Question 6

Does the hyperbolic space $\mathbb{H}^2$ (hyperboloid model) admit a global, smooth, orthonormal frame? I have tried to compute it explicitly. Let $\Phi: \mathbb{R}^2 \to \mathbb{H}^2\subset \mathbb{R}...

Question 7

Let $M$ be a $n$-dimensional Riemannian manifold, i.e. $M = \cup_{\lambda \in \Lambda}\, (U_{\lambda}, g_{\lambda})$ with each $U_{\lambda}$ being an open ball around the origin of ${\Bbb R}^n$ and $...

Question 8

I have the following situation: Suppose that we have two pseudo-Riemannian manifolds M and N, and fixed points $p$ $\in M$, $q \in N$, and a linear isometry between the corresponding tangent spaces ...

Question 9

Let $(M, \tilde{g})$ be a Riemannian manifold and let $g:=e^{2u}\tilde{g}$ be a conformal change. I'm trying to find a resource (ideally a book, or a paper where it's mentioned or derived) for the ...

Question 10

Let $g$ be a Riemannian metric on a domain $\Omega$ with boundary $\sigma$, and the Ricci curvature tensor is given by $$ R = [\, \partial \Gamma \,] + [\, \Gamma \Gamma \,], $$ where $$ {[\, \partial ...

Question 11

I´m trying to find an identity for the operator $\star\mathcal{L}_\nu\star$ acting on k-forms where $\mathcal{L}_\nu$ is the Lie derivative in direction of a vector field $\nu$. Using Cartan's magic ...

Question 12

I've often heard people think about the Riemannian connection as the "derivative of the metric", the Riemann tensor as the "Hessian of the metric", and Ricci's tensor as a "...

Question 13

Let $G$ be a Lie group acting properly and isometrically on a Riemannian manifold $(M,\mathtt g)$. By the slice theorem, we know that the tubular neighborhood $U=\exp(\nu^\epsilon G(x))$ of an orbit $...

Question 14

I was reading this post about the space of Riemannian Metrics, and I would like to know more about the topology of the space of complete Riemannian Metrics on a manifold $M$. Is this space path ...

Question 15

Let ${\mathrm{S}}^1$ be a circle with the radius $\frac{1}{2}$, which is a one-dimensional Riemannian manifold. I would like to give the chart on ${\mathrm{S}}^1$ by the affine $x$-real line and the ...

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Questions tagged [riemannian-geometry]

Reference request: Bounds on laplacian of cutoff functions

Are irreducible Hyperkähler manifolds irreducible Riemannian Manifolds.

How does Riemann curvature tensor change under conformal transformation [closed]

Type changing $(1,1)$ to $(2,0)$ tensor

Are the motion equations of an optimal control problem geodesics on a manifold?

Does the hyperbolic space $\mathbb{H}^2$ admit a global, smooth, orthonormal frame? [closed]

Why is the connection necessary?

Conditions to a linear map between tangent spaces being the differential of a isometry

Reference for the formula for conformal change of sectional curvature

Existence and uniqueness of a solution to a Ricci tensor-based boundary value problem [closed]

Dualizing Lie derivative with hodge star

Is the connection and curvature really derivatives of the metric?

G-invariant metrics on a Tubular neighborhood

The space of complete Riemannian Metrics on a manifold $M$

Levi-Civita connection for ${\mathrm{S}}^1$.

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