**36**

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 ...

**24**

votes

**0**answers

248 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 (...

**23**

votes

**0**answers

1k views

**18**

votes

**0**answers

649 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

342 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

351 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 $\...

**13**

votes

**0**answers

551 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 ...

**12**

votes

**0**answers

209 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 ...

**11**

votes

**0**answers

203 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

374 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 non-...

**9**

votes

**0**answers

241 views

### Are harmonic mappings of Riemannian manifolds always non-singular outside a set of measure zero?

Let $(M,g)$ be an $n$-dimensional, connected, compact, oriented, smooth Riemannian manifold with boundary. Assume we are given an immersion $f \colon M \to \mathbb{R}^n$ (note that $n=\dim M$).
Let $...

**9**

votes

**0**answers

299 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

460 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 divergence-...

**8**

votes

**0**answers

227 views

### A question on a result of Colin de Verdiere

Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdiere (see here) says that if we fix $\gamma$ and select a finite sequence $\...

**8**

votes

**0**answers

136 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 ...

**8**

votes

**0**answers

374 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 ...

**8**

votes

**0**answers

175 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

234 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

102 views

### Alternative definitions of Sobolev spaces on non-compact Riemannian manifolds

SHORT VERSION: Does the Meyers-Serrin theorem hold on complete, non-compact Riemannian manifolds, i.e. $W^{k,p}(M) = H^{k,p}(M)$? My guess is that this holds for the special case $k=1$ (and all $p\geq ...

**7**

votes

**0**answers

468 views

### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...

**7**

votes

**0**answers

835 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 in the book ...

**7**

votes

**0**answers

311 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 ...

**7**

votes

**0**answers

531 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

64 views

### k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.
Question. Are ...

**6**

votes

**0**answers

92 views

### Curvature Characterization of Homogeneous Spaces

A (Riemnnian) symmetric space is locally characterized by the first covariant derivative of its curvature tensor, namely, a manifold $M$ is locally a symmetric space if and only if $\nabla R=0$ (where ...

**6**

votes

**0**answers

147 views

### Is there any exotic smooth structure on open hyperbolic manifold?

I edited my post to clarify some confusions as suggested by Igor.
Let $M$ be an open hyperbolic manifold, with or without finite volume, Is there any manifold $N$ which is homeomorphic to $M$ but ...

**6**

votes

**0**answers

157 views

### $2-$conformal vector fields on Riemannian manifolds

A vector field $\zeta$ is conformal on a Riemannian manifold $(M,g)$ if $$\mathcal L_\zeta g=\rho g$$These vector fields have a well known geometrical interpretation. The flow of a conformal vector ...

**6**

votes

**0**answers

77 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

226 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

211 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

266 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

235 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

215 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 ...

**6**

votes

**0**answers

300 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

470 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 ...

**6**

votes

**0**answers

175 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 ...

**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

42 views

### Why is it easy to compute the first and fourth moments for random chord length in a convex solid?

Recently I was led to some considerations in geometric probability, a field pretty far from any specialization of mine. (Context: I was working with a collaborator on a question about mean escape ...

**5**

votes

**0**answers

63 views

### Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...

**5**

votes

**0**answers

59 views

### Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...

**5**

votes

**0**answers

241 views

### On the curvature tensor with certain conditions

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace $ be a locally orthonormal frame for $M$($3\leq n,m$).
If we suppose the curvature tensor $R$ of $g$ ...

**5**

votes

**0**answers

107 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 $S$ ...

**5**

votes

**0**answers

100 views

### Regularity of the distance from the boundary in singular riemannian manifolds

I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds.
I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...

**5**

votes

**0**answers

192 views

### Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...

**5**

votes

**0**answers

90 views

### On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...

**5**

votes

**0**answers

93 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, x_j)>\...

**5**

votes

**0**answers

138 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:
http://math.stackexchange.com/questions/1356518/...

**5**

votes

**0**answers

151 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 ...

**5**

votes

**0**answers

687 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 ...

**5**

votes

**0**answers

129 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 ...