**0**

votes

**1**answer

291 views

### A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding
$$\phi: (M,\omega)\to (\mathbb CP^N, ...

**2**

votes

**1**answer

246 views

### examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$，where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and ...

**1**

vote

**1**answer

151 views

### de Rahm Laplace operator on forms bounded

Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...

**5**

votes

**1**answer

285 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

**8**

votes

**1**answer

171 views

### Besicovitch Covering Lemma on Manifolds

The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...

**3**

votes

**0**answers

301 views

### Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...

**3**

votes

**0**answers

164 views

### Can we obtain topology results using analysis in metric measures spaces?

Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I ...

**4**

votes

**0**answers

79 views

### Is this groupoid a model for the derived fixed-point locus of the free loop space?

In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...

**4**

votes

**1**answer

198 views

### What is an element of an iterated tangent bundle?

An element of the tangent bundle $T M$ of a manifold is called a "(tangent) vector". An element of its dual $T^* M$ is called a "covector" or a "1-form". An element of the exterior square ...

**6**

votes

**1**answer

165 views

### Chern-Weil Theory for $p_1$

I'm studying Riemannian manifolds that admits a almost-complex manifold, thus
$$3\tau+2\chi=c_1^2,$$
where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern ...

**1**

vote

**2**answers

341 views

### Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$,
i.e., each $p=(p(x_0),\dots,p(x_n))\in ...

**1**

vote

**1**answer

95 views

### Injectivity radius on complete manifolds with curvature decay

I am wondering that the following statement is true or not:
Let $(M,g)$ be a complete non-compact Riemannian manifold with $0 < Sect \leq C\cdot dist(O,x)^{-2a}$, $a\in(0,1]$. ($O$ is a point ...

**11**

votes

**3**answers

736 views

### Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?

If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...

**1**

vote

**1**answer

355 views

### Monge–Ampère type equation

Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function).
I am looking for the solutions among of ...

**0**

votes

**1**answer

104 views

### both convex and superharmonic function on manifold concave?

M is a non-compact Rimannian manifold without boundary.
$f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e.
$$
-\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M ...

**7**

votes

**2**answers

330 views

### Riemannian distance functions on the real line

A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties:
$d$ is a length metric ...

**0**

votes

**0**answers

38 views

### Expansion of the squared distance function from a submanifold

let $(M,g)$ be a compact Kahler manifold of dimension $m$ with the Kahler metric $g$ real-analytic. Let $N\subset M$ be a complex submanifold of dimension $0<k<m$. Let $$d_{g}(M,\cdot): ...

**2**

votes

**1**answer

133 views

### Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?

Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies ...

**0**

votes

**0**answers

74 views

### Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$

I'm looking for a proof resp. reference for a statement of the following form:
Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...

**1**

vote

**2**answers

105 views

### Right invariant Killing fields of Right invariant Riemanian metrics

Can there exist a right invariant killing field of a right invariant (but not bi-invariant) Riemannian metric on a Lie group?
I am especially interested in the case of $SU(N)$ with a metric of the ...

**4**

votes

**1**answer

481 views

### The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions ...

**7**

votes

**0**answers

227 views

### diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...

**3**

votes

**1**answer

170 views

### Displaceability of submanifolds

My question is motivated by the following question.
How transitive are the actions of symplectomorphism groups ?
A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there ...

**1**

vote

**0**answers

49 views

### Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* ...

**0**

votes

**1**answer

75 views

### Optimal Regularity for Invariance of Curvature under Isometries

It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that ...

**12**

votes

**2**answers

872 views

### Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...

**1**

vote

**0**answers

37 views

### Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold.
Explanation
Suppose we have a 3D smooth manifold ...

**2**

votes

**0**answers

64 views

### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...

**3**

votes

**1**answer

247 views

### Special Riemannian connections?

Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where $x,y,z\in E_p$ are arbitrary.
Is there a ...

