**10**

votes

**1**answer

307 views

### Is a manifold with flat ends of bounded geometry?

A Riemannian manifold $(M,g)$ is said to have flat ends if the curvature tensor of $g$ vanishes outside a compact set $K$. I was wondering if such manifolds are of bounded geometry. Recall that a ...

**3**

votes

**2**answers

276 views

### Uniqueness of Kähler form with same volume

Hallo,
Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure ...

**3**

votes

**0**answers

169 views

### What is known about analogous results of Kazdan and Warner in higher dimensions?

First let me state a Theorem due to Kazdan and Warner:
``Let M be a compact two dimensional orientable manifold. Let
$f: M \rightarrow \mathbb{R}$ be a function that has the same
sign as ...

**8**

votes

**3**answers

348 views

### twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors

Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable ...

**2**

votes

**1**answer

274 views

### Grassmannian of oriented real $k$-planes

The Grassmann manifold $\widetilde{Gr}(k,\Bbb{R}^n)$ of oriented $k$-planes in $\Bbb{R}^n$ is a double cover of the Grassmann manifold $Gr(k,\Bbb{R}^n)$ of non-oriented $k$-planes. We can give ...

**4**

votes

**1**answer

288 views

### Rigorous solution to Ricci Flow on dumbbell $S^3$

To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering ...

**6**

votes

**1**answer

186 views

### Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section.
Recall that an ...

**8**

votes

**1**answer

245 views

### Discretization of a complete manifold

Suppose $M$ is a complete Riemannian manifold with very large injectivity radius (say larger than $100$) and $\left\lbrace x_i: i \in I\right\rbrace$ is a maximal $1$-separated subset of $M$.
Is ...

**1**

vote

**1**answer

126 views

### Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold

Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...

**6**

votes

**0**answers

152 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 ...

**3**

votes

**3**answers

293 views

### Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold?

In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T_cM$ is a set of ...

**3**

votes

**2**answers

232 views

### Real analytic submanifolds of $\mathbb{R}^{n}$

Hallo,
Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...

**2**

votes

**1**answer

177 views

### Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$

Greetings,
Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...

**5**

votes

**1**answer

406 views

### What are the Dirac operators on $S^1$?

This is crossposted at stack exchange as http://math.stackexchange.com/questions/248391/dirac-operators-on-s1.
I am trying to understand the Dirac operators associated to the 2 spinor bundles on ...

**1**

vote

**1**answer

180 views

### Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

Hallo,
It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...

**8**

votes

**0**answers

302 views

### “Homogeneity” of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...

**4**

votes

**1**answer

207 views

### Are there countably many diffeomorphism classes of finite radius balls of complete Riemannian manifolds?

Suppose $M$ is a smooth complete Riemannian manifold and $x$ is a point in $M$. For any positive radius $r$ we consider the open ball $B(x,r)$ centered at $x$ with radius $r$.
If we ignore the ...

**7**

votes

**2**answers

407 views

### What are first eigenfunctions of Laplacian for $CP^n$ with Fubini-Study metric?

I know the round $n$-sphere has $f_i=\cos(dist(e_i, x))$ as the set of first eigenfunctions for $e_i=(0, \cdots, 1, \cdots, 0)\in \mathbb R^{n+1}$. i.e. $\Delta f_i=\lambda_1 f$, where $\lambda_1$ is ...

**2**

votes

**2**answers

535 views

### Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry"， an English version ...

**1**

vote

**0**answers

92 views

### About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...

**1**

vote

**1**answer

234 views

### Berger's theorem on Riemannian holonomy applied to the orthogonal frame bundle.

Let $M$ be a compact Riemannian manifold and $TM$ be its tangent bundle. Given a initial point-vector $(x,v) \in TM$ and a curve $\alpha:[0,1] \to M$ starting at $x$ we can parallel transport $(x,v)$ ...

**9**

votes

**1**answer

679 views

### Intuition for Levi-Civita connection?

Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or ...

**6**

votes

**1**answer

176 views

### volume of exceptional group orbits

Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by ...

**0**

votes

**1**answer

220 views

### Polarisation in a nighbourhood of a Lagrangian submanifold

Hallo,
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a Lagrangian submanifold such ...

**3**

votes

**2**answers

428 views

### Rotation in Hyperkähler manifolds

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...

**6**

votes

**1**answer

307 views

### Green functions on Riemann surfaces

Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...

**0**

votes

**0**answers

113 views

### Relation between Adpted Complex Structure and Hyperkaehler Structure

Hallo,
I am reading the paper "Hyperkaehler structures on total spaces of holomorphic cotangent bundles" by Kaledin where he puts a hyperkähler structure on a neigbourhood of the $0$-section in the ...

**4**

votes

**2**answers

352 views

### Do transvers foliations induce complex structure?

Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...

**1**

vote

**0**answers

136 views

### Differentiation of Logarithm Map in Riemannian Geometry

I have a simple question regarding the differentiation of the logarithm mapping in Riemannian manifolds:
Assume that $M$ is a compact Riemannian manifold, isometrically embedded into $\mathbb{R}^n$.
...

**10**

votes

**2**answers

507 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

181 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

267 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

377 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

475 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

263 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

183 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 ...

**4**

votes

**3**answers

379 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 ...

**2**

votes

**1**answer

163 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

249 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

272 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

381 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

470 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

258 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

283 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

337 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

181 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

255 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

243 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

135 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

212 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 ...