All Questions
Tagged with reference-request riemannian-geometry
105 questions with no upvoted or accepted answers
10
votes
0
answers
464
views
Reference for sets of locally finite perimeter on Riemannian manifolds
I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
- 7,817
10
votes
0
answers
284
views
Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
- 109
9
votes
1
answer
646
views
Explicit construction of a (the?) dual symmetric space
I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type.
...
- 113
8
votes
0
answers
1k
views
Classification of flat Riemannian three manifold
By a theorem of L. Bieberbach we know that that every closed flat Riemannian manifold is a quotient of a torus via action of a finite group $\Gamma$. In this question we are interested in the ...
- 1,915
7
votes
0
answers
192
views
Higher homotopy groups of an orbifold
Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group:
Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as ...
- 1,452
7
votes
0
answers
248
views
Does the Hodge decomposition hold for equivariant differential forms?
Let $M$ be a Riemannian manifold. The Hodge decomposition tells that
$$
\Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M)
$$
where $d^*$ is the adjoint operator of the ...
- 2,789
7
votes
0
answers
305
views
Generalizing Gromov Hausdorff distance using Vietoris topology
There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for ...
- 1,630
7
votes
0
answers
508
views
Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,942
7
votes
0
answers
695
views
Sasaki Metric of the Tangent Bundle over the Hyperbolic Plane
This is a reference request on what are surely well known facts.
Let $M$ be a compact hyperbolic surface and $S(M)$ its unit tangent bundle. It follows from facts about Möebius tranformations in the ...
- 4,304
6
votes
0
answers
218
views
Optimal configurations on the flat torus
I'm studying (with my colleagues R.Piergallini and S.Isola) configurations of points on the flat torus which minimize an attractive or repulsive potential depending on the distance. Two model cases ...
- 4,459
6
votes
0
answers
101
views
Shortest path on Riemannian manifold with boundary
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Let $x\in \partial M$. Let $v\in T_x(\partial M)$ be a unit vector tangent to the boundary. Assume
$$II_{\partial M}(...
- 21.8k
6
votes
0
answers
272
views
Gravity, connection, and curvature
Starting with Synge and Fock, many modern authors identify gravity with curvature. On the other hand, Einstein always emphasized that gravity should be equated with a connection, but not with ...
- 16.5k
6
votes
0
answers
122
views
Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?
Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ ...
- 969
6
votes
0
answers
355
views
Higher order variations of Riemannian geodesics
Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic.
There is a well established theory of the first order ...
- 8,086
6
votes
0
answers
393
views
Steklov eigenvalue problem for a planar region bounded by ellipse
The Steklov problem for a compact planar region $\Omega$ is
\begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$},
\end{...
- 141
6
votes
0
answers
386
views
Is there a Bishop-Gromov inequality for manifolds with boundary?
EDIT. Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $...
- 21.8k
6
votes
0
answers
578
views
Lie bracket of gradients
Related to Why symplectic geometry gives Poisson geometry
Is there any common use for the Lie bracket of the gradients of two functions? That is, $[\nabla f, \nabla g]$?
Here, $\nabla f$ denotes the ...
- 4,991
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
5
votes
0
answers
78
views
Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
- 51
5
votes
0
answers
159
views
On Sobolev spaces on domains in Riemannian manifolds
There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing ...
- 505
5
votes
0
answers
276
views
Fundamental group of compact globally symmetric spaces
The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient
$$(*) \quad \pi_1(...
- 1,123
5
votes
0
answers
261
views
Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary
In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...
- 409
5
votes
0
answers
307
views
Gradient estimate for Poisson equation on manifold
In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...
- 2,789
5
votes
0
answers
445
views
Why are functions with vanishing normal derivative dense in smooth functions?
Question
Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm?
Here I define $...
- 881
5
votes
0
answers
77
views
Gibbons-Hawking space over over two points is $\text{T}^\ast\mathbb{CP}^1$
Is there any direct way of seeing that the space obtained via the Gibbons-Hawking ansatz over $\mathbb{R}^3\setminus\{p_1,p_2\}$ with a suitable choice of complex structure is biholomorphic to $\text{...
5
votes
0
answers
179
views
Some questions on the nodal geometry of Dirac operators
Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
- 1,407
5
votes
0
answers
1k
views
"The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics"
The title is a quote from p.256 of Wilhelm Klingenberg's 1995
Riemannian Geometry (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
- 151k
4
votes
0
answers
72
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 ...