**0**

votes

**0**answers

279 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**1**

vote

**1**answer

238 views

### Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $LG(2,4)$?

On the paper
http://arxiv.org/abs/1009.1364
(published on Proc. London Math. Soc.) I've found an interesting statement:
The Lie quadric $Q^3$, i.e., the space of all points, lines and
circles ...

**0**

votes

**1**answer

189 views

### A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$
is integral, for some almost complex structure ...

**1**

vote

**0**answers

61 views

### coordinate system in AF manifold

we know that an end of $(M,g)$ is asymptotically flat if it is diffeomorphic to the complement of a compact set $K$ in $\mathbb{R}^3$, and the metric tensor $g$ satisfies ...

**3**

votes

**2**answers

233 views

### The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?
(My ...

**0**

votes

**1**answer

292 views

### How to prove this Weitzenbock formula?

In Hutchings and Taubes lecture note on Seiberg-Witten equation HERE, above equation (4.20) the authors claim that there is a version of Weitzenbock formula reads (where $\beta \in \Omega^{0,2}(M, ...

**0**

votes

**0**answers

42 views

### Special family of Metrics on Transitive Lie Algebroids?

Let $\rho:E\longrightarrow TM$ is a transitive Lie Algebroid, then $L=ker\rho$ is bundle of lie algebras. Suppose $\Gamma:TM\longrightarrow E$ be a linear splitting. Define
$$\nabla_X ...

**1**

vote

**1**answer

100 views

### Which book will discuss torsion tensor and affine connection in detail? [closed]

Which book about differential geometry will have these formula about torsion tensor?
$$\nabla_{j}T^{i}_{kl}+\nabla_{k}T^{i}_{lj}+\nabla_{l}T^{i}_{jk}=R^{i}_{jkl}+R^{i}_{klj}+R^{i}_{ljk}$$
...

**3**

votes

**1**answer

147 views

### Deformation of Hitchin-Simpson correspondence

Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern ...

**0**

votes

**0**answers

98 views

### positive eigenfunction on complete Riemannian manifold

Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation
$$
\Delta u = u
$$
where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...

**9**

votes

**2**answers

428 views

### Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...

**2**

votes

**0**answers

303 views

### The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...

**1**

vote

**0**answers

46 views

### Allen Cahn Equation with Dirichlet Data

consider the 'Allen-Cahn Equations' given by $ \epsilon^2\Delta_g u + u - u^3=0 $ It is known that if $\Gamma$ is a non degenerate minimal surface then there exists a sequence of solutions ...

**1**

vote

**1**answer

369 views

### A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way.
...

**1**

vote

**1**answer

105 views

### metric has morse index 2

I am reading Richard Schoen's classical example on the multiplicity of solutions of yamabe problem. He says on $S^1(T)\times S^{n-1}$, there exists a critical number $T_0$ such that if $T\leq T_0$, ...

**6**

votes

**1**answer

449 views

### On bi-invariant metrics on groups

I'm looking for a quick, snap-your-fingers proof of the following result:
A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm.
To be clear about ...

**5**

votes

**0**answers

151 views

### Non-trivial isometric embedding of the standard sphere into $\mathbb{R}^n$? [closed]

Let $S$ be the unit sphere in $\mathbb{R}^3$. Is it possible to embed $S$ isometrically into some $\mathbb{R}^n$, $n>3$, such that the image does not lie on any 3-dimensional affine subspace?
More ...

**3**

votes

**1**answer

131 views

### All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...

**6**

votes

**0**answers

127 views

### Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...

**2**

votes

**0**answers

173 views

### Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
...

**0**

votes

**0**answers

72 views

### Comparison of support of a divisor and its class in de Rham cohomology

Let $X$ be a smooth surface over $\mathbb{C}$ and $D$ be an effective divisor. Suppose that the linear system corresponding to $D$ is zero dimensional. Denote by $[D] \in H^2(X,\mathbb{Z})$ the ...