**7**

votes

**1**answer

227 views

### Recovering a smooth manifold from its tensor fields

1) Consider the algebra $\mathcal T (M) = \bigoplus \limits _{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with ...

**3**

votes

**2**answers

171 views

### Upper bound for Willmore energy

Good day to everyone! Does anybody know if there are upper bound estimates for Willmore energy for a given surface?

**1**

vote

**0**answers

112 views

### Harmonic and Primitive Forms on a Kaehler Manifold

Let $M$ be a compact Kaehler manifold, and $p$ a primitive form, which is to say it is contained in the kernel of the adjoint of the Lefschetz operator $L$ associated to the Kaehler form. If $p$ is ...

**3**

votes

**0**answers

83 views

### Differentiability of a map to the free loop space

While reading Morse theory, closed geodesics, and the homology
of free loop spaces, the author claims the following:
Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...

**2**

votes

**1**answer

195 views

### Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...

**0**

votes

**1**answer

151 views

### An example for affine function [closed]

I'm looking for an example of a non-Euclidean non-compact Riemannian manifold $(M,g)$ such that we could define a non-constant affine function $f:M\rightarrow \mathbb{R}$, namely its gradient vector ...

**0**

votes

**0**answers

92 views

### Stability for open manifolds of finite volume under lower Ricci curvature bound

By a theorem of Cheeger-Colding if $N_i\to M$, where $\to$ means Gromov-Haussdorff convergence, $N_i$ and $M$ are of the same dimension, compact and smooth Riemannian manifold with finite volume, as ...

**2**

votes

**1**answer

60 views

### Constant Harmonic Mean surfaces

For surfaces embedded in $\mathbb R^3$ with principal curvatures $ \kappa_1, \kappa_2 $ we know bending/isometric mappings conserve $ K= \kappa_1 \kappa_2 $ and CMC DeLaunay type minimal surfaces ...

**0**

votes

**0**answers

88 views

### Conformally flat with zero scalar curvature 2 [duplicate]

First, i am the one who asked about the existence of compact manifolds of dimension $n\geq 4$ which are conformally flat, non-flat, with zero scalar. Due to the fact that i couldn't comment because i ...

**-1**

votes

**1**answer

123 views

### Are all ideals I in the ring of smooth functions on a compact manifold M equal to a set of smooth functions that vanish in $Z \subset M$?

Since $M$ is compact, we know that maximal ideals are $m_x$, the set of functions vanishing in $ x \in M$. Thus by Zorn's Lemma we also have that $I$ must sit inside such a $m_x$ for some $x \in M$.
...

**7**

votes

**3**answers

324 views

### Conformally flat manifold with zero scalar

I would like to ask the following : Is there any example of a compact conformally flat Riemannian manifold $(M^n,g)$ with $n\geq 4$ which is not flat and has zero scalar curvature?

**0**

votes

**1**answer

69 views

### A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...

**0**

votes

**0**answers

63 views

### Hessian of composite function

Given a (globally, but not necessarily locally) surjective smooth map: $V:\mathbb{R}^n \rightarrow SU(4)$ (with $n >> 4$) and the function $J_{G}: SU(4) \rightarrow \mathbb{R}$ defined by:
...

**2**

votes

**0**answers

103 views

### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...

**3**

votes

**0**answers

145 views

### The projection of density $1$ point on a rectifiable set

I posted this question on MSE but I received no reply. So I repost this here for better luck. Thank you!
Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal ...

**0**

votes

**0**answers

64 views

### Horizontal vector fields and the push forward of the differential of the projection map

I am not very familiar with differential geometry but need to understand some aspects of it for my research. This includes in particular the notion of horizontal vector fields and I would like to ...

**1**

vote

**1**answer

213 views

### Soliton equation and non-killing potential vector field

I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that
$$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$
$$ \frak L_\zeta \rm Ric=\lambda ...

**3**

votes

**1**answer

136 views

### can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly?

Can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly? Do you have any idea of which paper has disscussed this topic? For example, what is the ...

**-1**

votes

**1**answer

115 views

### Tensor bundles as G structures [closed]

For a smooth, real surface $\Sigma$, its bundle of symmetric, bi-linear forms $S^2T\Sigma$ reduced to a $PGL(2,\mathbb{R})$ structure. A similar reduction(with different structure group) can be done ...

**-2**

votes

**0**answers

71 views

### Parallel bi-linear forms on $S^2$

Let $S^2$ denote the sphere endowed with its standard (curvature $k= 1$) metric $g$. How many linearly independent, parallel, symmetric bi-linear forms can we find on $S^2$ ? Clearly the metric $g$ ...

**5**

votes

**1**answer

117 views

### The action of $GL_{\infty}$ on the infinite wedge space

This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...

**0**

votes

**1**answer

153 views

### Triviality of certain vector bundles

Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...

**9**

votes

**1**answer

262 views

### embedding of quaternionic projective spaces

Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...

**2**

votes

**0**answers

77 views

### Heat kernel on manifold with boundary

Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. ...

**2**

votes

**0**answers

44 views

### Are singular critical points isolated for control systems on comapct semi-simple Lie groups

Given a control system on $SU(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the algebra (to ...

**6**

votes

**0**answers

133 views

### Is there any exotic smooth structure on open hyperbolic manifold?

I edited my post to clarify some confusions as suggested by Igor.
Let $M$ be an open hyperbolic manifold, with or without finite volume, Is there any manifold $N$ which is homeomorphic to $M$ but ...

