**2**

votes

**1**answer

143 views

### Derivative of (the length of) the Ricci tensor

I was wondering, have you ever seen a formula in the Riemannian (more specially Kahlerian but not essential) setting for the derivative $X \cdot |Ric|^2 = 2 g(\nabla_X Ric, Ric)$ for a vector field ...

**1**

vote

**0**answers

43 views

### Rational homogenous functions

I'm interested in the set $\mathcal{S}$ of rational functions $F \colon \mathbb{R}^3 \to \mathbb{R}$ verifying:
\begin{align}
\Delta F=0 \quad \text{et} \quad F(\lambda x)= \lambda^d F(x) \quad d \in ...

**6**

votes

**3**answers

200 views

### Is the set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...

**1**

vote

**0**answers

353 views

### Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...

**5**

votes

**2**answers

164 views

### Are ramified covering of negatively curved manifolds negatively curved?

Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, ...

**1**

vote

**0**answers

58 views

### Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
...

**2**

votes

**1**answer

38 views

### Prescribing finitely many unparameterised planar geodesics

Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing ...

**-1**

votes

**2**answers

166 views

### On the definition of convergence of a sequence of sections of a bundle

Convergence of a sequence of sections of a bundle is defined as follows:
Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on ...

**5**

votes

**1**answer

190 views

### Can an open manifold with positive Ricci curvature be non simply connected at infinity?

The question is in the title, I haven't been able to locate a discussion of these kind of properties.

**3**

votes

**1**answer

218 views

### Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...

**11**

votes

**3**answers

245 views

### Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us:
Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...

**2**

votes

**1**answer

208 views

### Curve on a surface defined by its geodesic curvature

Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...

**15**

votes

**1**answer

239 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...

**1**

vote

**1**answer

114 views

### What is “Berger's isembolic inequality”?

Googled the name, but almost all result pointed to Berger's preprint.
Is there any reference for this?

**1**

vote

**1**answer

240 views

### How to understand two examples of spin bundle

I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places:
(1) The spinor bundle $S = S_+ \oplus S_-$ associated to a spin or spinc structure of Riemannian ...

**2**

votes

**0**answers

120 views

### What are Euler density and Weyl invariants?

I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...

**1**

vote

**0**answers

100 views

### Expressing the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary

I want to express the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary. For this I use the Einstein-Hilbert action
$$S(g_{\mu ...

**0**

votes

**1**answer

140 views

### discrete subgroups of the isometries of a product

Suppose $X_1$ and $X_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G_i=Isom(X_i)$. Then $G_1\times G_2\subset Isom(X_1\times X_2)$.
Suppose $X_1\times X_2$ is not compact and ...

**3**

votes

**0**answers

85 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

**1**answer

252 views

### Interpetation of torsion and curvature in terms of families of nearby geodesics

Let $M$ be a Riemannian manifold with affine connection such that the metric is covariantly constant (so that the connection equals the Levi-Civita connection up to torsion).
I know the ...

**0**

votes

**1**answer

152 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

**2**

votes

**0**answers

113 views

### Manifold with a quasi-positive curvature

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature.
But ...

**1**

vote

**0**answers

200 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 ...

**7**

votes

**2**answers

260 views

### Does positively curved sphere admit an isometric embedding as hypersurface in Euclidean space?

Let $(S^n, g)$ be an $n$-dimensional positively curved sphere. Assume the smoothness of the metric, does it admits an isometric embedding into $\mathbb R^{n+1}$?
for $n=2$ it is proved by A.D ...

**3**

votes

**1**answer

324 views

### Reverse Ricci Flow and Longtime Existence

The usual Ricci flow and normalized Ricci flow for surfaces are
$$ \partial_t g = -2Kg $$
and
$$ \partial_t g = -2Kg + 2sg,$$
where $K$ is the Gaussian curvature and $s$ is its average.
The latter ...

**6**

votes

**0**answers

182 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 ...

