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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3 votes
0 answers
160 views

A higher-dimensional "line of curvature"?

Let $M$ be a submanifold of a Riemannian manifold $Q$, and let $\Gamma$ be a $d$-dimensional submanifold of $M$, i.e., $\Gamma^{d} \subset M \subset Q$. Suppose that, for all (unit) normal vectors of $...
8 votes
4 answers
649 views

Torsion of submanifolds

Studying curves in the Euclidean three dimensional space, one usually defines the curvature and the torsion of a curve. If I am not missunderstanding the thing, I guess that a curve has zero torision ...
0 votes
1 answer
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Curvature tensor of interpolation of two metrics

Let $\hat{g}$ and $\bar{g}$ be two smooth Riemannian metrics defined, say, on $\mathbb{R}^n.$ Consider a smooth function $\xi$ that acts as an interpolation function between the two metrics above on ...
4 votes
1 answer
147 views

(Reference request) higher order Hölder spaces on riemannian manifolds

I am looking for a reference regarding the higher (than the first) order Hölder spaces on Riemannian manifolds. I am aware that defining Hölder spaces of form $C^{0,\alpha}$ is not an issue even ...
3 votes
0 answers
59 views

Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field

Setting Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
2 votes
1 answer
103 views

Parametrization of $k$-ruled submanifold: can we choose the base to be orthogonal to the rulings?

A smooth $m$-dimensional submanifold of $\mathbb{R}^{d}$ is said to be $k$-ruled if it is foliated by $k$-dimensional planes, called rulings. Let $M$ be a $k$-ruled submanifold. Then $M$ can be ...
4 votes
1 answer
624 views

An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

Background Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...
8 votes
1 answer
506 views

$C^1$ regularity of harmonic functions on Riemannian manifolds

Consider a smooth, connected and complete Riemannian manifold $M$. It is well known that harmonic functions defined on some open subset of $M$ are $C^\infty$. I'm interested in knowing whether there ...
2 votes
1 answer
157 views

Decomposition of forms on a Spin$(7)$ manifold

Let's take a $G_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+*_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\...
4 votes
0 answers
168 views

Reference request: $ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $

I have a Riemannian manifold $M$ of dimension 2 on which I am considering the following equation: $$ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $$ on some patch $U$ of the manifold which is &...
5 votes
0 answers
340 views

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}{...
0 votes
1 answer
182 views

Compute distance between geodesics and perturbed geodesics on a Riemannian manifold via Jacobi field $\vert J \vert$

I would like to pose a question regarding the distance between a geodesic $\gamma(t)$ and a perturbed geodesic $\gamma_{\epsilon}(t)$ on a Riemannian manifold. Specifically, is the distance controlled ...
3 votes
1 answer
401 views

Riemann Hurwitz vs Gauss Bonnet

The Gauss-Bonnet theorem implies the Riemann Hurwitz theorem http://sma.epfl.ch/~troyanov/Papers/Prescribing.pdf Prop 1 => Cor 2 In what sense is the Gauss- Bonnet theorem stronger? Are these ...
0 votes
0 answers
164 views

A Lie group whose Lie algebra is the (Lie algebra?) of all functions with fibrewise polynomial growth

Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket ...
9 votes
1 answer
336 views

Do geodesics avoid regions where the curvature diverges?

Let $(M^2,g)$ be a Riemannian manifold, with manifold boundary $\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like $K \to +\infty$...
1 vote
0 answers
83 views

Obstruction for a manifold to admit a periodic Ricci flow

Let M be a (compact) smooth manifold. What kind of obstruction exist for M to admit a metric whose Ricci flow is a t-periodic flow?
1 vote
1 answer
373 views

Relationship between the Fisher distance and Kulback Leibler divergence

I am reading the 2017 book "Information geometry" by Ay, Jost, Lê, Schwachhöfer. The Fisher distance is given by $$ d^F(\mu, \nu) := \inf_{\gamma} L(\gamma) $$ for curves $\gamma:[0,1]\to P$ ...
3 votes
1 answer
135 views

Reference: parallel transport in the hyperboloid model

I'm reading the documentation of this package: Manopt, and they claim that in the hyperboloid model for $\mathbb{H}^d$ the parallel transport between tangent spaces $T_x$ and $T_y$ is given for any $u\...
0 votes
0 answers
35 views

Obtaining metric and compatible differential equations on codimension one foliation of $n$-cube

Essentially the same content as this post on Math stack exchange: https://math.stackexchange.com/q/4741835/460999. I don't expect an answer there and after waiting a few days I've decided to post here....
2 votes
0 answers
86 views

Tangent cones at infinity and the regularity of minimal submanifolds

In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal ...
4 votes
1 answer
244 views

Conformal maps between two given domains

Consider two domains $$ \begin{aligned} D_1&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq 0\},\\ D_2&=\{x=(x_1,x_2,...,x_n)\in\mathbb{R}^n:x_n\leq \psi(x_1,x_2,...,x_{n-1})\}, \end{aligned} $$ ...
5 votes
1 answer
360 views

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 ...
2 votes
0 answers
133 views

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 ...
2 votes
0 answers
65 views

$ \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,...
3 votes
0 answers
147 views

$N$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 ...
1 vote
0 answers
155 views

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 ...
1 vote
0 answers
63 views

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^...
1 vote
0 answers
151 views

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:\...
2 votes
0 answers
110 views

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-\...
1 vote
0 answers
27 views

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 ...
4 votes
1 answer
223 views

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 ...
0 votes
0 answers
103 views

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 ...
4 votes
0 answers
113 views

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 $x_0$...
0 votes
0 answers
54 views

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 ...
0 votes
1 answer
97 views

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) \...
3 votes
0 answers
87 views

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 ...
9 votes
0 answers
235 views

Systole of Riemann surfaces of genus $g$

In Buser and Sarnak's "On the period matrix of a Riemann surface of large genus", we get $$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
10 votes
0 answers
317 views

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 ...
3 votes
0 answers
190 views

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{...
11 votes
1 answer
472 views

About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves

In the paper ``Morse theory on Hilbert manifolds'' (1963), on page 326, Richard Palais makes a remark that if $\phi\colon V \to W$ is an isometry (of submanifolds of $\mathbb{R}^n$), then this does ...
0 votes
0 answers
157 views

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 $$...
15 votes
6 answers
2k views

Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology

I'm now attending a reading seminar on the algebraic topology. The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes). In those ...
0 votes
0 answers
47 views

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 ...
1 vote
0 answers
216 views

Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow

However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
2 votes
0 answers
144 views

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
4 votes
1 answer
264 views

Ricci curvature of totally geodesic submanifold

Let $M$ be a Ricci-flat Riemannian manifold and $N \subset M$ a totally geodesic submanifold. Is $N$ also Ricci-flat? A partial result in that direction is that the Ricci curvature of $N$ is given by $...
2 votes
1 answer
931 views

Weak derivatives and Sobolev spaces on Riemannian manifolds

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them. Let $(M,g)$ be smooth ...
7 votes
2 answers
2k views

The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
4 votes
1 answer
94 views

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{...
4 votes
0 answers
67 views

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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