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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Geodesic distance on $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
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An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
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$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
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Ricci-flat metrics on complex tori of dimension $n \geq 3$
Let $\mathbb{T}^n = \mathbb{C}^n /\Lambda$ be a complex torus of (complex) dimension $n$. If $n=2$, it is a theorem of Berger that the Ricci-flat metrics on $\mathbb{T}^2$ are flat. This follows from ...
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Integrability (and hence regularity) of $\alpha$-harmonic maps
To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^...
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Bi-$M$-invariant measure on a Riemannian symmetric space
Let $G$ be a noncompact connected semi simple Lie group. Let $K$ be a maximal compact subgroup and $G=K\overline{A_{+}}K$ be a Cartan decomposition of $G.$ Let $M=Z_{K}(\mathfrak{a})$. Then how to ...
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Local smoothness of harmonic heat flow
Assume that $ (M,g) $ is a smooth closed manifold and $ \mathbb{S}^{L-1} $ is a unit sphere with dimension $ L-1 $ in $ \mathbb{R}^{L} $. Consider the equation of harmonic heat flow
$$
\partial_tu-\...
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Can a laplacian-beltrami operator have negative eigenvalues?
Is it possible for an Laplace-Beltrami operator for Riemannian manifold to have negative eigenvalues?
If not, are there any non-riemannian manifolds where one may observe negative eigenvalues for heat ...
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Homotopy type / Homology of the free loop space of aspherical manifolds
Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
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Geodesics on orthogonal matrix
Let $ O(n) $ be the manifold of orthornormal matrix, i.e.
$$
O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}.
$$
Then $ O(n) $ is a submanifold of $ \mathbb{R}^{n\times n} $. On $ O(n) $, there is a ...
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Projection of Fréchet mean(s) to tangent space of a Riemannian manifold versus mean of the projection of these points
Let $\{x_1\dots x_n\}\subset (M,g),$ where $(M,g)$ is a complete finite $d$ dimensional a Riemannian manifold. Let us denote by $\bar{x}$ a Fréchet mean of $\{x_1\dots x_n\},$ i.e. a minimizer of the ...
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Local isometric embedding right inverse to a Riemannian submersion
Let $M$ and $N$ be Riemannian manifolds such that $\pi:M\to N$ is a surjective Riemannian submersion, i.e. for each $x\in M$,
$$\langle \pi_{*x}(v),\pi_{*x}(w) \rangle_{\pi(x)} = \langle p(v), p(w) \...
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$N$th-order approximation of point stabilizing diffeomorphisms by $N$th-order jet group?
NOTE: migrated from math SE.
I was wondering if ever higher jet groups of frames on a (possibly pseudo) Riemannian manifold $M$ approximate the point stabilizing subgroup of diffeomorphisms on $M$ as ...
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Cycloid on manifolds
Inspired by differential equation
$$y(1+y'^2)=c$$
which generates the cycloid we consider the following differential equation on a Riemannian manifold:
$$f(1+|\nabla f|^2)=c$$
On the other hand ...
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Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
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Why are conformal transformations so relevant?
I have been studying construction of initial data in general relativity for many years now and it turns out that the most efficient methods to construct such data rely at some point on conformally ...
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Relation between two gradient dynamics
If $f:\mathbb{R}^n\rightarrow\mathbb{R}_+$ is a nonnegative real analytic function and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ is a strongly convex smooth function with a surjective gradient $\nabla g:\...
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Curvature of an affine system
I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, ...
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Cheeger constant and isoperimetric ratio
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric. Consider any curves $\gamma$ on $S^2$ dividing the total area $A$ into two parts $A_1+A_2 =A$. The isoperimetric ratio is
$$
C_s(\gamma)=\frac{...
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A Multiplicative Average of Positive Operators
Let $G$ be a finite group. I have an action of $G$ on a matrix algebra of positive operators, $\mathcal{M}$. In particular, $\mathcal{M}$ has a $G$-module structure, yielding a linear representation ...
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A question on generalized Einstein manifold
Let $(M,g)$ be a Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold. The zero section is denoted by $Z$.
We define a Hamiltonian on $T^0 M=TM\setminus Z$ via $$...
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Active areas of Research in Riemannian Geometry? [closed]
I've taken a course in Riemannian Geometry and would like to know which topics in Riemannian Geometry are nowadys topic of research
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Riemannian manifolds with a unique distance property
Let $M$ be a compact Riemannian manifold with geodesic distance function $d$, of (normalised) diameter $1$.
Some of my favourite manifolds $M$ have the property that there exists an integer $k$ such ...
