**9**

votes

**1**answer

337 views

### Scalar curvature notion for Cartan connections

In Riemannian geometry, there is a well-known notion of the scalar curvature on a Riemannian manifold $M$, which is a function on $M$ given by a suitable contraction the Riemannian curvature tensor. ...

**6**

votes

**1**answer

419 views

### How submanifolds evolve under Ricci flow?

This may be very naive, since I just started trying to learn Ricci flow; but I couldn't really find any answer after looking for a while in all the textbooks and lecture notes I found online...
...

**3**

votes

**1**answer

243 views

### First eigenvalue of $\Delta$ on Kaehler manifold with $Ricci\ge k$.

Let $M$ be a Kaehler manifold of complex dimension $n$. Let $\Delta$ be the real Laplacian of the underline Riemannian manifold. Let's assume the Ricci curvature of $M$ satisfies $\text {Ric}\ge ...

**3**

votes

**0**answers

150 views

### Is there a way to metricize the notion of $C^\infty$ convergence of pointed Riemannian manifolds?

A sequence of pointed Riemannian manifolds $(M_n,p_n,g_n)$ is said to converge $C^\infty$ to pointed Riemannian manifold $(M,p,g)$ if for each positive radius $R$ there exists sequence of embeddings ...

**7**

votes

**1**answer

228 views

### Smoothing of piecewise Euclidean Riemannian metrics

Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...

**4**

votes

**2**answers

367 views

### What is the Weitzenböck formula for the $\bar\partial$-Laplacian

It is well-known that the Weitzenböck formula for the real Laplacian is
$$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$
where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the ...

**8**

votes

**6**answers

2k views

### Tensor contraction and Covariant Derivative

What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?
In addition, I see that one of the requirements for a covariant ...

**13**

votes

**1**answer

335 views

### Algebraic characterization of the curvature operator of symmetric spaces

My question is the following :
Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the ...

**4**

votes

**2**answers

387 views

### Cutlocus and conjugate points

I am thinking about the following questions about the cutlocus of a point in a Riemannian manifold or of a hypersurface in the Euclidean space:
1) If all the points of the (nonvoid) cutlocus of a ...

**4**

votes

**0**answers

212 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 :
...

**6**

votes

**1**answer

199 views

### Fattening of totally convex sets

Suppose $(M, g)$ is an open complete nonnegatively curved Riemannian manifold with $d$ its distance.
A totally convex set $C\subset M$ has the property that for any two point $x, y \in C$ any ...

**4**

votes

**0**answers

143 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 ...

**1**

vote

**2**answers

220 views

### Examples on small cut radius of totally convex set in non-negatively curved manifold

Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...

**5**

votes

**3**answers

868 views

### Totally Geodesic Submanifolds

Suppose that $N$ is a totally geodesic submanifold of a complete Riemannian manifold $(M,g)$. Is it the case that a geodesic segment that minimizes length in the submanifold $N$ also minimizes length ...

**11**

votes

**1**answer

354 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

289 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

174 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

356 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 ...

**3**

votes

**1**answer

341 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

311 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

191 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

263 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

140 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

181 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

300 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

237 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

188 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

425 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

207 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

342 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

238 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

462 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

552 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

96 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

257 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 $(x,v)$ ...

**10**

votes

**2**answers

851 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

187 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

229 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

441 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

313 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 $G_g$ ...

**4**

votes

**2**answers

366 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

149 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 $\mathbb{R}^n$.
...

**11**

votes

**2**answers

537 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

228 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

286 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

425 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

503 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 ...

**5**

votes

**1**answer

277 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

206 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

428 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 ...