**0**

votes

**0**answers

18 views

### Systole of a flat surface

Is the systole (length of the shortest saddle connection) of a flat surface $(X,\omega)$ ($X$ is a Riemann surface and $\omega$ an abelian differential on it with zeros in the points $\Sigma=\{p_1,\...

**1**

vote

**0**answers

113 views

### Fibration when central fibre is a Calabi-Yau variety

Let $f\colon X\to Y$ be a surjective proper holomorphic fibre space such that X and Y are projective varieties and central fibre $X_0$ is Calabi-Yau variety and Y is also Calabi-Yau variety, then can ...

**-2**

votes

**0**answers

69 views

### An Example to Marsden-Weinstein Theorem [on hold]

Suppose that the action of a compact Lie group $G$ on the closed symplectic manifold $(M,\omega)$ is Hamiltonian, with moment map $\mu : M\to \mathfrak{g}^*$. From the Hamiltonian condition it ...

**3**

votes

**1**answer

95 views

### Elementary question: Curvature change under Complexified Gauge Transformation

Forgive me for this elementary question.
Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ ...

**-3**

votes

**0**answers

64 views

### Advice on Family Index theorem [on hold]

I am reading Bismut's paper Family Index and Heat equation, but I have knowledge on the probability or stochastic.
Could anyone give some advice or introduce some ref. on probability to understand ...

**3**

votes

**2**answers

66 views

### Is there a full-rank map with connected graph and simply connected image that is not injective?

I want to find a continuously differentiable function $F:X\to Y$, where $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$ are open ($n\le m$) with
${\rm rk}\, \frac{\partial F}{\partial x}(x) = ...

**13**

votes

**1**answer

945 views

### What should I cite for the Poincaré conjecture?

I'm writing a paper that, rather unexpectedly, needs the Poincaré conjecture for one of the results. (The paper has almost nothing to do with differential geometry!)
The conjecture was famously ...

**1**

vote

**0**answers

110 views

### The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...

**1**

vote

**1**answer

123 views

### Lift Lie group action on a small neighborhood

Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$.
Q: Why in a small neighborhood of $N$, $G$ also action ...

**4**

votes

**1**answer

92 views

### Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature

Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...

**6**

votes

**0**answers

71 views

### Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n=dim M$

For a connected $n$ manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a pointe $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\} $....

**0**

votes

**1**answer

103 views

### Right inverse of the Seiberg-Witten functional

For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e.
$$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...

**3**

votes

**0**answers

127 views

### Research topics in Curves and Surfaces [on hold]

I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry ...

**0**

votes

**1**answer

117 views

### Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold.
Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.

**0**

votes

**0**answers

93 views

### Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...

**6**

votes

**1**answer

209 views

### The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$

Assume that $M$ is an arbitrary manifold.
Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
If not, what is a counter ...

**2**

votes

**0**answers

79 views

### Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is
ample. then there is a Kahler metric $\omega_E$ such that
$Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then
$h=\frac{1}{\...

**1**

vote

**0**answers

46 views

### Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that
$$S^°=\{s\in S| f\text{ is ...

**1**

vote

**0**answers

67 views

### Singular canonical hermitian metric

Let $M$ be a complex manifold , take
$$K_{M,m}:=\sup \{|\sigma|^{\frac{2}{m}}; \sigma\in H^0(M,\mathcal O_M(mK_M)), |\int_M(\sigma\wedge\bar\sigma)^{\frac{1}{m}}|\leq 1\}$$
Let $$K_{M,\infty}:=\lim\...

**0**

votes

**1**answer

69 views

### Heat kernel upper bounds on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds:
$$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ...

**7**

votes

**2**answers

212 views

### Curves embedding in plane

Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve can be embedded inside the other one, here embedding ...

**4**

votes

**0**answers

76 views

### Symbol of differential operator and change of variables [closed]

Recently I posted the following question on stack exchange, but it remained with no answer http://math.stackexchange.com/questions/1863658/symbol-of-differential-operator-and-change-of-coordinates
I ...

**1**

vote

**1**answer

113 views

### Example of bundle-mapping over $S^4$ with singularity $S^2$

Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E_0,E_1$ over $S^4$ and a bundle mapping
$$0\to E_0\overset{v}{\to}E_1\to0$$
such ...

**3**

votes

**1**answer

179 views

### Rank 2 vector bundles over $\mathbb CP^2$

Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism?
Thank you.

**1**

vote

**1**answer

132 views

### Rank 2 complex vector bundles over $S^2\times S^2$

How could people classify all rank $2$ complex vector bundles over $S^2\times S^2$ up to isomorphism?
Could you give a rank 2 complex vector bundle which cannot be split as a sum of two line bundles?

**8**

votes

**0**answers

136 views

### Penrose transform and general wave equations

In the late 1960's Penrose developed twistor theory, which (amongst other things) lead to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose ...

**5**

votes

**2**answers

220 views

### Rank 2 complex vector bundles over $S^4$

On $S^4$, we know that rank 2 complex vector bundles are classified by $\pi_3(U(2))=\mathbb Z$. Any element $g\in\pi_3(U(2))=\mathbb Z$ determines a complex vector bundle $E$ over $S^4$. Can we say ...

**2**

votes

**0**answers

70 views

### relative quantization on fibration

Let $\pi:X\to B$ be a holomorphic submerssion of two Kaehler varieties $X,$ $B$ and $(B,\omega)$ be quantizable and fibres $X_s$ also are quantizable, then $X$ is quantizable?. I want to define ...

