**0**

votes

**0**answers

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

**2**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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

**3**answers

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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 ...