**6**

votes

**1**answer

240 views

### Making the identification $\tau M\approx TM\oplus (TM\odot TM)$

Given a smooth manifold $M$, there is a vector bundle over $M$, denoted $\tau M$, known as the second-order tangent bundle. The fiber $\tau_mM$ at $m\in M$ is the collection of linear operators ...

**1**

vote

**1**answer

79 views

### Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...

**5**

votes

**0**answers

308 views

### Correspondence between line bundles and $U(1)$-bundles: a possible mistake from physicists? [closed]

I am reading a paper written by physicists and they say the following:
Let $(L,h,\nabla)$ be an holomorphic Line bundle equipped with a hermitian metric $h$ and Chern connection $\nabla$. If ...

**1**

vote

**1**answer

213 views

### A subalgebra of the Virasoro algebra

Let $L_n$ ($n\in\mathbb{Z}$) and $c$ be the standard generators of the Virasoro algebra ${\rm Vit}$. In the literature one usually considers the involutive authomorphism given by $\tau(L_n)=-L_{-n}$, ...

**1**

vote

**0**answers

43 views

### sharp conditions characterizing the vanishing of scalar Jacobi fields

Let $T>0$, let some function $\kappa(t)$ smooth on $[0,T]$, and let $b$ the unique solution to the ODE $\ddot b + \kappa(t) b = 0$ with initial conditions $b(0)=0$ and $\dot b(0) = 1$.
I was ...

**0**

votes

**0**answers

72 views

### Compact locally conformal Kahler manifolds with non-zero Euler characteristic

I would like to know if there exist eight-dimensional compact manifolds such that:
It has SU(4)-structure (and hence it is spin).
It is locally conformal Kahler (and not Kahler).
Its Euler ...

**1**

vote

**1**answer

124 views

### Chern classes of three (two) dimensional complex vector bundles

Let $M$ be a manifold.
Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$.
Let $S_3$ be the symmetric group of order $3$.
Let $S_3$ act on $F(M,3)$ by ...

**0**

votes

**0**answers

110 views

### Moduli space of holomorphic sections

Let $(L,M,\omega,\nabla)$ be an holomorphic line bundle over a Kahler manifold $(M,\omega)$ equipped with the Chern connection $\nabla$. Let $\Gamma(L)$ denote the space of holomorphic sections of ...

**0**

votes

**0**answers

75 views

### Equivalence of holomorphic line bundles from Kahler potentials

Let $(M.\omega)$ be a Kahler manifold with fundamental form $\omega$. Then $\omega$ is closed and by the $\partial\bar{\partial}$-lemma on every contractible open set $U\subset M$ we can write
...

**8**

votes

**3**answers

505 views

### Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...

**1**

vote

**0**answers

68 views

### Characterization of locally conformally flat manifolds: strange application of Frobenius theorem

(Crossposted from math.SE because of the lack of replies)
In Hamilton's Ricci Flow (by Chow, Lu, Ni, pp. 29-31, see here) they show that a Riemannian manifold $(M^n,g)$ is locally conformally flat ...

**7**

votes

**1**answer

135 views

### Boundary values of boundary value problems

Let $M$ be a manifold with smooth boundary. We can consider the Dirichlet or the Neumann problem on $M$. Let $(\phi_k)$ be an orthonormal basis of eigenfunctions to the Dirichlet problem and let ...

**9**

votes

**0**answers

206 views

### Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...

**0**

votes

**0**answers

93 views

### Relative canonical sheaf of an Abelian Scheme over a noetherian base

Let $A$ be an abelian scheme over a noetherian base $S$ of relative dimension $r$, i.e., $A$ is a smooth, proper group scheme over $S$ with geometrically connected fibers of dimension $r$. The ...

**1**

vote

**1**answer

127 views

### Image of any curve can be parametrized without zero derivative?

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Is it true that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets ...

**1**

vote

**0**answers

130 views

### Products between metrics in a product of manifolds

In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...

**2**

votes

**2**answers

169 views

### Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result:
Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$
be a non ...

**0**

votes

**0**answers

60 views

### Cohomology operators inducing local basis of $1-$forms

Suppose that $\partial$ is a non-trivial ($\partial \neq 0$) cohomology operator on an $m-$dimensional manifold $M$ (that is: $\partial:\Omega(M)\to\Omega(M)$ is a degree $1$ derivation such that ...

**15**

votes

**4**answers

892 views

### Equations satisfied by the Riemann curvature tensor

It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
...

**3**

votes

**1**answer

147 views

### The complex of Kahler differentials and de Rham complex

Let $A$ be a unital commutative algebra (say over complex numbers). Consider the multiplication map $m:A \otimes A \to A$ and put $\Omega^1_u(A)=\ker m$ to be the space of universal differential ...

**2**

votes

**1**answer

127 views

### acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...

**0**

votes

**1**answer

256 views

### Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space

This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...

**4**

votes

**2**answers

373 views

### How many metrics of constant curvature exist on a Riemannian surface?

I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...

**2**

votes

**0**answers

107 views

### Heat semigroup estimate on complete Riemannian manifold

Consider a complete noncompact Riemannian manifold $M$ such that the heat kernel $h_t(x, y)$ satisfies
$h_t(x, y) \leq Ct^{-n/2}$. Consider a function $u \in L^p(M)$. How can we prove that ...

**7**

votes

**3**answers

310 views

### Classification of natural invariants of Riemannian structures

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.
By ...

**3**

votes

**1**answer

124 views

### convex decompositions of the sphere

Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of ...

**1**

vote

**1**answer

139 views

### Metric, torsion free connections on principal bundles

I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers.
Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold ...

**3**

votes

**1**answer

137 views

### Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence

Thank you for spending time on the following question.
I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic ...

**2**

votes

**1**answer

192 views

### Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected.
Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...

**0**

votes

**0**answers

90 views

### When does the integral of a Dolbeault-exact form vanish?

What conditions (if any) can be imposed on a Kahler manifold $M$ so that we get a Dolbeault analogue of Stokes' theorem on a closed manifold, i.e.
$\int_M \partial ( ... ) =0$
The trivial solution ...

**2**

votes

**1**answer

112 views

### Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a ...

**3**

votes

**1**answer

177 views

### A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$
Does for any ...

**1**

vote

**0**answers

40 views

### Invariant Girsanov Theorem on a Riemannian manifold

This is somewhat a follow-up on this post.
Let $X_t$ be the stochastic process on a compact Riemannian manifold generated by the (possible time-dependent) second-order elliptic operator $L_t$. Let ...

**1**

vote

**1**answer

243 views

### Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$.
We have the ...

**3**

votes

**1**answer

87 views

### Null tetrad transformation

I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism.
I have a question about what he calls a "class III ...

**6**

votes

**2**answers

351 views

### Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...

**13**

votes

**2**answers

649 views

### (Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...

**2**

votes

**1**answer

139 views

### Curvature of a principal bundle and the exterior covariant derivative

I am sorry if this is too elementary; I had posted it on math.stack but no one answered.
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...

**24**

votes

**2**answers

747 views

### Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...

**0**

votes

**0**answers

55 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?

**2**

votes

**0**answers

74 views

### Lower boundedness of the Ricci curvature [closed]

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 ...

**1**

vote

**3**answers

260 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 $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...

**7**

votes

**2**answers

530 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 ...

**-6**

votes

**1**answer

185 views

### Stiefel-Whitney class of complex projective spaces [closed]

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

171 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

**1**answer

55 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

245 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

96 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

173 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

78 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 ...