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

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0
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0answers
35 views

Existence of local frames in Hilbert manifolds

Thank you for looking at my queation. I'm studying about Hilbert manifolds. I would like to ask you if there exists some concept like a local frame for every Hilbert manifold. I'm refering to the ...
13
votes
2answers
429 views

How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
10
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0answers
150 views

A simple proof that parallelizable oriented closed manifolds are oriented boundary?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
1
vote
1answer
88 views

Cut locus, conjugate points and smoothness of distance function

I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance ...
2
votes
1answer
163 views

almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
3
votes
1answer
176 views

Newlander-Nirenberg in dimension 2

What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions ...
7
votes
1answer
203 views

Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded Lie algebra" as explained first in ...
0
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0answers
27 views

When is Dirichlet solution from disk to Riemannian manifold Holder continuous near the boundary?

$D$ is the two-dimensional unit disk. $X$ is a compact Riemannian manifold. Let$\phi \in W^{1,2}(D,X)$ and define $$ W^{1,2}_{\phi}=\{v \in W^{1,2}(D,X):Trace(v)=Trace(\phi)\} $$ Let $$ ...
2
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0answers
77 views

Laplacian on manifolds with corners

So far I've studied smooth riemannian manifolds and the Laplace-Beltrami operator associated to it. Therefore I know basic theorems like Stoke's theorem, divergence theorem, green's identities etc. ...
11
votes
3answers
662 views

Sobolev spaces and geometry

This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study ...
1
vote
1answer
145 views

The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...
1
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0answers
81 views

Euler class and self-intersection number of a surface in a 4-manifold [duplicate]

In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that For a compact oriented surface $X$ in a 4-dimensional oriented ...
2
votes
1answer
156 views

Chern class of Hopf fibration over elliptic curve

Let $N = \{ (z_0,z_1,z_2) \in S^5 \mid z_0^3+z_1^3+z_2^3 = 0 \}$, where we consider $S^5\subset\mathbb{C}^3$. The circle $U(1)$ acts on $N$ by $$e^{i\theta} \cdot (z_0,z_1,z_2) = ...
1
vote
0answers
66 views

Boundary of fibers of submersions

Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result: ...
2
votes
2answers
147 views

Does $(M\times M, \omega\times -\omega)$ admit a Lagrangian fibration?

Let $(M,\omega)$ be a manifold endowed with symplectic form. Then the product manifold $M\times M$ with symplectic form $\omega\times -\omega$ is symplectic, and the diagonal submanifold ...
3
votes
0answers
62 views

tangent developable surface in $\mathbb{R}^3$

Let $C$ be a regular curve embedded in $\mathbb{R}^3$ (i.e. a real 1-dimensional manifold embedded in $\mathbb{R}^3$). Let $S$ be the union of its affine tangent lines: $$S=\bigcup\limits_{p\in ...
1
vote
1answer
108 views

A question about the “size” of the neighbourhoods in which bundles are trivializable

My question is about domains of trivializability of distributions on a smooth compact manifold. Assume that you have a sequence $\{E^k\}_k$ of $C^r$ distributions of rank $n$ which is $C^0$ close to a ...
2
votes
1answer
124 views

Killing constant in Killing spinor equation

This is a simple question, but I can't find explicit discussion in literatures that I can find. Real/imaginary Killing spinor equation \begin{equation} \nabla_\mu \psi = \lambda \gamma_\mu \psi ...
7
votes
0answers
183 views

quantitative version of the rigidity of the 2-sphere

I am looking for a quantitaive version of the following theorem: A compact surface with $K\equiv 1$ is isometric to the round sphere. Of course I get the Berger, Brendle-Schoen Theorem which insures ...
15
votes
2answers
397 views

Does a small-area sphere in a 3-manifold bound a small ball?

Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out. For every $\varepsilon>0$ there ...
3
votes
1answer
183 views

A question on differential forms and integral invariants

The following question comes up in the study of metrics with the same unparameterized geodesics in Riemannian and Finsler geometry: Question. Let $M$ be a closed manifold of dimension $2n+1$ and let ...
4
votes
1answer
163 views

Curvature for $C^{1,\alpha}$-metrics

According to Gromov's Metric Structures for Riemannian and Non-Riemannian Spaces every limit $V_0$ of sequences in the class of manifolds $V$ with $|K(V)| \leq 1$ and $\mathrm{InjRad}(V) \geq \rho ...
1
vote
2answers
159 views

Methods of probability theory in differential geometry fruitful? [closed]

