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

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3
votes
0answers
21 views

Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$. Place $K$ on an inclined plane, and let it roll down the plane, under some reasonable assumptions of friction between $K$ and the plane, ...
6
votes
1answer
243 views

When a symplectic manifold is formal?

Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, then we have the symplectic Hodge operator $$*:\Omega^{k}(M)\rightarrow\Omega^{2n-k}(M)$$ Furthermore, we can define a differential ...
2
votes
0answers
36 views

Positivity of Intersections in higher dimensions

"Positivity of Intersections" is a phenomenon in 4-dimensions, due to Gromov: Given two embedded $J$-holomorphic curves in an almost-complex 4-manifold $(X^4,J)$, all intersection points are isolated ...
3
votes
1answer
55 views

Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...
6
votes
1answer
144 views

The moduli space of special Lagrangian submanifolds

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures on the base space $B$. A theorem of Hitchin says that there ...
-2
votes
0answers
34 views

Information geometry divergence

on http://en.wikipedia.org/wiki/Information_geometry How to derive this equation. I tried but always got 0 for each item. $$ D[\partial_i\partial_j||\cdot]= ...
1
vote
1answer
184 views

Why tangent vector of statistical manifold is a function?

In differential geometry, tangent vectors are considered operators. At point p, the local tangent space is defined as $$ T_p(M)=\{X^i\partial_i|X\in R^n\} $$ This is quite easy to understand for me. ...
0
votes
0answers
6 views

Contraction between basis vectors and basis one-forms [migrated]

Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms. I always thought by definition, $dx^i (e_j) =\delta^i_j $. But, I am confused ...
5
votes
1answer
162 views

Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
0
votes
1answer
72 views

Normal coordinates near the boundary

Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...
2
votes
3answers
175 views

Diffeomorphism with prescribed behaviour

If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...
0
votes
0answers
40 views

Total differential of Lipschitz submanifolds embedding

My interest is analysis on Lipschitz manifolds. I want to define traces of differential forms on a Lipschitz submanifold $N$ of a Lipschitz manifold $M$. In other words, I want to push-forward ...
7
votes
1answer
375 views

Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form. My question is as ...
3
votes
0answers
77 views

Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...
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 ...
3
votes
1answer
259 views

Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic? For example, is this true for small ...
2
votes
1answer
108 views

Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...
5
votes
5answers
817 views

Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?

Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat? I am especially interset in the case ...
0
votes
1answer
134 views

Kahler structure on holomorphic principal bundles

Let $G$ be a compact complex Lie group and $M$ be a compact Kähler manifold. Does there exist any example of a holomorphic principal $G$-bundle over $M$ admitting Kähler structures?
0
votes
0answers
66 views

Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature". The complex is referenced as "Calabi complex" in various citing ...
2
votes
1answer
291 views

Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint ...
0
votes
0answers
29 views

How can we get the area element ' expansion with respect to the induced metric defined in this way? [closed]

Let $Σ_r$ be a topological sphere in a 3-dimensional asymptotically flat Riemannian manifold $M$ with metric g, $\{\frac{\partial}{\partial x^i}\},1≤i≤3$ is the standard coordinate frame in ...
0
votes
0answers
104 views

Fiber bundle trivialization. Transition functions

Depending on the authors, trivialization is considered either as a diffeormorphism from $U\times G$ to $\pi^{-1}(U)$ or from $\pi^{-1}(U)$ to $U\times G$. The result leads to transition functions ...
1
vote
0answers
123 views

Cotangent bundle of symmetric space is symmetric space?

Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...
0
votes
1answer
201 views

curvature and volume growth

Let $M$ be a non-compact connected Riemannian manifold with $\mathrm{sec}_g=0$ and $\operatorname{vol} B(x,r)\geq c(n)r^n$ for any $r$, where $c(n)>0$. How to prove that $(M,g)$ is isometric to ...
7
votes
1answer
202 views

The cones for Bochner–Lichnerowicz–Weitzenböck formula

The Bochner–Lichnerowicz–Weitzenböck formula can be written the following way $$ \Delta \phi-\nabla^*\nabla \phi= R(\phi),$$ here $\phi$ is a section in a Dirac bundle and $R$ the something which can ...
4
votes
0answers
61 views

Is the $L^2$ metric on the space of unit volume Riemannian metrics on a closed, oriented surface Kahler?

