# Tagged Questions

**11**

votes

**2**answers

528 views

### A riemannian manifold with finitely many closed contractible geodesics

By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if ...

**3**

votes

**1**answer

196 views

### Holonomy of a Kähler manifold

Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla ...

**6**

votes

**1**answer

280 views

### Fundamental groups of compact manifolds with non-negative Ricci curvature.

I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...

**3**

votes

**1**answer

410 views

### Curvature as metric invariant

This is quite well-known: the ONLY metric invariants are curvature, its higher
derivatives, and any possible contractions between them.
The meaning of an invariant is, to put it simply, a tensor ...

**4**

votes

**1**answer

494 views

### Perelman's example on nonuniqueness of tangent cones at infinity

Perelman has an example on manifolds with nonunique tangent cones at infinity. The paper is here. It is a complete manifold with positive Ricci curvature, Euclidean volume growth, and quadratic ...

**4**

votes

**1**answer

273 views

### Open problems about CMC hypersurfaces with symmetries?

Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...

**8**

votes

**0**answers

201 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 ...

**5**

votes

**3**answers

412 views

### Degeneration of riemannian metrics with curvature bounds

In short, I'm curious to know what modes of degeneration of metric might still keep the curvature bounded. More precisely, assume we are keeping the total volume of the manifold fixed and deform the ...

**3**

votes

**1**answer

189 views

### Star-shaped domain in a space form

Let $M$ be either $\mathbb R^n$, $\mathbb H^n$ or $\mathbb S^n$ and $p\in M$, by a star-shaped domain w.r.t $p$ I mean a connected open subset $\Omega$ in $M$ containing $p$ such that its boundary is ...

**4**

votes

**2**answers

260 views

### Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for ...

**4**

votes

**1**answer

278 views

### Collapsing of Riemannian manifolds with a group action

Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold ...

**4**

votes

**2**answers

416 views

### $J$-holomorphic curve as a minimal surface

The following is a part of the proof of Gromov nonsqueezing theorem.
The existence of a $J$-holomorphic curve gives an upper bound for the radius of a symplectically embedded ball.
Let $\psi: B(r) ...

**8**

votes

**2**answers

505 views

### Almost constant bump function

I ran into the following situation and it turned out to be more subtle than it looked.
I have a complete Riemannian manifold $M$ and my objective is to construct a sequence of functions $f:M \to ...

**1**

vote

**2**answers

292 views

### Complete metric on a Riemann surface with punctures

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?
I know that in this case the universal cover is the hyperbolic plane ...

**1**

vote

**2**answers

290 views

### Volume of a Riemannian manifold and its relation to fundamental group

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem):
If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...

**6**

votes

**3**answers

349 views

### Nearly constant curvature implies “nearly isometric” to a space form?

It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally.
...

**1**

vote

**1**answer

195 views

### Sobolev Norm of distance function on Riemannian manifold

Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and left invariant metric (like ...

**1**

vote

**1**answer

271 views

### Extension of groups in Bieberbach's theorem

I am reading de la Harpe's book "Topics in Geometric Group Theory".
On page 145, there is a theorem:
Let $V$ be a complete $n$-Riemannian manifold with sectional curvature satisfying $K\ge 0$. Then ...

**2**

votes

**1**answer

248 views

### holomorphic extension of forms

hallo,
I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore ...

**2**

votes

**1**answer

138 views

### estimate over simply-connected Riemannian manifold with non-positive sectional curvature

Let$M$ be a Complete simply-connected n-dimensional Riemannian manifold with nonpositive curvature,$\Omega $is a open subset of $M$ ...

**9**

votes

**2**answers

220 views

### Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?

Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...

**5**

votes

**1**answer

286 views

### Minimal distance spheres in complex projective spaces

My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold ...

**3**

votes

**1**answer

594 views

### Totally geodesic submanifold of round sphere

Let $S^n$ be the $n$-dimensional round sphere (i.e. with Riemannian metric of constant curvature +1). Is there any classification result of totally geodesic embedded submanifolds? Are they all round ...

**15**

votes

**0**answers

294 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 ...

**0**

votes

**1**answer

238 views

### G-structures and complete riemannian manifolds

what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a ...

**2**

votes

**1**answer

449 views

### Parallel translation on surfaces

Parallel translation of a vector along a geodesic in a surface is characterized by the following three properties:
The vector being transported moves continuously.
It has constant norm.
It maintains ...

**5**

votes

**1**answer

164 views

### cone angle at infinity for product of cones

Let $A=\lim_{r \rightarrow +\infty} \frac{Vol(B(o,r))}{\omega_{n} r^{n}}$ for any Riemannian manifold $(\mathbb{M}^{n},g)$ with nonnegative Ricci curvature. Here $\omega_{n}$ is the volume of unit ...

