**2**

votes

**1**answer

213 views

### Coordinates on Riemannian manifolds

Let $M$ be an n-dimensional manifold endowed with a Riemannian metric. Suppose we have a coordinate chart, say $(U,\varphi)$ where $U\subset M$ and $\varphi\colon U\rightarrow {\mathbb R}^n$, and let $...

**3**

votes

**1**answer

91 views

### On the $\omega$-limit set of a trajectory converging to a submanifold

Let $X$ be a $C^1$ vector field on $\mathbb{R}^n$. Let $S$ be a compact submanifold of dimension $s(<n)$. Suppose $S$ is invariant under the flow of $X$ and that we know everything about the ...

**7**

votes

**0**answers

111 views

### Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices

Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...

**2**

votes

**0**answers

115 views

### question about currents

I have a question in the field of currents:
Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...

**6**

votes

**0**answers

187 views

### Is the space of holomorphic maps a manifold

To be more specific:
Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...

**0**

votes

**1**answer

153 views

### Normal bundle of a fiber of the family of curves

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\...

**6**

votes

**1**answer

354 views

### Techniques to solve a non linear differential equation related to curvature

Many years ago, I considered the following non linear differential equation:
$y=y''.(1+y'^{2})^{-3/2}$
This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and ...

**2**

votes

**0**answers

110 views

### Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...

**1**

vote

**0**answers

68 views

### Splitting of totally geodesic Riemannian foliations

Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...\...

**2**

votes

**1**answer

155 views

### projectivity with assumption of big and semi-amplness

Let $X$ be a compact Kaehler manifold with $D$ be an effective divisor on $X$ such that $K_X+D$ is semi-ample and big then $X$ is projective?

**5**

votes

**0**answers

69 views

### Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?

Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...

**14**

votes

**2**answers

508 views

### List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds:
Although it looks like a rather innocent technical statement, it is
crucial for ...

**13**

votes

**1**answer

256 views

### Does a Kähler manifold always admit a complete Kähler metric?

Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case?
Does a ...

**1**

vote

**1**answer

191 views

### Question on Harmonic maps between Riemannian manifolds

In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows:
$$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$
In many Books such ...

**2**

votes

**0**answers

51 views

### On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$
\begin{equation}
(\...

**0**

votes

**0**answers

86 views

### Two questions about Li-Yau-Hamilton estimate

This question is from my question on mathematics.
Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$,
$(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...

**4**

votes

**1**answer

325 views

### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...

**1**

vote

**0**answers

58 views

### Covering rough boundaries of closed sets in manifolds by charts

This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible.
Consider a Riemannian ...

**1**

vote

**0**answers

109 views

### Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it.
Intuitively, an orbifold, from what I understand, is a "...

**5**

votes

**0**answers

63 views

### Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...

**8**

votes

**2**answers

172 views

### Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...

**6**

votes

**1**answer

355 views

### The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...

**2**

votes

**0**answers

110 views

### Parametric surfaces in $\mathbb{R}^4$ via quaternion multiplication of curves

In this question, $\mathbb{R}^4$ is identified with the quaternions $\mathbb{H}$ via the map $$(a_1, a_2, a_3, a_4) \mapsto a_1+ia_2+ja_3+ka_4,$$
where $1,i,j,k$ are the standard basis elements of ...

**2**

votes

**1**answer

174 views

### The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the ...

**0**

votes

**0**answers

48 views

### Small perturb a continuous map

I am reading the book: Convex Integration Theory by D. Spring and encounter a
question which I subtract as follows.
Let $p: X \rightarrow V$ be a smooth fibre bundle and $\mathcal{R} \subset X^{(1)}$...

**0**

votes

**0**answers

204 views

### Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function?
I know that this metric is a special kind ...

**25**

votes

**3**answers

1k views

### What's the supersymmetric analogue of the Monster group?

Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...

**4**

votes

**1**answer

171 views

### Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page.
I am asking if it ...

**0**

votes

**0**answers

33 views

### A C(B)-module structure on the function algebra of the total space of a vector bunlde $\pi:V \to B$

For a continuous vector bundle $\pi:V \to B$ vector bundle over a compact Hausdorff space $B$, and $C(B)$, $C(V)$ the continuous complex valued functions on $B$ and $V$ respectively, we can give $C(...

**4**

votes

**0**answers

70 views

### Geometrically-explicit upper bound for on-diagonal heat kernel

Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form
$$K(t;z,z) \leq \frac{C_M}{f_z(t)...

**2**

votes

**2**answers

295 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**5**

votes

**0**answers

74 views

### In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context.
Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...

