All Questions
51 questions with no upvoted or accepted answers
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
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
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
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
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
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
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
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
347
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
256
views
Uniqueness of scalar curvature
I'm reading Gromov's notes
http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf
and at page 7 they say that there is a unique second order differential operator $S$ from the space of Riemannian ...
- 585
2
votes
0
answers
113
views
What is known about warped product metrics satisfying conditions more general than conformal flatness?
In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...
- 659
2
votes
0
answers
664
views
Reference request - Texts on geometric analysis with exercises
I’ve recently been studying some Riemannian geometry and geometric analysis, however I have found it difficult to find resources with exercises to practice. It seems that many textbooks past the ...
2
votes
0
answers
127
views
Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?
Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$.
Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric ...
- 969
2
votes
0
answers
65
views
Where can I find a proof of the main properties of Weyl Curvature for semi-Riemannian manifolds?
Most of the references I've seen deal with Riemannian geometry, rather than semi-Riemannian geometry. Chens monograph, Pseudo-Riemannian Geometry, $\Delta$-Invariants and Applications is one of the ...
- 2,369
2
votes
0
answers
141
views
For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, $...
- 969
2
votes
0
answers
67
views
Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
- 5,405
2
votes
0
answers
134
views
Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$
Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property:
For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
- 965
2
votes
0
answers
66
views
One-parameter group of nonvanishing vector field
Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
- 1,915
2
votes
0
answers
195
views
Reference for connection of a Hessian metric
Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a ...
- 1,163
2
votes
0
answers
305
views
"Riemannian" collar theorem
Let $(M,g)$ be a compact manifold of dimension $n$ with boundary. If $\partial M$ is smooth then one has a control on the determinant of the Jacobian of the diffeomorphism in the collar theorem, i.e....
- 143
2
votes
0
answers
127
views
Functional inequality under mean curvature flow
Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
- 143
2
votes
0
answers
123
views
Mean value operator on Riemannian manifold
Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...
- 21
2
votes
0
answers
168
views
Sources on evolution of submanifolds subject to Ricci flow
I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...
- 133
1
vote
0
answers
122
views
Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
- 111
1
vote
0
answers
138
views
References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
- 11
1
vote
0
answers
213
views
Injectivity radius bounds for Riemannian manifolds of low regularity
In their seminal paper, Jeff Cheeger, Mikhail Gromov, and Michael Taylor derivated bounds on the injectivity radius of Riemannian manifolds with bounded sectional curvature of the form:
$
inj(p)\geq r ...
1
vote
0
answers
159
views
The Laplacian of a tubular neighborhood
Let $(M,g_M)$ be a compact submanifold of $\mathbb{R}^n$. Are there any known results relating the spectrum of the Laplace-Beltrami operator of M to the spectrum of the Laplace-Beltrami operator of a ...
- 409
1
vote
0
answers
272
views
A cohomology associated to a Riemannian manifold
Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(...
- 356
1
vote
0
answers
162
views
Gromov-Hausdorff relative compactness without curvature restrictions
A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
- 1,407
1
vote
0
answers
133
views
Reference for example of gradient steady Ricci solitons
Recently I read a paper about Ricci solitons. I quote a paragraph of it here:
In dimension three, the classification of complete gradient steady Ricci solitons is still open. Known examples are ...
- 4,195
1
vote
0
answers
156
views
Proof of Berger in "Sur les variétés d’Einstein compactes"
I would appreciate any reference that contains either a translation or proof of the following interesting observation of Berger (Sur les variétés d’Einstein compactes, M. Berger paper (in French)).
...
- 4,195
1
vote
0
answers
82
views
Scattering in (pseudo-)Riemannian spaces
I will ask my question in a broad way, leaving a lot of freedom for answers.
Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
- 1,461
1
vote
0
answers
463
views
Reference request for parallel transport
I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
- 119
0
votes
0
answers
32
views
Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
0
votes
0
answers
425
views
Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405