**10**

votes

**1**answer

484 views

### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

**2**

votes

**0**answers

135 views

### Geometric meaning of a certain form in almost-Kähler geometry

I have difficulties finding an appropriate reference for the following question:
Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of ...

**11**

votes

**1**answer

209 views

### Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...

**1**

vote

**1**answer

148 views

### Regarding Ricci curvature of Markov chains

In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: ...

**4**

votes

**1**answer

304 views

### Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)

Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that.
Cheers

**2**

votes

**1**answer

188 views

### Regularity of metric of the double of a Riemannian manifold

Let $M$ be a Riemannian manifold with totally geodesic boundary $\partial M$. We let $\check{M}$ be its double, i.e. the disjoint union of $M$ with itself under identification of corresponding ...

**-1**

votes

**1**answer

216 views

### Buseman function on manifolds with $Ric \ge - \left( {n - 1} \right)$

It's well known that if M is a Riemannian manifold with $Ric \ge 0$ and contains a line $\gamma $. Set ${\gamma _ + } = \gamma \left| {_{[0, + \infty )}} \right.$, ${\gamma _ - } = \gamma \left| {_{[ ...

**3**

votes

**1**answer

289 views

### Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...

**2**

votes

**1**answer

226 views

### iwaniec's conjecture

Does anyone know whether there is any geometric applications of the iwaniec's conjecture on $ l^p $ bound of beurling alfhors transform( or the complex hilbert transform). One application could have ...

**4**

votes

**3**answers

510 views

### Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...

**6**

votes

**2**answers

179 views

### Isometric embedding as a graph

Question
Let $M$ be a (finite dimensional) smooth manifold and $g,\bar{g}$ be Riemannian metrics on $M$.
Under what conditions can we guarantee that there exists another finite dimensional ...

**4**

votes

**3**answers

438 views

### Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically
immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to ...

**0**

votes

**0**answers

46 views

### volume of a submanifold in a bounded region implies bounds on curvature

I would like to ask the following question: Suppose an $m$-dimensional submanifold in $\mathbf{R}^n$, such that there is a constant $l$, representing the largest number allowing an open normal bundle ...

**0**

votes

**1**answer

140 views

### volume of a submanifold implies bounds on curvature

I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...

**5**

votes

**1**answer

319 views

### Proof of the general expression for anomaly in a CFT and its partition function

I think the statement is that for any dimensional CFT the following is true,
$$\langle T^{\mu}_\mu \rangle = \sum B_n I_n - 2(-1)^{d/2}AE_d,$$
where $E_d$ is the `"Euler density" and $I_n$ are ...

**5**

votes

**1**answer

269 views

### Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?

First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...

**1**

vote

**3**answers

181 views

### Geometric means of matrices beyond the positive definite cone

Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for ...

**5**

votes

**1**answer

185 views

### Voronoi cells and the dual complexes in Riemannian manifolds

I would like to use some "intuitively clear" properties of Voronoi cells in general Riemannian manifolds, but I have trouble finding references.
Let $(X,d)$ be a connected Riemannian manifold and ...

**1**

vote

**1**answer

256 views

### The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...

**2**

votes

**1**answer

115 views

### The measure on the harmonic spectrum from Selberg trace formula

One can see the following two equations,
Theorem 6.1 (Selberg Trace formula) on page 26 of these notes.
Equation 3.19 and 3.20 on page 11 of this paper.
I vaguely feel that these two are the ...

**0**

votes

**0**answers

67 views

### Minkowski functional on the sphere

Let $\mathbb{S}$ be the unit sphere, and let $v_1\in\mathbb{R}^d$ be the north-pole. Let $\mathcal{E}$ be a symmetric convex body (with respect to the geodesics on the sphere) and let the minkowski ...

**0**

votes

**2**answers

287 views

### Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?

**0**

votes

**1**answer

175 views

### Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces

For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...

**4**

votes

**0**answers

195 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 ...