- 1,949
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.8k
4
votes
0
answers
880
views
Scalar curvature in terms of second fundamental form, reference request
I would like to cite a reference for the following formula for scalar curvature:
If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...
4
votes
0
answers
116
views
$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities
Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
- 6,853
4
votes
0
answers
108
views
"Uniqueness" of the Yamaguchi submersions of Riemannian manifolds
The Yamaguchi submersion theorem says the following. Let $\{M_i\}$ be a sequence of $n$-dimensional smooth connected closed Riemannian manifolds of diameter at most $D$ and sectional curvature at ...
- 21.8k
4
votes
0
answers
247
views
Canonical connections on Hermitian manifolds
A Hermitian manifold $(M,J,g)$ can be defined as a complex manifold $(M,J)$, with integrable complex structure $J$, equipped with a Riemannian metric satisfying $g\circ (J\otimes J) = g$. In this ...
- 2,816
4
votes
0
answers
343
views
Riemannian metrics on a manifold with corners
For a smooth manifold with corners (although maybe there is no universally agreed definition of it), is there always a Riemannian metric making every face totally geodesic?
Is there any reference ...
- 803
4
votes
0
answers
244
views
A simple proof that all the symmetries of the Dirichlet energy are conformal
This is a cross-post.
It seems to be folklore knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps.
Specifically, I have found this nice proof for the following ...
- 6,741
4
votes
0
answers
140
views
Has this functional been studied?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
This is a cross-post from MSE.
Let $\M,\N$ be Riemannian ...
- 6,741
4
votes
0
answers
95
views
Laplacian Spectra on Nearly Nodal Riemann Surfaces
Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...
- 1,207
4
votes
0
answers
152
views
Faster (than normal) convergence of the normalized Ricci flow on surfaces
Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...
- 41
3
votes
0
answers
247
views
Fibre metrics on non-linear bundles
Usually what is meant under a fibre metric is that one is given a (smooth) vector bundle $\pi:Y\rightarrow X$, and on each fibre $Y_x$ an algebraic inner product $g_x$ that varies smoothly from point ...
- 2,792
3
votes
0
answers
188
views
References and results for the eigenvalues of Ricci tensor
I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
- 1,774
3
votes
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
3
votes
0
answers
151
views
Manifolds with positive sectional curvature almost everywhere
On the paper
Manifolds with positive sectional curvature almost everywhere
Burkhard Wilking asks the following (Question $2$ pg. 121):
Let $(M^n,g)$ be a compact Riemannian manifold with non--...
- 1,774
3
votes
0
answers
68
views
Diffusion generators with gradient vector fields
Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as
$$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$
where $X_0,X_1,...,X_k$ are ...
- 1,675
3
votes
0
answers
105
views
metric with curvature bounded in $L^2$
My question is about the regularity of a metric whose curvature is bounded in $L^2$. Of course, this question doesn't really make sense since the regularity of the metric depends on the coordinates ...
- 914
3
votes
0
answers
348
views
The uniqueness of Poincaré metric
The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance.
...
- 31
3
votes
0
answers
112
views
Is the square root of curl^2-1/2 a natural (Dirac-)operator?
In current computations on a particular $3$-dimensional Riemannian manifold, a first order differential operator $D:\Gamma^\infty(TM,M)\to \Gamma^\infty(TM,M)$ acting on vector fiels shows up, with ...
- 1,942
3
votes
0
answers
74
views
Deforming a non-positively curved Riemannian manifold into a negatively curved one
Cheeger deformations can be used to deform some non-negatively curved Riemannian manifolds into positively curved manifolds (e.g., sectional curvatures strictly positve), see
What is a Cheeger ...
- 1,942
3
votes
0
answers
68
views
Brownian motion on a $\mathbb{Z}$-cover
Let $(M,g)$ a smooth closed Riemannian manifold with non trivial first homology group $H^1(M,\mathbb{R})$. Any element of $H^1(M,\mathbb{Z})$ will define a riemannian $\mathbb{Z}$-cover of $M$ ...
user50806
3
votes
0
answers
563
views
On the Cheeger's estimate of injectivity radius
EDIT: The Cheeger's theorem says that if $M^n$ is a compact smooth Riemannian manifold such that the absolute value of its sectional curvature is less than $\kappa$, diameter at most $D$, and volume ...
- 21.8k
3
votes
0
answers
96
views
Invariant Lagrangians of a connection and its derivatives: how do they look like?
Let
$$
L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma)
$$
be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
- 2,527