Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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200 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
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1answer
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 ...
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0answers
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 ...
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0answers
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
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2answers
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 ...
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1answer
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
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2answers
201 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 ...
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1answer
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: ...
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164 views

Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type? [closed]

Using Stiefel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?
2
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0answers
98 views

Estimate for the first eigenvalue of the Laplacian

I was studying the paper of S. T. Yau - Seminar on Differential Geometry - and there asks if the first eigenvalue is equal to $ n $, if we have a embedded oriented Riemannian manifold and closed ...
6
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1answer
294 views

Can a complete manifold have an uncountable number of ends?

Let $M$ be a complete and noncompact Riemannian manifold. Fix a point $p$ in $M$. Let $\gamma$: $[0, L]\rightarrow M$ (parametrized by its arc length) be a geodesic starting from $p$. Denote by ...
2
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2answers
140 views

Commutator 2-forms on Lie groups

Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra. For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto ...
4
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2answers
174 views

How many minimal surfaces do we have if the metric in the target space is not flat

It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point, and any two othogonal vectors in this plane, and any ...
4
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1answer
75 views

$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
3
votes
1answer
146 views

Immersing spaces in $\mathbb{R}^{n+1}$, Stiefel-Whitney classes

Where can I find references to proofs/can anyone supply me a quick proof of the following facts? If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$, then each $w_i(M)$ is ...
7
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1answer
100 views

$\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
5
votes
1answer
181 views

Compact Eucledean hypersurfaces with “almost” constant H_k curvature

Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there ...
6
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1answer
326 views

Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
9
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1answer
451 views

What is the spin connection in 9 dimensions as opposed to 5 dimensions?

From Spin Connection in 5 dimensions I can define a massless fermion's covariant derivative on a curved manifold as $$ \nabla_\mu \psi = (\partial_\mu - {i \over 4} \omega_\mu^{ab} \sigma_{ab}) \psi ...
0
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0answers
85 views

Unit sphere of a norm is a submanifold implies the norm is smooth?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map. Suppose the unit sphere of a norm $\| \cdot \|$ is an ...
8
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2answers
289 views

Volume form on a hyperbolic manifold with geodesic boundary

Let $M$ be a compact connected orientable Riemannian $n$-manifold with boundary $\partial M\ne\emptyset$. Since $H^n(M,\mathbb R)=0$, the connecting morphism $\delta: H^{n-1}(\partial M,\mathbb R)\to ...
1
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1answer
81 views

What is the Fano index for Hermitian symmetric spaces of compact type?

As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...
6
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1answer
171 views

Exotic line arrangements

I would like to discuss about the following problem. Hopefully, you will suggest me some ideas and bibliography. At first I provide some basic definitions to set up the notation. Let us consider a ...
4
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0answers
166 views

Lie group structure over diffeomorphisms group

Let $M$ be a smooth manifold. Is it true that for every subgroup $G$ of $diff(M)$ there is at most one Lie group structure on $G$ such that the natural left $G$-action on $M$ is smooth? Edit: by "Lie ...
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3answers
938 views

Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

A particle physicist asked me the above question. Let me try to make it more precise. Suppose $M$ is a 3-dimensional Calabi-Yau manifold: that is, a compact Kähler manifold of complex dimension ...
2
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2answers
205 views

How to prove Liouville measure is invariant under geodesic flow?

Let $M$ be a complete n dimensional Riemannian manifold. $vol$ denotes the n dimensional Hausdorff measure. Let $$ SM=\{(x,v)|x\in M, v\in T_xM, \|v\|=1\} $$ be the unit tangent bundle of $M$. Then ...
4
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1answer
152 views

Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?