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A formula in harmonic heat flow
Assume that $ (M,g) $ and $ (N,h) $ are two smooth closed manifold and $ N $ is embedded isometrically into $ \mathbb{R}^K $ for some $ K\in\mathbb{Z}_+ $. Assume that $ u\in C^{\infty}(M\times\mathbb{...
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Is squared geodesic distance a Morse-Bott function on simply-connected Hadamard manifolds?
Let $M$ be a simply connected Hadamard manifold. That is, $M$ is a complete Riemannian manifold with sectional curvature bounded above by 0. Let $f(x,y)=\frac{1}{2}d^2(x,y)$, where $x,y \in M$, and $d(...
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Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
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Conformal Killing vector fields on compact surface of genus \ge 1
Let $(M, g)$ be a compact 2-dimensional Riemannian manifold with genus $\ge 1$. Can $M$ has a conformal Killing vector field $X$ other than Killing vector fields? That is, $L_X g = (\mathrm{div} X) g$ ...
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Existence of normal and harmonic coordinates around a point
Let $(M^n,g)$ be a Riemannian manifold with a fixed point $p$. Can we find a local coordinate system $(x_1,x_2,\cdots, x_n)$ around $p$ satisfying the following two at the same time:
(1) $(x_1,x_2,\...
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Maximizing the first Neumann eigenvalue on disks
Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that
$$\mu_1(g) \operatorname{...
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Steklov eigenvalue for circle valued functions
Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization:
$$\sigma_1(M,g)...
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Pythagorean theorem in Riemann metrics of non constant curvature
I already asked the same question here, but received no answer. I was reading this interesting article by Givental
Givental, Alexander. "The Pythagorean theorem: What is it about?" Amer. ...
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What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?
Note: I asked this on Mathematics SE and even though @TheSimpliFire offered a bounty on it, no-one had a good answer
Find the optimal shape of a coffee cup for heat retention. Assuming
A constant ...
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Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?
Let $(M,g)$ be a (semi-)Riemannian manifold, and let $(M,[g])$ be the conformal class thereof. The following two sets are naturally torsors for the collection of positive-valued functions on $M$:
The ...
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Symmetric cones and symmetric spaces
I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian ...
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Do we have $d(P_{+}(\omega\wedge \theta))=d_{+}\omega\wedge \theta-\omega\wedge d_{+}\theta$ on a self-dual manifold?
Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of self-...
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Are these the only first eigenfunctions on a hemisphere?
Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
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Show $\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle $ for all $R,S \in \mathrm{SO}(3)$
$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(...
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Is every linear Lie group of bounded geometry?
$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...
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Is there a good description of harmonic maps from $\mathbf{C}$ to $\mathbf{H}$?
Given a non-constant holomorphic quadratic differential $\phi \, \mathrm{d} z^2$ on the complex plane $\mathbf{C}$, there is a harmonic diffeomorphism $u: \mathbf{C} \to \mathbf{H}$ into the ...
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Finding a hyperbolic metric with geodesic boundary on a given Riemann surface
Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics ...
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Approximate isometric embeddings of surfaces
The fundamental theorem of surfaces states that if symmetric matrices $g_{ij}$, $l_{ij}\colon U\subset R^2\to R$, where $U$ is open and $g_{ij}$ is positive definite satisfy the Gauss and Codazzi ...
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Integral geometric meaning of diameter
Let $X\subset \mathbb CP^n, n>2$ be a complex smooth algebraic hypersurface.
Any hyperplane section $H\cap X$ is connected and has diameter $Diam(H\cap X)$
in the inner metric induced from the ...
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$1$-parameter family of metrics preserving the normal direction
Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
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A Question about an article by Birman, Series
Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON
SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
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Finding a real-analytic diffeomorphism
Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
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Distance between two geodesics originating from separate but nearby points
Let $(M,g)$ be a complete Riemannian manifold with sectional curvatures constrained within $[\kappa_{\min},\kappa_{\max}]$. Suppose $x,y\in M$ are two points in $M$ and $v_x\in T_{x}M$ is a tangent ...
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Euler-Poincaré characteristic of even-dimensional Einstein manifolds with nonnegative sectional curvature
My question is about whether there are some known conditions on the sign of the Euler-Poincaré characteristic for Einstein manifolds in even dimensions.
In dimension $4$ some conditions on the sign of ...
2
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Teichmuller interpretation of unbounded holomorphic quadratic differentials
For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...
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Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...
3
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0
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Implicit function theorem in Riemannian manifold and Wasserstein space
My question is about to what extent can we extend the implicit function theorem to Riemannian manifolds. In the Euclidean space, consider a bivariate function $F \colon \Theta \times \mathcal{X} \...