Questions tagged [riemannian-geometry]
For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.
8,530 questions
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Smooth functions with arbitrarily large regular level sets
Let $f:M\to\mathbb{R}^k$ be smooth, where $M$ is a closed manifold of dimension $n>k$. I am thinking about the continuity properties of the map $c\mapsto\mathcal{H}^{n-k}(f^{-1}(c))$. Clearly this ...
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What mathematical structure do the manifolds described by color models and color spaces have? [closed]
As I understand it, a color model (like the RGB model) is a way of mapping the space of all human-perceptible colors to a certain manifold -- typically, but not always, either three- or four-...
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The orbit of 1-parameter group generated by Killing field on compact Riemannian manifold [closed]
Let (M,g) be a compact Riemannian manifold, X is nonzero Killing field, $\phi_t:M \longrightarrow M$ is the 1-parameter group generated by X. $O(p)=\{\phi_t(p)|t\in \mathbb{R}\}$, I want to prove that ...
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Tensor Algebra Problem [duplicate]
Can anyone tell me why this question was closed ?? I was genuinely stuck and was looking for tips to advance the problem:
How does Riemann curvature tensor change under conformal transformation
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Exterior covariant derivative of a vector valued $k$-form [closed]
Given a connection on the tangent bundle $TM$ of $M$, how can we define an exterior derivative of a vector valued (with values in a vector bundle) $k$-form? Thank you for your answers.
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geodesic distance in embedded manifolds
I wish to describe a problem encountered in my research and am seeking advice, or just pointers on where to look.
My setting is as follows. Given a scatterplot of data in $\mathbb{R}^D$, we wish to ...
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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-...
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Bound on Jacobi field
Edit: I got an answer on Mathoverflow, see https://mathoverflow.net/questions/504580/bounding-norm-of-jacobi-fields-given-end-point-values
I'm reading the paper on the generalized sphere theorem by ...
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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 ...
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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 $f$-...
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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 ...
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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} ...
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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, ...
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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 $...
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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 ...
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