**1**

vote

**1**answer

95 views

### A question about curvature for linear connections

Let $M$ be a manifold and $\pi : E \to M$ a rank $n$ vector bundle on $M$. We can define a connection on $E$ in two ways:
We can specify the covariant derivatives $\nabla_X s$ or
We can choose a ...

**1**

vote

**2**answers

50 views

### change of the residue of meromorphic differential by a covering map.

Let $C_1$ and $C_2$ denote complex algebraic curves, and let
$f : C_1 \rightarrow C_2$ be a non-constant map. Let $x$ be a point on $C_1$ and its ramification degree is $e_x$.
I want to compute the ...

**6**

votes

**3**answers

269 views

### Smoothness of the fourth power of the geodesic distance in a Finsler geometry

The simplest form of Finsler metric is:
$ds (X, dx)=\sqrt[4]{g_{\alpha, \beta, \gamma, \delta}(X).dx^{\alpha}dx^{\beta}dx^{\gamma}dx^{\delta}}$, where $g_{\alpha, \beta, \gamma, \delta}$ is a fourth ...

**4**

votes

**0**answers

72 views

### Hodge-Weil Formula for Quaternionic-Kähler manifold

Let $M$ be a quaternionic-Kähler manifold, with fundamental form $\omega$, and let $L$ be the Lefschetz operator of $\omega$. In the Kähler and, more generally, symplectic cases, there is a mysterious ...

**22**

votes

**1**answer

679 views

### How and why did mathematicians develop spin-manifolds in differential geometry?

First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that ...

**4**

votes

**1**answer

90 views

### Exponential of approximate quadratic variation of Brownian motion

Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all ...

**4**

votes

**2**answers

229 views

### triviality of Whitney sums of a vector bundle

Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid\sum_{i=1}^4x_i=0\}.$$ The alternating group $A_4$ acts on $V$ by
...

**8**

votes

**1**answer

184 views

### Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...

**2**

votes

**0**answers

311 views

### Weil Petersson metric on moduli space of Calabi Yau manifolds

Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...

**9**

votes

**1**answer

215 views

### Tori in three-space

Recently I was talking to an alien who does not know complex function theory. I was trying to convince her that the set of conformal equivalence classes of smooth embedded tori in $R^3$ is two ...

**3**

votes

**3**answers

192 views

### Compact surfaces with arbitrary gaps in spectrum

Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...

**3**

votes

**0**answers

62 views

### Proof of asymptotic non-flatness

in a few papers I came across a statement that the Kerr-NUT metric
$g_{uu}=\rho\overline{\rho}(r^{2}-2mr-l^{2}+a^{2}\cos^{2}x)$
$g_{ur}=1$
$g_{uy}=-2\rho\overline{\rho}l\cos ...

**3**

votes

**1**answer

74 views

### Closed leaves on foliations of $\mathbb{R}^n$

I want to know if there exists a characterization of k-foliations of $\mathbb{R}^n$ which have all the leaves closed.
Do exists a $k$-foliation of $\mathbb{R}^n$ with a non-closed leaf?
In general, ...

**4**

votes

**0**answers

147 views

### Comparing spectra of Laplacian and Schrödinger operator

Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators ...

**6**

votes

**1**answer

187 views

### Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...

**4**

votes

**0**answers

48 views

### Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action

Given a Riemannian manifold $M$ and a group of isometries $G$ of $M$, I am interested in when there exists a isometric embedding $\iota : M \to H$, where $H$ is a Hilbert space and a representation ...

**7**

votes

**2**answers

308 views

### first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles

Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to ...

**3**

votes

**1**answer

141 views

### Norm on space of metrics

I recently heard a differential geometry talk where the speaker constructed a one-parameter family of metrics $g(t)$ on a smooth manifold and said that $g(t)$ is real analytic in the Banach space ...

**2**

votes

**0**answers

72 views

### Rellich Embedding Theorem for the $2$-Sphere

I'm trying to understand the Rellich-Embedding Theorem in the non-flat case by looking at the $2$-sphere. To be precise, for $S$ the spinor bundle of $S^2$; $L^2(S^2)$ the space of square integrable ...

**3**

votes

**0**answers

134 views

### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the ...

**4**

votes

**2**answers

142 views

### Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...

**2**

votes

**2**answers

316 views

### $dd^\mathbb{C}$-lemma on pair $(X,D)$

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma ...

**0**

votes

**1**answer

157 views

### A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction,
$$\Delta u=-\lambda u.$$
I was wondering under what condition (for example, spaces ...