Let $\Sigma$ be a closed, oriented, smooth surface. Denote by $\mathcal{M}^{1}(\Sigma)$ the deformation space of unit volume Riemannian metrics on $\Sigma:$ here we consider two metrics equivalent if ...
10
votes
1answer
208 views

Do there exist non-totally geodesic isometric minimal immersions $\mathbb{H}^2\rightarrow G/K.$

Suppose $G$ is a non-compact, semi-simple Lie group, of rank at least two, with maximal compact subgroup $K$ and $G/K$ the corresponding Riemannian symmetric space. Let $\mathbb{H}^2(-c^2)$ be the ...
3
votes
2answers
136 views

Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
2
votes
0answers
45 views

Complete gradient shrinking Ricci soliton with nonnegative Ricci curvature?

Besides the product of a positive Einstein manifold with the Euclidean Gaussian shrinker, does there exist other complete (nonconpact) gradient shrinking Ricci soliton with nonnegative Ricci ...
1
vote
0answers
77 views

Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$. For $m$ large, let ...
4
votes
1answer
335 views

What manifolds can have a (non-piecewise) linear structure?

By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert ...
0
votes
0answers
32 views

Buseman function is regular on manifolds without boundary containing a line?

For an n-dim noncompact manifold M without boundary. Assume M contains a line $\gamma$. For every point $p \in M$, let $\widetilde{p\gamma(t)}$ be the geodesic from p to $\gamma(t)$. Choose a ...
0
votes
1answer
313 views

why quintics are Calabi-Yau?

Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?
2
votes
1answer
138 views

Extending Reeb field from contact submanifold to ambient contact manifold

Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb ...
5
votes
1answer
285 views

A question on generalized Einstein metrics on four-dimensional manifolds

I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds, \begin{equation*} ...
5
votes
2answers
932 views

Space with 720° / not 2$\pi$ rotational symmetry?

Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)? The reason I am asking is because in quantum ...
1
vote
2answers
297 views

Isomorphism of connections on a complex line bundle

Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact? Theorem. Let $E ...
2
votes
1answer
112 views

Existence of planar orthogonal curvilinear coordinates on a surface embedded in $R^3$

We consider a surface (co-dimension 1) in $R^3.$ I read from the book of Stoker that for any surface there always exist patches of orthogonal curvilinear coordinates that cover the surface. I want ...
2
votes
1answer
188 views

Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits. A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold: $X$ is holomorphically convex, ...
3
votes
1answer
211 views

Given Gaussian curvature, can one construct a metric to fulfill the Gauss-Bonnet theorem?

Consider a compact surface $\mathcal{S} \subset \mathbb{R}^3$ without boundary and assume we are given the Gaussian curvature $K(x), x\in \mathcal{S}$. It is know that Gaussian curvature does not ...
6
votes
1answer
468 views

Yang-Mills flow, Ricci flow and the holonomy

Is the holonomy group (based at some point) preserved along the Yang-Mills flow/ Ricci flow? (1) For Yang-Mills case, we know that the centralizer of the holonomy $H_x$ is the isotropy group of the ...
1
vote
0answers
85 views

Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...
0
votes
0answers
54 views

Fill radius and fundamental group

I am reading M. Ramachandran and J. Wolfson's article Fill radius and fundamental group, whose main result is: Theorem. Let $N$ be a closed Riemannian manifold. If its universal cover has fill ...
-1
votes
0answers
73 views

If there is a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be hypersurfaces of dimension $n$ in $\mathbb{R}^{n+1}$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Denote the Laplace-Beltrami operator by $\Delta_{S(\cdot)}$. Let ...
-2
votes
0answers
20 views

Why the Oloid is developable surface?How to prove it? [migrated]

It is well known that the Oloid is developable surface,but why?How to prove it?
0
votes
0answers
134 views

Symplectic submanifolds in $\mathbb{R}^4$

Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...
8
votes
3answers
356 views

Number of disjoint simple closed geodesics

According to Jairo comment on the first version of this question I revise the question as follwos; Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that: There are at most ...
0
votes
1answer
64 views

CR Structures as Integrable G-Structures

Let $M$ be a closed manifold, with dimension $2n+1$. Let $F(M)$ be the frame bundle, a principal $GL(2n+1,\mathbb{R})$-bundle over $M$. An almost CR structure $P$ on $M$ is a structure group reduction ...
1
vote
1answer
84 views

Connection on canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on symplectic manifold

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic 4-manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} ...