**1**

vote

**0**answers

133 views

### Stronger version of Bertini's theorem

In char 0, is there a generalised version of Bertini's theorem that will ensure that for a proper map $f: Y\rightarrow X$ between smooth projective varieties and for every point $x\in X$ we can find ...

**0**

votes

**1**answer

112 views

### 3-form torsion and Cartan structural equations

First, my level of math isn't very high as I come from the physics world.
I am trying to understand the derivation of Cartan's 3-form torsion.
I've read Robert Bryant's answer in this thread:
Relating ...

**2**

votes

**1**answer

75 views

### Set of singular points for momentum map (with coisotropic action)

Let $G$ be a Lie-group acting on a connected symplectic manifold $M'$ in a hamiltonian way, with an $\operatorname{Ad}^*_G$-equivariant momentum map. Assuming $G$ acts properly on $M'$, we can ...

**11**

votes

**2**answers

471 views

### Unexpected regularity of the distance from a $C^2$ submanifold

Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...

**2**

votes

**1**answer

133 views

### Conformal vector field on the sphere

Let's $\mathbb{S}^d$ be the unit sphere with it's standard metric $g$. A vector field $X \in \mathfrak{X}(\mathbb{S}^d)$ is conformal if and only if there is a function $f \in C^{\infty}(\mathbb{S}^{d}...

**0**

votes

**0**answers

33 views

### Reparametrisation of a PDE with arclength

Suppose I have the following PDE:
$\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$
I notice ...

**0**

votes

**0**answers

143 views

### Problem of Weakly closed

Goodmornig everyone, my problem is:
Let $X$ a complex analytic $n$-manifold, for $p,q$ positive integers with $p,q$$<=$$n$, let $K^p$$^,$$^q$$(X)$ the bi-grade $(p,q)$ current space on $X$. i.e. ...

**3**

votes

**1**answer

149 views

### Definition of Levi-Civita connection map and a theorem about it?

Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem:
Theorem: If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $...

**1**

vote

**4**answers

180 views

### On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

[This question was asked on MSE, but got no answers, I thought it could be more appropriate here]
Let $M$ be a parallelizable manifold.
Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$...

**1**

vote

**1**answer

97 views

### Applications of Hessian operator in the Riemann manifold. Simple samples $S_{2}(f)$

Study article R. C. Reilly is entitled Applications of Hessian operator in the Riemann manifold had a doubt in the remark, shortly after the theorem 2 of that Article.
The theorem is stated as: ...

**3**

votes

**1**answer

90 views

### Sufficient conditions for a curve on the sphere to be the Gauß map of a closed path

I was wondering which curves on the $n-1$ sphere arise as the Gauss maps of closed paths in $\Bbb R^n$. Necessary conditions are obviously that the path on the sphere is the image of some smooth $S^1\...

**2**

votes

**0**answers

89 views

### Volume comparison under Ricci curvature upper bounds

Say I have a Hadamard $d$-manifold $M$ with an upper Ricci curvature bound of $-b^2$. Write the volume form in polar exponential coordinates at $p\in M$ as $V(r,\theta) \, dr \, d\theta$, and ...

**3**

votes

**0**answers

75 views

### Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...

**1**

vote

**1**answer

128 views

### Open non-parallelizable 4-manifolds

Let $M$ be a connected orientable open 4-manifold (noncompact, without boundary).
Is it possible for $M$ to be non-parallelizable ?
If yes, what example of such $M$ is there ?
[EDIT : The answer ...

**7**

votes

**2**answers

266 views

### Surfaces contained in a ball

In this Paper there is a proof that a closed plane curve of length
$L$ and curvature bounded by $K$ can be contained inside a circle of radius
$L/4 - (\pi - 2)/2K$. Are there similar results for ...

**9**

votes

**0**answers

241 views

### Are harmonic mappings of Riemannian manifolds always non-singular outside a set of measure zero?

Let $(M,g)$ be an $n$-dimensional, connected, compact, oriented, smooth Riemannian manifold with boundary. Assume we are given an immersion $f \colon M \to \mathbb{R}^n$ (note that $n=\dim M$).
Let $...

**3**

votes

**1**answer

67 views

### Hessian of distance function from a Jordan curve

Assume that $\gamma$ is $C^2$ Jordan curve in $\mathbf{R}^3$ and let $f(x)=d^2(x,\gamma)$. Is there any known formula for the hessian of $f$ near $\gamma$?

**1**

vote

**0**answers

58 views

### Are affine maps (wrt to a connection), which preserve a tensor field, given by a PDE?

Let $(M, \nabla)$ be a manifold together with a connection on $TM$ and let $T$ be a tensor field on $M$. Suppose the pseudogroup $\Gamma$ of locally defined smooth maps, that simultanously preserve ...