Questions tagged [riemannian-geometry]
For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.
8,528 questions
7
votes
0
answers
54
views
Reference request of Calabi-Yau theory
I and my friend want to run a representation-theory-oriented as well as geometry-motivated reading seminar on Calabi-Yau theory.
To be precise, we want to learn the Kähler-Einstein aspect of Calabi-...
- 131
2
votes
0
answers
25
views
Bound on Jacobi field
I'm reading the paper on the generalized sphere theorem by Grove and Shiohama, and there is an observation they made that I'm struggling to prove.
We have two unit geodesics $\tau,\sigma:[0,L]\to M$, ...
- 315
0
votes
0
answers
127
views
Why do vectors behave as derivation on functions
I am learning about differentiation in the context of Manifolds and Tangent Space and am struggling with the idea that a vector operates as a differentiation operation on a function $f$. Here is a ...
- 21
2
votes
0
answers
65
views
Reference request: Bounds on laplacian of cutoff functions
I am reading the paper "Liouville Theorems, Volume growth, and Volume comparison for Ricci Shrinkers" by Li Ma.
Notation:
$Ric_f=Ric+Hess f$ and $Δ_f u= Δu −g(∇f,∇u)$. We call $u$ to be $f$-...
- 100
0
votes
0
answers
50
views
Are irreducible Hyperkähler manifolds irreducible Riemannian Manifolds.
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 ...
0
votes
0
answers
63
views
How does Riemann curvature tensor change under conformal transformation [closed]
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:
$$a_{ij}=\nabla_i\...
- 1,955
1
vote
1
answer
62
views
Type changing $(1,1)$ to $(2,0)$ tensor
In Petersen's Riemannian Geometry, we have the following passage.
The Ricci tensor: For now this is simply an abstract $(1,1)$-tensor: $\mathrm{Ric}(E_i) = \mathrm{Ric}\,_i^j E_j$; thus $$\mathrm{Ric} ...
- 2,306
4
votes
1
answer
60
views
Are the motion equations of an optimal control problem geodesics on a manifold?
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, ...
- 534
0
votes
0
answers
86
views
Does the hyperbolic space $\mathbb{H}^2$ admit a global, smooth, orthonormal frame? [closed]
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}...
- 619
-1
votes
0
answers
80
views
Why is the connection necessary?
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 $...
- 1,067
3
votes
1
answer
105
views
Conditions to a linear map between tangent spaces being the differential of a isometry
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 ...
- 31
0
votes
0
answers
45
views
Reference for the formula for conformal change of sectional curvature
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 ...
- 2,826
1
vote
0
answers
40
views
Existence and uniqueness of a solution to a Ricci tensor-based boundary value problem [closed]
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 ...
1
vote
0
answers
27
views
Dualizing Lie derivative with hodge star
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 ...
- 11
3
votes
1
answer
85
views
Is the connection and curvature really derivatives of the metric?
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 "...
- 1,170