**0**

votes

**0**answers

19 views

### Derivative of Log_p map

If $\mathscr{M}$ is a $(d-dimensional)$ Riemann Manifold and $p$ is a point therein. What is the derivative of $Log_p$ function?

**0**

votes

**0**answers

40 views

### Does a (non-closed) differential 1-form define a curve?

I am trying to understand under what conditions the following procedure properly defines a curve. Take a manifold $M$ with a (non-closed) 1-form $B$ and an exact 1-form $dA$. Define the functions ...

**2**

votes

**0**answers

40 views

### Lower boundedness of the Ricci curvature [on hold]

I know that this is going to be slightly vague, but it's itching me: why is lower boundedness of the Ricci curvature required by some of the very important theorems in Riemannian geometry? The ...

**0**

votes

**0**answers

43 views

### orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1})$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...

**4**

votes

**2**answers

442 views

### Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...

**-5**

votes

**1**answer

129 views

### Stiefel-Whitney class of complex projective spaces [on hold]

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$?
Let $a_m$ be the maximal integer such that the $a_m$-th dual ...

**1**

vote

**1**answer

147 views

### Moduli space of flat connections over a torus

Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...

**0**

votes

**0**answers

41 views

### Two surfaces with zero gaussian curvature [on hold]

According to Hartman's article every surface $f(s,t)$ with zero gaussian curvature locally admits parametrization
$f = a_1(u) v + b_1(u),$
$s = a_2(u) v + b_2(u),$
$t = a_3(u) v + b_3(u).$
Now let's ...

**0**

votes

**1**answer

47 views

### Principal bundle associated to a Courant algebroid

A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition ...

**4**

votes

**1**answer

208 views

### Differential forms along the fiber

Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on ...

**1**

vote

**1**answer

65 views

### Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`

Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong ...

**0**

votes

**1**answer

135 views

### Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
Wirtinger's Inequality.
Let $L$ be a complex linear space and let $M$ be a real
even-dimensional ...

**4**

votes

**0**answers

67 views

### Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...

**5**

votes

**1**answer

178 views

### Strange problem about triplets of differential forms

Suppose we have the following map:
$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$
...

**3**

votes

**2**answers

110 views

### distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that
$$
\int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...

**0**

votes

**1**answer

88 views

### Existence of a special Kahler structure on the contangent bundle

It is known that the total space of the cotangent bundle $T^{\ast}M\to M$ of any manifold $M$ can be equipped with a Kahler structure. My question is, it is known when $T^{\ast}M$ admits a Special ...

**2**

votes

**2**answers

148 views

### Connections having the same holonomy along loops at a point

I find myself stuck with the following question, which seems very classical but for which I have not been able to find a reference.
Consider a smooth vector bundle $E$ of rank $r$ over a compact ...

**2**

votes

**0**answers

56 views

### A question about lagrangian submfd of Calabi-Yau manifold

In the paper 'Special Lagrangians, stable bundles and mean curvature flow' by R. P. Thomas and S.-T. Yau, page 2. They said
A Lagrangian submanifold $L$ of the Calabi-Yau manifold $(X,\Omega)$, we ...

**3**

votes

**3**answers

163 views

### Are linearizations of involutive PDEs locally solvable?

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing ...

**13**

votes

**2**answers

599 views

### Are there any natural differential operators besides $d$?

Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ and $\mu = (\mu_1, \ldots, \mu_r)$ be partitions such that $\mu_j = \lambda_j +1$ for one index $j$ and $\mu_i = \lambda_i$ for all other $i$. Then there ...

**2**

votes

**1**answer

69 views

### distance formula of warped products

Given a warped product, I want to compute the ditance of any two points. First get the equation for the geodesic, then compute the length of the geodesic.
Consider the two-dimensional surface
$$
...

**1**

vote

**1**answer

185 views

### Is there a way to define a Lie derivative of a connection?

I've been reading a little bit about the definition of symmetries on General Relativity, and they are related with the concept of Killing vector, i.e., vectors along which the Lie derivative of the ...

**2**

votes

**0**answers

55 views

### Analogue of the Euler class of a circle bundle and the global angular form

This is a general question that asks whether there is geometric significance to differential forms that restrict to G-invariant volume elements on the fibers of a principal G-bundle.
For an SO(2) ...

**3**

votes

**1**answer

117 views

### structure of metrics on a compact manifold

is there a reference on the structure of the space of metrics on a compact manifold that induce a given measure $\mu $?
i have a given manifold $M$, a given measure $\mu$ with an everywhere positive ...

**0**

votes

**0**answers

65 views

### Functional involving Ricci curvature: convex and coercive?

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, volume form $\mu_g$, and Ricci curvature $\text{Rc}_g$.
Question: Given a fixed vector field $V\in\Gamma(TM)$, under what ...