**3**

votes

**0**answers

78 views

### Plurisubharmonic functions on Kähler manifolds, intuition?

As the question suggests, what is the intuition for working with plurisubharmonic functions on Kähler manifolds?

**7**

votes

**1**answer

415 views

### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...

**6**

votes

**1**answer

194 views

### Minimal immersions of the 2-sphere

Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his ...

**4**

votes

**0**answers

171 views

### Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...

**0**

votes

**0**answers

109 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**2**

votes

**1**answer

200 views

### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...

**4**

votes

**1**answer

115 views

### normal form of currents?

(this question did not get any answers on math.SE, so I am reposting it here)
Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space ...

**1**

vote

**0**answers

64 views

### Uniform bounds on affine invariants of a family of manifolds

Suppose $\{M_i\}$ is a family of $C^{\infty}$ smooth, strictly convex hypersurfaces in $\mathbb{R}^n$ such that $B_r\subseteq M_i\subseteq B_R$, where $B_r,B_R$ are balls centred at the origin of ...

**2**

votes

**1**answer

95 views

### Curvature computations of globally symmetric spaces of rank $1$

I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.
More specifically, I need to use results about the eigenvalues of the operator $R:T_pM ...

**3**

votes

**1**answer

128 views

### about transverse complete intersection

There are several questions about transverse complete intersection arising from L. Guth's paper:
http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html
We say a polynomial ...

**9**

votes

**3**answers

747 views

### Is each closed convex set a manifold with corners?

Assume that $C$ is a convex set in $\mathbb{R}^{n}$ with non empty interior.
Then consider its closure, is it a smooth manifold with corners?
Edit:
1) The closure of $C$ should be a smooth manifold ...

**3**

votes

**1**answer

67 views

### compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21:
Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...

**8**

votes

**2**answers

132 views

### Tangent fields spanning the distribution of principal directions on a surface

Suppose $S$ is an orientable regular surface in $\mathbb R^3$ without umbilical points (not necessarily compact, and with no boundary). There are two well-defined smooth $1$-dimensional tangent ...

**5**

votes

**2**answers

438 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**2**

votes

**1**answer

164 views

### Are the Sasaki metrics on tangent and cotangent bundle isomorphic?

Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
...

**3**

votes

**0**answers

57 views

### If $M$ is a globally symmetric space, do we need it to be compact to prove that all the critical submanifolds in $\Lambda M$ are nondegenerate?

In "The free loop space of globally symmetric spaces" by Ziller, he proves the following theorem:
Theorem 2. For a globally symmetric space the critical submanifolds in $\Lambda M$ are all ...

**14**

votes

**4**answers

944 views

### Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**7**

votes

**1**answer

270 views

### Area of square to wrap a torus

The Nash-Kuiper
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
has recently been spectacularly visualized by the
Hevea Project.
This suggests two questions.
Q1. What is the area of ...

**2**

votes

**0**answers

87 views

### What is the metric on the Fuchsian model? [closed]

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model ...

**4**

votes

**2**answers

147 views

### Obtaining Killing fields from the tetrad

I'm reading the following article by Newman
http://scitation.aip.org/content/aip/journal/jmp/4/7/10.1063/1.1704018
about the generalization of the Schwarzschild metric. My question is the following: ...

**6**

votes

**3**answers

329 views

### Examples of Stiefel-Whitney classes of manifolds

Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, ...

**8**

votes

**2**answers

295 views

### Isometric imbedding of a sphere with positively curved metric

QUESTION. Given a Riemannian metric on the sphere $S^n$ with positive sectional survature. Can it be isometrically imbedded into $\mathbb{R}^{n+1}$ (of any class of regularity) as a boundary of a ...

**4**

votes

**1**answer

88 views

### A property of geodesic triangles in manifolds with lower bounds on curvature and injectivity radius

Does there exist a function $\tau(\varepsilon)=\tau(\varepsilon,n,K,\mu)$ such that $\lim_{\varepsilon\to +0}\tau(\varepsilon)=0$ and for any $n$-dimensional complete Riemannian manifold $M^n$ with ...

**12**

votes

**2**answers

498 views

### formula for Eta invariant

Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...

**1**

vote

**1**answer

75 views

### Definition o branched 1-manifold [closed]

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:
"A finite graph whose edges are smoothly embedded arcs in a surface, ...

**6**

votes

**0**answers

72 views

### Convexity of Isoperimetric Domains

I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...

**19**

votes

**0**answers

429 views

### Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...