The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix $$ \left\|\frac{\partial^2u}{\partial ...
3
votes
1answer
137 views

Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem: Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
3
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0answers
85 views

Growth of norm of curvature under direct sum or existence of universal connection

For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by: $$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ...
3
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1answer
284 views

Tangent bundle of $S^2 \times S^1$ trivial or not [closed]

Is the tangent bundle of $S^2 \times S^1$ trivial or not?
6
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0answers
143 views

$2-$conformal vector fields on Riemannian manifolds

A vector field $\zeta$ is conformal on a Riemannian manifold $(M,g)$ if $$\mathcal L_\zeta g=\rho g$$These vector fields have a well known geometrical interpretation. The flow of a conformal vector ...
5
votes
2answers
238 views

3-sphere bundles over 4-sphere bound smooth disc bundles

I saw in the answer of this post: Is it true that all sphere bundles are boundaries of disk bundles? that a $S^3$-bundle over $S^4$ bounds a disc bundle over $S^4$ iff $O(4)\rightarrow Diff(S^3)$ is ...
3
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0answers
83 views

Spectral theory of differential forms over a circle bundle

Here is the set up : I consider the unitary tangent bundle of a surface $(S,g)$ endowed with the Sasakian metric ; $(T^1S, g_s)$, in fact we have the following fibration : \begin{equation*} ...
0
votes
2answers
141 views

Diameter of immersed surfaces with bounded from above mean curvature

Is the following true? I cannot see a counterexample and it seems very intuitively clear, at least in the embedded case. Claim: Consider the set $S$ of closed immersed Riemann surfaces $\Sigma ...
1
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1answer
159 views

Ricci flow on Kähler manifold

Knowing the Ricci flow on Riemann surfaces, see e.g. Ricci flow on Riemann surfaces How could we write the Ricci flow on Kähler manifold? Thanks for the reply!
7
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2answers
257 views

Spectrum of Ring of Smooth Functions on $\mathbb{R}^n$

When we define smooth manifold, we starting with topological space $M$ which localy homeomorphic to $\mathbb{R}^n$ and setting up sheaf $\mathscr{F}(M)$ of functions on it which localy isomorphic to ...
5
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1answer
126 views

geometric interpretation of derivation between two algebras

Given a smooth manifold $M,$ it is well known that any derivation of the algebra of smooth functions $C^{\infty}(M)$ can be seen (or it is associated to) a smooth vector field on $M.$ I am looking for ...
2
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0answers
184 views

Some examples of non trivial principal bundles

1.Is there a nontrivial pricipal bundle $P(M,G)$, with $G$ connected, such that the total space $P$ admit a foliation such that each leaf is diffeomorphic to $M$(Not necessarily via projection ...
18
votes
3answers
602 views

All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open: Given ...
2
votes
3answers
254 views

Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this Theorem ? Let ...
1
vote
1answer
138 views

Schauder estimate for the heat equation on compact manifolds

I asked this question on math.stackexchange.com, however I didn't get any answers so I'll try it here. Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial_tu-\Delta u$. Let $f\in ...
2
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0answers
122 views

Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold

$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
4
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0answers
143 views

Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$). But what do we know about ...
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66 views

conformal deformation with fixed boundaries

For a flat plane with certain boundary, e.g., a rectangular patch, is it possible to conformally displace or deform such patch to a curved bump with exact same boundary? In this thesis, Dr. Keenan ...
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0answers
144 views

Shape-related vector fields

Assume that $M$ is a surface in $\mathbb{R}^{3}$. We denote its shape operator by $S$. A vector field $X$ is shape related to $Y$ if $S(X)=Y$. (of course it is not an equivalent relation). ...
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0answers
36 views

Relation between Aitchison Distance on a Simplex and Geodesic distance on the multinomial manifold [closed]

I am trying to understand the difference/relation between the Aitchison distance on a simplex $$\left[ \sum^D_{k=1} (\log{\frac{x_{ik}}{g(\mathbf{x}_i)}} - \log{\frac{x_{jk}}{g(\mathbf{x}_j)}})^2 ...
2
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0answers
161 views

Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?

It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers My question is about the moduli space of ...
0
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0answers
79 views

Divergence free vector field on compact surface

I get a free divergence field $X$ on a compact surface $(\Sigma, g)$ and I would like to integrate it. On the sphere $X=\nabla^\bot f$ since the spher is simply connected.($\nabla^\bot =J\circ ...
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0answers
70 views

a question on warped product

This question is on J.Cheeger and Tobias H.Colding's paper "lower bound on Ricci curvature and the almost rigidity of warped products". For a warped product $M=(a,b)\times_f N^{n-1}$ with metric $g$. ...
3
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1answer
130 views

A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...