**1**

vote

**0**answers

66 views

### Gaussian heat kernel bounds on Riemannian manifolds

I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$
t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}},
$$
on a ...

**3**

votes

**0**answers

47 views

### Does Alexandrov space satisfy a reverse doubling condition?

Let $X$ be an $n-$dim Alexandrov space with curvature $\geq k$. Does $X$ satisfy a reverse doubling condition? That is, does there exist a constant $C>1$, s.t., for any $x\in X$, ...

**3**

votes

**0**answers

56 views

### Minimal volume of manifold homotopic to a hyperbolic manifold with finite volume

Let $(X, g_0)$ be a $n$-dimensional open manifold with finite volume hyperbolic metric. Suppose $(Y, g)$ is another $n$-dimensional manifold and $f: Y\to X$ a proper degree one map. Then by Storm's ...

**6**

votes

**1**answer

134 views

### Can the potential of a complete Kahler metric be bounded?

Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...

**7**

votes

**1**answer

151 views

### Does $S^n\times H^k$ have non-isometric conformal transformations?

Question: Consider the product Riemannian manifold $(M,g)=(S^n\times H^k,g_{round}\oplus g_{hyp})$ of a round sphere $(S^n,g_{round})$ of curvature $1$ and hyperbolic space $(H^k,g_{hyp})$ of ...

**4**

votes

**0**answers

103 views

### Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
...

**0**

votes

**0**answers

78 views

### Metric calculation from tetrad gives wrong answer

I'm reading the following article by Kinnersley
http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958
and cannot reproduce one (rather trivial) result.
On page 5 of the paper, in ...

**4**

votes

**0**answers

82 views

### Is positively curved Alexandrov surface isometrically embeddable in $\mathbb R^3$?

I guess it is not. The example I have in mind is: $X^2$ is the spherical suspension of a circle $S^1(t)$ of length $0<t<2\pi$. Then $X$ has constant curvature =1 except at two suspension points, ...

**8**

votes

**2**answers

131 views

### On the positive isotropic curvature in higher dimensions

Is it true that any simply connected compact $n$-dimensional Riemannian manifold with positive isotropic curvature is diffeomorphic to the standard sphere $S^n$? I know that it is true for the case ...

**7**

votes

**1**answer

391 views

### How is Ricci flow related to computer graphics?

I recently came across the book Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications by Wei Zeng and Xianfeng David Gu. Because, I just saw the book on the ...

**5**

votes

**1**answer

168 views

### Connection, compatible with type (1, 1) tensor field

I met with the following problem. Consider real manifold $M^{2n}$ with operator field $R$ (that is the tensor field of type $(1,1)$). We are to find a symmetric connection $\Gamma^k_{ij}$ such that ...

**6**

votes

**1**answer

242 views

### Jets in synthetic differential geometry

As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$
where $$D_k(n) = \{(x_1, \ldots, ...

**0**

votes

**0**answers

133 views

### First Chern class of the tautological line bundle over $\mathbb{CP}^n$

I'm trying to understand the following example in which the first Chern class of the tautological line bundle $L^{taut} \to \mathbb{CP}^n$ will be calculated and then it is shown that these bundles ...

**10**

votes

**2**answers

512 views

### Who was the first to discover that the curvature of an embedded surface is the product of the principal curvatures?

The invention of intrinsic differential geometry is usually attributed to Gauss in the context of his theorema egregium but the notion of the curvature of an embedded surface existed before. Who was ...

**2**

votes

**1**answer

107 views

### A question on anti-self-dual Weyl curvature of Kaehler surfaces

It is well known (see Derdzinski) that for a Kaehler metric on a four-manifold, its self-dual Weyl curvature has only two distinct eigenvalues:
$$-\frac{R}{12},\ -\frac{R}{12},\ \frac{R}{6}.$$
I was ...

**5**

votes

**0**answers

168 views

### Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...

**1**

vote

**0**answers

199 views

### Weil-Petersson metric is quasi isometric with which model?

Let $\mathcal M_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a ...

**11**

votes

**1**answer

298 views

### Extrinsic Geometry: Connections

The Idea:
Every vector bundle admits a realization as a subbundle of a trivial bundle of sufficiently-high rank. The trivial bundle has a canonical connection, and this connection induces a unique ...

**1**

vote

**0**answers

58 views

### Isotropically conjugate points in reductive spaces

Is it the case that a naturally reductive Riemannian homogeneous space that contains a pair of conjugate points necessarily contains a pair of isotropically conjugate points? That is, there is a pair ...

**0**

votes

**0**answers

67 views

### Kahlerness of the projectivized cotangent bundle [duplicate]

Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient ...

**8**

votes

**2**answers

464 views

### Vector bundles, finitely generated projective module?

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...

**1**

vote

**1**answer

123 views

### Intuition behind the “Lapse Function”

I came across the following definite of the Lapse Function:
$N=\sqrt{\frac{1}{2}g(L,\overline{L})}$
where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...

**5**

votes

**2**answers

200 views

### Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...

**0**

votes

**1**answer

86 views

### A connection on fibred manifold always exists?

May be $\pi:Y \mapsto X$ a general fibred manifold. Is it true that in fibred manifold a connection always exists? This wiki article states this: ...