**31**

votes

**0**answers

2k views

### Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...

**21**

votes

**0**answers

1k views

**19**

votes

**0**answers

186 views

### Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...

**17**

votes

**0**answers

619 views

### Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...

**16**

votes

**0**answers

328 views

### Negative Einstein manifolds

In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values.
For ...

**13**

votes

**0**answers

519 views

### Applications of Berger's Curvature Estimate

I'm interested in applications of the following estimate of Berger on the Riemann curvature tensor:
Let $(M,g)$ be a Riemannian manifold of dimension $n \geq 4$, let $p \in M$, and assume that the ...

**11**

votes

**0**answers

190 views

### A variation on the local Günther inequality

This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a ...

**11**

votes

**0**answers

150 views

### Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...

**10**

votes

**0**answers

278 views

### Vector field built from connection and metric

Consider a smooth finite-dimensional manifold $M$ with metric $g$ and connection $\nabla$. For some local coordinate system, denote by $g^{\alpha \beta}$ the inverse of the metric tensor and by ...

**10**

votes

**0**answers

358 views

### Best metrics on exotic R^4

What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have ...

**9**

votes

**0**answers

280 views

### Is it overkill to invoke Kirszbraun theorem to prove the following fact ?

Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...

**9**

votes

**0**answers

435 views

### Killing spinors and symmetric tensor fields.

Hi all,
I have a question of the following form: Let $(M,g)$ be a Riemannian spin manifold which admits a Killing spinor $\sigma$ and let $h:T M \to T M$ be a symmetric, trace-free and ...

**8**

votes

**0**answers

137 views

### Closed geodesics avoiding points in hyperbolic surfaces

Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...

**8**

votes

**0**answers

299 views

### Reference - Asymptotic geodesics on compact surfaces without conjugate points

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...

**8**

votes

**0**answers

166 views

### Tangent space, metrics etc. on simplicial sets

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...

**8**

votes

**0**answers

224 views

### Exhaustion of an open manifold of bounded curvature and finite volume

In the Cheeger-Gromov paper "On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume",
http://www.maths.ed.ac.uk/~aar/papers/cheegergr1.pdf,
the authors make the ...

**7**

votes

**0**answers

67 views

### configuration space of Riemannian manifolds with a parameter on the distance of distinct points

Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, ...

**7**

votes

**0**answers

285 views

### Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...

**7**

votes

**0**answers

520 views

### Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...

**6**

votes

**0**answers

65 views

### local definability of geodesics in an o-minimal structure

Let $R$ be an o-minimal expansion of the reals, and let $(M,g)$ be a Riemannian manifold, such that $M$ and $g$ are definable in $R$. Let $\gamma: [0,1] \to M$ be a geodesic, i.e. a curve such that ...

**6**

votes

**0**answers

127 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 ...

**6**

votes

**0**answers

170 views

### Smooth morse theory of Riemannian distance functions

Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, ...

**6**

votes

**0**answers

251 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 ...

**6**

votes

**0**answers

228 views

### Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...

**6**

votes

**0**answers

205 views

### Different complexifications of a real analytic Riemannian manifold

Hi,
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...

**6**

votes

**0**answers

289 views

### Compactness of solutions to parabolic equations (parabolic regularity)

I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.
For each $s>0$, I have a ...

**6**

votes

**0**answers

1k views

### Geometric Proof that Fubini-Study Metric is Round

The Fubini-Study metric d(x,y) on $CP^1$ is defined as follows: for x and y in $CP^1$ let v and w be unit vectors in $C^2$ representing x and y. Then $d(x,y)=2arccos(\langle v,w\rangle)$. The round ...

**5**

votes

**0**answers

119 views

### Nice applications of Liouville's theorem

I need a big list of nice-looking and simple applications of Liouville's theorem on geodesic flow in Riemannian geometry.
Please help.
Examples:
A Riemannian manifold with finite volume does not ...

**5**

votes

**0**answers

76 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...

**5**

votes

**0**answers

173 views

### average Riemannian distance between Identiity and a random point in SO(n) or SU(n)

I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...

**4**

votes

**0**answers

134 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 ...

**4**

votes

**0**answers

33 views

### Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action

Given a Riemannian manifold $M$ and a group of isometries $G$ of $M$, I am interested in when there exists a isometric embedding $\iota : M \to H$, where $H$ is a Hilbert space and a representation ...

**4**

votes

**0**answers

50 views

### symmetric points on symmetric spaces

Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on ...

**4**

votes

**0**answers

98 views

### Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...

**4**

votes

**0**answers

107 views

### Difference between parallel transport composed with exponential maps along two different geodesics starting at the same point?

I asked this question on math.stackexchange too: it's not a homework problem, but something that came to my mind while thinking of commutation:
...

**4**

votes

**0**answers

101 views

### Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...

**4**

votes

**0**answers

161 views

### Curvature estimates for Kaehler-Einstein and Hermitian, Einstein four-manifolds

Four-dimensional Kaehler-Einstein manifolds and Hermitian, Einstein manifolds with positive scalar curvature have been classified by Tian and LeBrun respectively. I was wondering are there any ...

**4**

votes

**0**answers

116 views

### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...

**4**

votes

**0**answers

57 views

### Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions.
By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...

**4**

votes

**0**answers

286 views

### Isometries of hyper-Kähler manifolds

For the purposes of this question, a hyper-Kähler manifold will be a complete connected Riemannian manifold $(\mathcal{M},g)$ whose holonomy representation is isomorphic to the natural representation ...

**4**

votes

**0**answers

92 views

### Generalized Hawking Mass

This is a fairly general question. Let $(M^3,g)$ be a Riemannian 3-manifold. Let $\Sigma^2$ be a dimension-2 submanifold of $M$. The Hawking mass of $\Sigma^2$ is defined as
$m(\Sigma^2) := ...

**4**

votes

**0**answers

106 views

### tangent developable surface in $\mathbb{R}^3$

Let $C$ be a regular curve embedded in $\mathbb{R}^3$ (i.e. a real 1-dimensional manifold embedded in $\mathbb{R}^3$). Let $S$ be the union of its affine tangent lines:
$$S=\bigcup\limits_{p\in ...

**4**

votes

**0**answers

523 views

### Second derivative of Riemannian Exponential Map

Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$.
The derivative of the exponential map can be expressed in ...

**4**

votes

**0**answers

116 views

### The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...

**4**

votes

**0**answers

243 views

### non compact riemannian manifolds

According to Gromoll and Meyer:
Let M be a complete non-compact Riemannian manifold of positive sectional curvature. Then M is diffeomorphic to $\mathbb{R}^n$.
Thus, I think to classify ...

**4**

votes

**0**answers

272 views

### Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements
$\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...

**4**

votes

**0**answers

178 views

### Gromov-Haussdorf and Lipschitz convergence of a non-collapsing sequence of manifolds with Ricci curvature bounded below

There is a theorem from Cheeger-Colding saying the following:
Let $n$ be an integer. If a sequence of $n$-dimensional Riemannian manifolds $(M_i,g_i)$ converges with respect to the Gromov-Hausdorff ...

**4**

votes

**0**answers

254 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**

votes

**0**answers

148 views

### Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?

Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...

**4**

votes

**0**answers

807 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 ...