**9**

votes

**1**answer

316 views

### Length spectrum for Riemannian metrics in the projective plane

Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum?
This question is related to MO questions Length spectrum and Zoll surfaces ...

**3**

votes

**1**answer

208 views

### Length spectrum and Zoll surfaces of revolution

The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces,
all of whose ...

**9**

votes

**0**answers

254 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 ...

**7**

votes

**1**answer

654 views

### Integration By Parts on Non-compact Manifolds

This is undoubtedly a very easy question, but perhaps there are some subtleties. Under what circumstances can we integrate by parts over a non-compact Riemannian manifold? I am aware that having ...

**7**

votes

**2**answers

471 views

### Full isometry groups of Stiefel and Grassmann manifolds

Hi,
I'm looking for a reference for the full isometry groups of the
(i) complex Stiefel manifolds $U(m)/U(m-l)$, either for the Euclidean metric (i.e. identifying it with orthonormal $m \times ...

**4**

votes

**2**answers

335 views

### Conformal structure does not see conical singularities

the conformal structure does not see the conical singularities of a polyhedral surface.
This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli).
The sentiment is ...

**10**

votes

**0**answers

329 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 ...

**1**

vote

**0**answers

230 views

### What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise:
Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...

**10**

votes

**3**answers

676 views

### Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?

I do not know if this question is appropriate for this site, but I posted here without having answers, so now I make this attempt.
Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the ...

**4**

votes

**1**answer

806 views

### Fubini Study Metric and Einstein constant

Hi all,
it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold?
Moreover, I would ...

**0**

votes

**0**answers

246 views

### einstein metrics on the tangent bundle

hi,
i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ?
marco

**3**

votes

**0**answers

268 views

### Cheeger's Finiteness Theorem and Lipschitz Constant

Cheeger's Finiteness Theorem states that
For each positive numbers $D,v,n$, the
number of diffeomorphism classes of Riemannian manifolds $M$ with
$Diameter(M)\le D$, $Vol(M)\ge v$, and $|K(M)|\le 1$ ...

**3**

votes

**0**answers

183 views

### Seek “typical examples” for the structure of spaces with two-sided Ricci bounds

By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds ...

**3**

votes

**4**answers

1k views

### space of geodesics

hallo,
i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | ...

**4**

votes

**1**answer

471 views

### How the Jacobi metrics may be useful in mechanics with or without constraints?

A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$
If ...

**5**

votes

**1**answer

463 views

### Partitions of Unity

Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has ...

**4**

votes

**2**answers

436 views

### Which vector bundle are the Christoffel symbols sections of?

The collection of Christoffel symbols $\Gamma_{ij}^k$ of a connection (or of a metric) on a smooth manifold $M$ is not the collection of components of a tensor field in some local chart, i.e. they ...

**4**

votes

**1**answer

211 views

### Symmetries vs. Bound in codimension of Nash isometric embedding

Let $(M^m,g)$ be a compact smooth Riemannian manifold of dimension $m$. From the celebrated Nash Embedding Theorem, we know there exists a (smooth) isometric embedding $M\hookrightarrow\mathbb R^n$ on ...

**2**

votes

**2**answers

732 views

### Metric Connections on a Lie Group

A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...

**7**

votes

**2**answers

860 views

### Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold

Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a ...

**6**

votes

**0**answers

265 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 ...

**3**

votes

**1**answer

326 views

### eigenspinors of Dirac operator

$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...

**8**

votes

**4**answers

736 views

### Riemannian metric on a flag variety

$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = ...

**3**

votes

**0**answers

167 views

### Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds

I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...

**6**

votes

**3**answers

280 views

### Large geodesically convex subsets of tori

Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is geodesically convex if for any $x,y\in E$ the shortest geodesic connecting $x$ and $y$ lies in ...

**10**

votes

**2**answers

516 views

### Behavior of sectional curvature under metric deformations

Metric deformation:
Let $(M,g_0)$ be a Riemannian manifold and consider a (sufficiently smooth) deformation of $g_0$, $$g_t=g_0+th+O(t^2), \quad 0< t<\varepsilon $$ where $h$ is some symmetric ...

**2**

votes

**1**answer

306 views

### Conformally-flat

Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$.
Is there a way to know if this is always a non-positive (sectional) curvature manifold?
Note ...

**5**

votes

**1**answer

918 views

### Good Surface,Bad Surface-Surface classification

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...

**1**

vote

**2**answers

341 views

### intersection of geodesiques

Let $(M,g)$ be a closed riemannian surface . let $\alpha$ be a simple closed geodesique . does there is exist a simple closed geodesic $\beta$ that intersect alpha at only 1 point p such that ...