**1**

vote

**1**answer

131 views

### Symplectic reduction: from indefinite signature to Riemannian signature

Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ ...

**3**

votes

**1**answer

224 views

### 'Unitary' charts on odd-dimensional spheres

Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific ...

**2**

votes

**1**answer

250 views

### Perimeter of ellipse: Combination of two geometries

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter ...

**10**

votes

**1**answer

502 views

### Associativity of Kontsevich's star product up to second order

In Deformation quantization of Poisson manifolds, Kontsevich gives the quantization formula
$$f \star g = \sum_{n=0}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma,\alpha}(f,g).$$
He gives ...

**2**

votes

**0**answers

130 views

### Formula for the distance in noncommutative geometry

Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data:
$$(1) \ \ ...

**0**

votes

**0**answers

68 views

### Diffusion on a semi-Riemannian manifold?

A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ...

**1**

vote

**0**answers

126 views

### How the exceptional simple Lie groups/ algebras were first discovered and by whom?

I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...

**2**

votes

**1**answer

132 views

### Is there a characterization of Riemannian manifolds that split off two factors?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...

**4**

votes

**0**answers

149 views

### Curvature estimates for Kaehler-Einstein and Hermitian, Einstein four-manifolds

Four-dimensional Kaehler-Einstein manifolds and Hermitian, Einstein manifolds with positive scalar curvature have been classified by Tian and LeBrun respectively. I was wondering are there any ...

**1**

vote

**1**answer

85 views

### Uniqueness affine curvature

Let $\gamma_1,\gamma_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their (special)affine curvature, $\mu_1,\mu_2$ are equal, that is $\mu_1(\theta)=\mu_2(\theta)$, for ...

**3**

votes

**0**answers

178 views

### Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...

**0**

votes

**0**answers

58 views

### Bispinors, polyforms, bilinears and supersymmetric manifolds

Let $(V,q)$ be a regular quadratic vector space, and let us denote by $Cl(V,q)$ the corresponding Clifford algebra. Then there exists an isomorphism of $\mathbb{Z}_{2}$-graded algebras:
...

**2**

votes

**1**answer

136 views

### Closed geodesics that cross one another frequently

Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$.
$S$ has at least three simple (non-self-intersecting), closed geodesics by
a theorem of Lyusternik and Shnirel'man.
Alternatively, ...

**1**

vote

**1**answer

117 views

### Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles

As explained in:
Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds
principal $SU(2)$ bundles $P_{SU(2)}$ over a four-dimensional manifold $M$ are classified by their ...

**1**

vote

**0**answers

97 views

### Estimates of eigenvalues of elliptic operators on compact manifolds

The classical Weyl law says that if $\Delta$ is the Laplace operator on functions on a compact Riemannian manifold $(M^n,g)$, $n>2$, then its $k$th eigenvalue satisfies the asymptotic formula
...

**1**

vote

**0**answers

138 views

### Geometric representatives of homology classes of manifolds

Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?

**3**

votes

**0**answers

48 views

### Geodesic rays in a toric variety

Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic ...

**4**

votes

**3**answers

248 views

### Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...

**1**

vote

**0**answers

104 views

### Classification of $SU(n)$-principal bundles over a four-dimensional base

It is well-known that a principal $SU(2)$-bundle $P$ over a four-dimensional manifold $M$ is topologically classified by its second Chern-class $c_{2}(P)\in H^{4}(M,\mathbb{Z})$, as explained for ...

**1**

vote

**1**answer

120 views

### Is there a name for the function on $TTM$ swapping the 2nd and 3rd coordinates?

I'm not so good on geometry, so I fear this is a relatively basic question.
For any $N \in \mathbb{N}$, let us identify the tangent bundle of $\mathbb{R}^N$ with $\mathbb{R}^{2N}$ in the obvious ...

**0**

votes

**0**answers

50 views

### Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...

**0**

votes

**0**answers

62 views

### Ricci solitons on non-metric spaces

A Ricci soliton is a generalization of the Einstein metric and is defined on a
Riemannian manifold (M,g) by
$\mathcal{L}_v g_{ij} + 2 R_{ij} + 2λ g_{ij} = 0$,
for some constant λ, a vector field $v$ ...

**2**

votes

**1**answer

135 views

### length comparison on negatively curved surfaces

Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy ...

**0**

votes

**1**answer

86 views

### Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact?

let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and
...

**2**

votes

**0**answers

60 views

### How to check if a manifold can be foliated by strictly convex hypersurfaces?

Let $M$ be a compact Riemannian manifold with boundary.
How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?
An exact definition is given below.
If the ...