**6**

votes

**0**answers

71 views

### k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.
Question. Are ...

**4**

votes

**1**answer

164 views

### Does every smooth manifold admit a metric with bounded geometry and uniform growth?

Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...

**2**

votes

**0**answers

113 views

### Chern character (form) of a Gauss-Manin connection

Consider the trivial fibration $\mathbb{T}^2\to\mathbb{S}^1$, where $\mathbb{T}^2$ is the two-torus. Denote by $\mathbb{C}\to\mathbb{T}^2$ the trivial line bundle over $\mathbb{T}^2$, and equip it ...

**3**

votes

**1**answer

210 views

### Lie algebra of invariant polynomials or invariant smooth functions

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb{...

**2**

votes

**0**answers

74 views

### Parabolicity, equivalent definitions

How to prove an equivalence between the following two definitions of parabolic manifolds:
(1) A Riemannian manifold $M$ is called parabolic if for each compact subset $K\subset M$ and each $\epsilon &...

**7**

votes

**1**answer

154 views

### Geometric Construct for Integrating Symmetric Tensors?

I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.
The motivation comes ...

**3**

votes

**1**answer

74 views

### Does every Zoll metric on $\mathbb{S}^2$ arise from a perturbation of the round metric?

The introduction here states 'A formal perturbation argument
of Funk later indicated that, modulo isometries and rescalings, the general Zoll
metric on $\mathbb{S}^2$ depends on one odd function $f:\...

**0**

votes

**0**answers

182 views

### Differential rotations in Chebychev Net

A Chebychev net obeying Sine-Gordon equation is drawn on a surface of constant negative Gauss Curvature $K$ so that the asymptotic differential rhombic element corners lie on lines of maximum/minimum ...

**1**

vote

**0**answers

84 views

### Riemann surface defined by a Beltrami differential

Let $R$ be a Riemann surface, $\omega$ and $\tau$ respectively a holomorphic and an antiholomorphic 1-form on $R$. Locally $\omega=fdz$ and $\tau=gd\overline{z}$ with $\partial_{\overline{z}}f=0$ and $...

**1**

vote

**0**answers

43 views

### Morphing between constant Gauss curvature surfaces

Requesting responses on two related questions basically. First,
the Riemann sphere has umbilical points everywhere. Including this case, can a
bending morph parameter be defined to continuously ...

**2**

votes

**0**answers

48 views

### Thom form of holomorphic bundle over Kaehler manifolds/orbifolds

Consider a holomorphic vector bundle $\pi:E\rightarrow X$ of complex rank $m$ over a Kaehler manifold $X$. Can we find a Thom form $\Theta$ of $E$ such that as a form on the complex manifold $E$, it ...

**2**

votes

**1**answer

146 views

### How to increase the injectivity radius function of a hyperbolic 3 manifold of finite volume?

Let $N$ be an oriented hyperbolic 3-manifold of finite volume and let $\Delta \subset N$ be a smooth connected compact subdomain such that the restriction of the injectivity radius function of $N$ to $...

**1**

vote

**0**answers

24 views

### Formalization of adaptive sampling [closed]

The notion on adaptive sampling or adaptive plotting is fairly popular, but I have not found a formal definition.
I have developed an algorithm for plotting implicit algebraic curves in the plane. ...

**2**

votes

**0**answers

47 views

### Multiplicativity of the analytic index (or of kernel bundle)

What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...

**1**

vote

**0**answers

38 views

### covariant derivative of Ricci operator [closed]

I am going through a calculation where we have in hand...
$\nabla_Y Du = -QY + \lambda Y,$ where $u$ and $\lambda$ are smooth real valued functions on a manifold $M$ and $Q$ is the Ricci operator. ...

**2**

votes

**0**answers

102 views

### Distance metric on Riemannian quotient manifold

It's well known that if I have a Riemannian manifold $M$ and a Lie group of isometries $G$ that acts freely and properly on $M$, then the quotient $M/G$ is a manifold and inherits the Riemannian ...

**0**

votes

**0**answers

23 views

### Property of solid spanned by optimal functions

I can only solve this in a two-dimensional world. I want to characterize a solid that is spanned by cutoff functions $g$ that are optimal in the following sense.
Let us suppose we have $n$ random ...

**2**

votes

**1**answer

154 views

### Decomposition of pullback metric

Let $(M^3,g)$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the immersion $\phi: \Sigma \times [0,\varepsilon)\to M$ given by
$$\phi(p,t)=\...

**1**

vote

**0**answers

151 views

### Symplectic structures on the total space of vector bundles

What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure?
To what extent all ...

**1**

vote

**0**answers

129 views

### Bending Beltrami Pseudosphere

The Beltrami Pseudosphere
$$[x = a \sin p \cos t , y= - a ( \cos p + \log \tan p/2 ) , z= b+ a \sin p \sin t \; ], (.1 <p<\pi/2), (0< t< 2 \pi), \; (b>a) $$
is bent to a non-...

**3**

votes

**0**answers

140 views

### Geometrical point of view of the harmonic constraints ($\Delta g_{ij}=0$) in General Relativity

What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-...

**1**

vote

**1**answer

104 views

### A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$

According to the answer of Sebastan and previous edit of Ben McKay I revise my post as follows:
Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$.
Is there a (...