**4**

votes

**5**answers

501 views

### Exponential and Logarithm Mapping on Stiefel Manifold

The Stiefel Manifold is defined as
$$
\mathrm{St}(p,n):= \{ X\in \mathbb{R}^{n\times p} :\ X^T X = I_p \}.
$$
Recall that the tangent space at a point $X\in \mathrm{St}(p,n)$ is given by
$$
...

**3**

votes

**0**answers

107 views

### rigidity of eigenvalues of circular ensemble

Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...

**10**

votes

**3**answers

497 views

### volume of compact simple Lie groups under the natural Euclidean embedding

I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I ...

**4**

votes

**1**answer

617 views

### Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?

As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,
A complete connected Riemannian manifold ...

**3**

votes

**1**answer

283 views

### extended forms from foliations [closed]

hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...

**3**

votes

**2**answers

197 views

### non commutative elements in the fundamental group of a closed hyperbolic surface

Let $(M,hyp)$ be a closed hyperbolic surface. fix a point $m$ in $M$ and denote by $G=\pi_1(M,m) $.
now let $\alpha$ and $\beta$ in $G$ such that $\alpha$ and $\beta$ does not commute . my first ...

**9**

votes

**2**answers

659 views

### Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...

**9**

votes

**0**answers

385 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 ...

**1**

vote

**1**answer

102 views

### Minimal representative of the elements of the fundamental group of a negatively curved manifold

Let (M,g) be a negatively curved manifold , let p be any point of M and denote by G=π1(M,p) . the minimal representative (by minimal i mean the smallest length representative ) of every α in G is a ...

**10**

votes

**1**answer

953 views

### Theorem of Bryant in higher dimensions

hallo,
i have the following question. i read about Bryant's theorem which sais that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically ...

**1**

vote

**2**answers

461 views

### Riemann surfaces with bounded curvature

Say there are metrics $g_n$ on a compact Riemann surface $\Sigma$ with bounded curvature and bounded area, or even with the same area element . What can we say about the 'limit' of $(\Sigma, g_n)$? ...

**0**

votes

**2**answers

294 views

### norm of n-th covariant derivative of smooth function

The question is how define the norm of n-th covariant derivative of smooth function f on a manifold M. The manifold is two dimensional so maybe I can do it in the following way: thing about n-th ...

**23**

votes

**4**answers

2k views

### What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...

**2**

votes

**1**answer

251 views

### The Tubular Neighborhood of a Closed Geodesic

Suppose $M_{g}$ is the mapping torus $\Sigma_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma_{g} \to \Sigma_{g}$ is an ...

**2**

votes

**1**answer

572 views

### Isometry groups of Riemannian submersions with totally geodesic fibers

Suppose $F\to M\stackrel{\pi}{\to} B$ is a Riemannian submersion with totally geodesic fibers, all manifolds compact. In general, unless $M=B\times F$ is a Riemannian product, the isometry groups of ...

**4**

votes

**1**answer

236 views

### Dimension of certain subgroup of isometry group of positively curved manifold

Let $M$ be a closed $n$-dimensional Riemannian manifold with positive sectional curvature.
Let $G$ be a close subgroup of isometry group ${\rm Iso}(M)$. Suppose the action of $G$ on $M$ is not ...

**18**

votes

**2**answers

1k views

### Area of distance sphere in manifold with Ricci $\ge 0$.

Let $M$ be a open complete manifold with Ricci curvature $\ge 0$.
By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear.
I am wondering whether the following statement is ...

**10**

votes

**3**answers

616 views

### Why the quarter in the $\frac{1}{4}$-pinched sphere theorem?

Is there any hope of a high-level explanation of why the fraction $\frac{1}{4}$
plays such a prominent role as a
sectional curvature
bound in Riemannian geometry?
My (dim) understanding is that the ...

**2**

votes

**1**answer

236 views

### Minimum set of subharmonic function in $\mathbb R^n$

Let $f :\mathbb R^n\to \mathbb [0, \infty)$ be a (continuous, $C^2$, or smooth) subharmonic function with minimum value $0$. Then we know the sublevel set $f^{-1}((-\infty, c])$ is mean convex for $c ...

**5**

votes

**2**answers

450 views

### Example for Busemann function is not an exhaustion when Ricci $\ge 0$

For an open complete Riemannian manifold $M$ with non-negative sectional curvature, the Busemann function defined below is a convex exhaustion function (by Cheeger-Gromoll's proof of soul theorem)
...

**18**

votes

**1**answer

604 views

### Is the following a sufficient condition for asphericity?

I recently came across the following question while working on some problems on manifolds with lower Ricci curvature bounds.
Given $n$ does there exist a large $R>0$ with the following property:
...