Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
0
votes
1answer
19 views
If there exists a vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat?
If there exists a vector field $V$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be flat?
That is, if ...
0
votes
0answers
13 views
Factor of 2 In the Definition of Metric Contact Structure
In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
...
4
votes
2answers
137 views
When distance nonincreasing map is an isometry
Let $f: M \to M$ be a distance nonincreasing map between a closed Riemannian manifold $M$
and $f$ is homotopic to the idendity map. Is it then $f$ an isometry?
4
votes
1answer
151 views
Quotienting $SU(3)$ by $U(1)$?
As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
0
votes
0answers
81 views
Jacobian change of basis matrix for different dimensions
I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: U \rightarrow \mathbb{R}$ ...
4
votes
0answers
37 views
Euler number of the complex of basic forms
Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...
1
vote
1answer
102 views
Kaehler form on weighted projective space
The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum_{i=0}^n |z_i|^2)).$$
What is the analogous formula for a Kaehler ...
2
votes
0answers
41 views
DGBV algebra of symplectic manifold
Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...
6
votes
1answer
393 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, ...
3
votes
1answer
95 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 ...
-2
votes
0answers
38 views
Information geometry divergence [on hold]
on http://en.wikipedia.org/wiki/Information_geometry
How to derive this equation. I tried but always got 0 for each item.
$$
D[\partial_i\partial_j||\cdot]= ...
0
votes
0answers
6 views
Contraction between basis vectors and basis one-forms [migrated]
Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms.
I always thought by definition, $dx^i (e_j) =\delta^i_j $.
But, I am confused ...
7
votes
1answer
222 views
The moduli space of special Lagrangian submanifolds
Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...
5
votes
1answer
188 views
Can eta invariant be written in terms of topological data?
The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
1
vote
1answer
77 views
Normal coordinates near the boundary
Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...
0
votes
0answers
41 views
Total differential of Lipschitz submanifolds embedding
My interest is analysis on Lipschitz manifolds. I want to define traces of differential forms on a Lipschitz submanifold $N$ of a Lipschitz manifold $M$. In other words, I want to push-forward ...
1
vote
1answer
189 views
Why tangent vector of statistical manifold is a function?
In differential geometry, tangent vectors are considered operators.
At point p, the local tangent space is defined as
$$
T_p(M)=\{X^i\partial_i|X\in R^n\}
$$
This is quite easy to understand for me.
...
3
votes
0answers
84 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 ...
2
votes
1answer
109 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 ...
0
votes
0answers
69 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
3answers
179 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 ...
1
vote
0answers
124 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
0answers
105 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 ...
4
votes
0answers
63 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 ...
2
votes
0answers
47 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 ...
3
votes
2answers
137 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 ...
5
votes
5answers
823 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 ...
10
votes
1answer
230 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 ...
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
314 views
why quintics are Calabi-Yau?
Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?
4
votes
1answer
215 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 ...
2
votes
1answer
189 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, ...
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 ...
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
56 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
75 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
1answer
115 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 ...
6
votes
1answer
257 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
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
136 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 ...
2
votes
1answer
139 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 ...
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
1answer
65 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 ...
0
votes
1answer
66 views
Compatible connection on the associated vector bundle
Assume $E\rightarrow X$ is a holomorphic vector bundle of rank $n$ with a linear connection $\nabla=\nabla^{1, 0}+\bar\partial_E$ which is compatible with the complex (somewhere the literature says ...
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^*} ...
0
votes
1answer
202 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 ...
1
vote
0answers
57 views
Recovering Spherical Harmonics from Discrete Samples
Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...
2
votes
1answer
125 views
Atiyah classes of holomorphic vector bundles with trivial Chern classes
Let $X$ be compact Kahler and $E \to X$ a holomorphic vector bundle. Then $E$ has an Atiyah class, $At(E)$, valued in the sheaf cohomology $H^1(\Omega_X \otimes \operatorname{End} E)$. Suppose the ...
6
votes
1answer
230 views
Are there nontrivial real functions of 2 real variables with gradient having constant euclidian norm on each level line?
Let $F$ be the class of locally Lipschitz continuous functions $z=f(x,y)$, from $\mathbb R \times\mathbb R \to\mathbb R,$ such that the euclidean norm $|\ \mathrm{grad}\ f (x,y)\ |$ of its gradient ...
0
votes
0answers
104 views
Third variation of area of a minimal surface
There is a formula for the third variation of area on page 96 of Nitsche's book,
Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the
page it is good for normal ...