**6**

votes

**1**answer

467 views

### Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...

**9**

votes

**1**answer

524 views

### A strange question about closed geodesics on a closed manifold

I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me
to know the answer to the following kinda weird question:
Does there exist a closed Riemannian manifold $M$ ...

**0**

votes

**0**answers

177 views

### Understanding Paul Lévy Theorem [Riemannian Geometry]

In Gromov's paper regarding this proof, he says that by taking the best possible $H$ , we can get the following result:
Let $k>0 , x$ be such that $ 1-x \leq K \int_0^{\infty} J_{k,H} (t) dt $ and ...

**1**

vote

**1**answer

412 views

### Heisenberg group: research themes

I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...

**2**

votes

**2**answers

364 views

### How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...

**4**

votes

**3**answers

434 views

### Manifold-Valued Sobolev Spaces

I have the following basic question about Sobolev-spaces which take their values in a
Riemannian manifold $(M,g)$, i.e.
functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that ...

**5**

votes

**2**answers

554 views

### Kähler potentials that depend only on geodesic distance

Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are ...

**3**

votes

**0**answers

486 views

### Short time existence on Hyperbolic Ricci flow in non-compact case

We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...

**3**

votes

**1**answer

210 views

### estimate of metric tensors in terms of curvatures

I would appreciate if someone knows how to get the following estimates:
Let $\rho_m$ is a sequence of real numbers approaching $\infty$. Consider a sequence of Riemannian metrics $g^{(m)}$ on $S^3$ ...

**2**

votes

**0**answers

132 views

### Deforming isometric embeddings in low codimension

Let $F:M\to \mathbb R^N$ be an embedding. This embedding induces a metric $g_F=dF\cdot dF$ on $M$, that turns $F$ into an isometric embedding. Probably the hardest part of the proof of the Nash ...

**2**

votes

**1**answer

401 views

### Geometric conditions for isoperimetric, Sobolev, Poincar\'e inequalities on a riemannian manifold

By a theorem of Lichnerowicz, on a riemannian manifold $M^{(m)}$ with positive Ricci curvature, the reciprocal of Sobolev constant(ie. the first eigenvalue of laplacian) can be bounded from below by ...

**6**

votes

**2**answers

627 views

### Constant scalar curvature metrics in a conformal class

Let $(M,g)$ be a compact Riemannian manifold, then by the resolved Yamabe-problem, there exists a metric $\tilde{g}$ of constant scalar curvature in the conformal class $[g]$ of $g$. By normalizing ...

**19**

votes

**2**answers

2k views

### Does the curvature determine the metric?

Hello,
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...

**3**

votes

**1**answer

223 views

### Constant Mean Curvature hypersurfaces “condensing” onto a minimal submanifold

Let $M$ be Riemannian manifold and $S\subset M$ a minimal submanifold, with $\dim S<\dim M-1$. According to a few references (e.g., Mahmoudi, Mazzeo & Pacard), it should not be hard to see ...

**2**

votes

**1**answer

607 views

### recognizing Kahler manifolds of complex dimension n

Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...

**7**

votes

**2**answers

637 views

### Tweetable way to see that Willmore energy is Möbius invariant?

Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional
$$\mathcal{W} = \int_M H^2 dA$$
...

**0**

votes

**1**answer

316 views

### embedding torus [closed]

could anyone please help me?
why is it impossible to embed a torus in R^3 with index 1 ( usual euclidean space with index 1 as a semi-riemannian manifold) as a semi-riemannian submanifold?
thanx.
...

**6**

votes

**1**answer

325 views

### Preissmann and Byers Theorems

I'm starting to study at the elementary level the relationship between topology and geometry of a Riemannian manifold of negative curvature. The first two theorems, simple and interesting in this ...

**0**

votes

**1**answer

167 views

### relation with jacobifields in a small neighbourhood

hi,
I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...

**3**

votes

**1**answer

374 views

### Geodesic circles on riemannian manifolds

Can one always find, in a compact riemannian manifold, a closed geodesic isometric to a usual circle when endowed with the ambient distance ? For instance, in the usual flat torus, the only geodesics ...

**6**

votes

**2**answers

373 views

### Metric Deformations from Non-Negative to Positive Curvature

Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) ...

**2**

votes

**1**answer

530 views

### Conformal Killing spinors

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).
If $\epsilon$ and the $\bar{\epsilon}$ ...

**4**

votes

**2**answers

416 views

### Is the exponential map of a $C^{1,1}$ Riemannian metric a local homeomorphism?

Suppose that $g$ is a $C^{1,1}$ (i.e., continuously differentiable with locally Lipschitz first derivative) Riemannian metric on a smooth manifold $M$. It seems to be known that locally the ...