I am trying to understand how the two paradigms of differential geometry and probability theory can fruitfully be applied to each other. The more suggestive direction is to use methods of ...
18
votes
1answer
271 views

Integral formula for Euler class

The tangent bundles of closed hyperbolic surfaces have flat $PSL(2,\mathbb{R})$ connections showing that there can be no integral formula for the Euler class such connections. This contrasts the ...
1
vote
1answer
105 views

Bakry-Emery Laplacian and Hodge Decomposition

I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian ...
1
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0answers
175 views

Second derivative of Riemannian Exponential Map

Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$. The derivative of the exponential map can be expressed in ...
4
votes
1answer
183 views

Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
8
votes
1answer
238 views

counterexample to the Chern number inequality on Fano manifold

We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality $$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$ My question is whether there ...
2
votes
1answer
119 views

On the balanced metric on complex manifold

Let $(M,h)$ be a Hermitian manifold and denote the Kahler form of the Hermitian metric by $\omega$. From the definition of Kahler manifold we know that $M$ is a Kahker manifold if $d\omega=0$. A ...
7
votes
0answers
193 views

Surfaces with many (but not solely) closed geodesics?

Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random ...
2
votes
1answer
90 views

Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group

Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It ...
3
votes
1answer
195 views

Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...
1
vote
1answer
118 views

Entropy of Negatively pinched manifolds

Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...
0
votes
1answer
74 views

harmonic maps from cone to $S^2$ locally lipschitz?

Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
8
votes
1answer
296 views

Are there more “types” of derivatives than “symmetric” or “alternating”?

The $kth$ derivative of a function $f:\mathbb{R}^n \to \mathbb{R}$ can be thought of as a symmetric $k$-tensor. (Well, almost. It is not invariant under coordinate changes, and should really be ...
0
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0answers
90 views

Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)

Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
1
vote
0answers
74 views

PL or projective PL map on the links of a PL manifold

Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
1
vote
0answers
149 views

Brakke's theorem for gap in entropy between self-shrinkers

In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any ...
5
votes
2answers
270 views

A good metric for transversal intersections

Let $V_1,\ldots,V_k$ be a transversal set of smooth compact orientable sub-manifolds of a compact orientable manifold $M$, and set $V=\bigcap V_i$. Is it always possible to equip a neighborhood $U$ ...
2
votes
1answer
237 views

Euler characteristic of Cauchy surface in Lorentz manifold

Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler ...
3
votes
2answers
437 views

A Converse to the Gauss Bonnet Theorem

Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent ...
2
votes
1answer
124 views

reference of extension of flat bundle

Does any one know where one can find a reference about the following fact? Let $X$ be a smooth projective variety over an algebraically closed field $k$. Fix two flat bundles $(L_i,\nabla_i)$ over ...
0
votes
1answer
105 views

model compact coisotropic submanifold

$\{x_1,\ldots,x_n,\xi_{m+1},\ldots,\xi_n\in\mathbb{R},\xi_1=\ldots=\xi_m=0\}$ is a "model" codimension $m$ coisotropic submanifold of $T^*\mathbb{R}^n$, and is of course noncompact. Is there such a ...
0
votes
2answers
102 views

Normal vector field associated to deformations of Riemannian submanifolds

Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $X$ be a immersed submanifold in $M$ of dimension $k$ i.e there is a immersion $F_{0}:X \longrightarrow M$. A deformation of the submanifold ...
1
vote
1answer
99 views

Is there any geometric and intuitive interpretation of Newton-like iterative steps in numerical optimization?

Are the iterative steps in optimization affected by the intrinsic and extrinsic curvatures of the objective functions ? and How? Is there any geometric and intuitive demo show illustrating the ...
9
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0answers
311 views

How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
6
votes
1answer
352 views

Explicit Kodaira-Spencer map of hyperelliptic curves

Let $g\geq 2$, and $$\mathcal T=\{(t_1,\cdots,t_{2g+2})~|~t_i\neq t_j,\forall i\neq j\}.$$ For any $t=(t_1,\cdots,t_{2g+2})\in \mathcal T$, let $$Y_t=\left\{y^2=\prod_{i=1}^{2g+2}(x-t_i)\right\}.$$ ...
0
votes
1answer
89 views

What is a “normal curvature matrix” of a specific surface? and how to calculate it? [closed]

It is the first time that I meet the concept "normal curvature matrix". By google searching, it seems like a concept in differential geometry. But I know little of differential geometry and do not ...
-1
votes
2answers
160 views

Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$? [closed]

Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?
7
votes
1answer
189 views

Monograph or rich survey on infinite-dimensional Riemann manifolds

I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...