**3**

votes

**1**answer

196 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

**3**

votes

**0**answers

134 views

### Hilbert's syzygy theorem in the analytic setting

If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite global resolution by locally free modules.
By GAGA, I believe this ...

**1**

vote

**0**answers

121 views

### Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on ...

**1**

vote

**1**answer

129 views

### manifolds whose charts are maps to Minkowski space

I'm doing a project involving tilings of Minkowski space. For instance in 2d I have rectangular tiles determined by a spacelike line segment: the rectangle is the region caused by the line segment. ...

**5**

votes

**3**answers

204 views

### Constructing a vector field with given zeros on a torus

By the Hopf-Poincaré theorem, the sum of the indices of the zeros of a vector field on the d-dimensional torus must equal zero. Given an even number of points $x_i$ on a d-dimensional torus, and ...

**0**

votes

**2**answers

97 views

### Derivations of algebra of smooth $g$-valued function?

Let $M$ is a smooth n-manifold and $g$ is a $Z_2$-graded Lie algebra, we denote the algebra of smooth $g$-valued function on $M$ by $C^{\infty} (M,g)$. I wanna find all graded derivation of ...

**2**

votes

**1**answer

168 views

### Is this sphere bundle over SL3/SO3 trivial?

The space $SL_3(\mathbb{R})/SO_3(\mathbb{R})$ can be though of as some kind of 5-(real)dimensional generalized upper half-space.
Fix any copy of $SO_2(\mathbb{R})$ sitting inside $SO_3(\mathbb{R})$, ...

**2**

votes

**2**answers

163 views

### Probability of random geodesics on the round sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim round sphere. Here uniformly is in the obvious sense of the volume form ...

**1**

vote

**2**answers

334 views

### Quantization by cohomology

Ok, so I have heard some cool stuff here and there about how to Quantize Yang-Mills via cohomology, can anyone refer any texts in the literature that have shed some light on this, I mean I have some ...

**4**

votes

**2**answers

364 views

### The example of mechanical system that has a Mobius strip as their configuration space

Can you give examples for mechanical system that has a Mobius strip as their configuration space?

**0**

votes

**0**answers

137 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

**5**

votes

**2**answers

397 views

### Does for every vector field there always exist a volume form for which the vector field is a homothety?

Let $v$ be a vector field. Does there exists a volume form $\Omega$
such that its Lie derivative is proportional to itself with a constant coefficient:
$$\mathcal{L}_v \Omega= C \cdot \Omega? \ \ \ ...

**3**

votes

**1**answer

200 views

### Existence and Properties of 3D Curves with unusual 2D $(\kappa(s),\tau(s))$ Trajectories

This question is inspired by Surface in 3D that realizes all pairs of principal curvatures
While one can imagine, that a 3D surface could exist, that realizes all pairs of principal curvatures, ...

**5**

votes

**1**answer

527 views

### What is the geometric interpretation of this quantity?

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic
structure on the tangent bundle, $(TM, ...

**3**

votes

**1**answer

400 views

### How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.
...

**7**

votes

**1**answer

169 views

### Flows of vector fields and diffeomorphisms isotopic to the identity

Let $M$ be a compact manifold and $\varphi : M \longrightarrow M$ be a diffeomorphism which is isotopic to the identity. Does there exist a vector field $ X $ on $M$ such that $\varphi$ is the flow at ...

**0**

votes

**1**answer

153 views

### An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
...

**0**

votes

**0**answers

93 views

### Dimension of the set of polarized sections

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, ...

**6**

votes

**1**answer

244 views

### What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...

**1**

vote

**1**answer

268 views

### The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation.
Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...

**1**

vote

**1**answer

98 views

### Bitangent locus of torus knots

Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...

**4**

votes

**0**answers

79 views

### are smooth homotopic open embedding of Hilbert manifolds smoothly isotopic?

A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert ...

**4**

votes

**2**answers

179 views

### Some facts about cut-locus

Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$.
S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree."
How the set of point can be a tree? ...

**2**

votes

**1**answer

77 views

### An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar ...

**6**

votes

**2**answers

461 views

### Obstructions for a metric to be conformally equivalent to a product metric

Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformally equivalent to a product metric?
More generally, assume that $M$ and $N$ are two manifolds. ...

**1**

vote

**1**answer

131 views

### If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?

Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function.
When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...

**1**

vote

**1**answer

121 views

### family of metrics with same geodsics

For every bi-invariant metric on a lie group we know geodesics are flow of left invariant vector fields, so this question naturally arise:
are there family of metrics on manifolds that have same ...

**1**

vote

**2**answers

151 views

### Space curves and torsion

Consider smooth closed space curve (parametrised by its arc length) in 3 dimensional space.Does there exist atleast one point at which both the curvature & torsion attain extremum values?

**1**

vote

**1**answer

104 views

### Group of CR automorphisms

Let $(M, D, J)$ be a strictly pseudoconvex hypersurface type CR manifold with $J$ integrable.
Let $D$ be the kernel of a $1$-form $\eta_0$.
As known the automorphism group is defined to be
$$
CR = \{ ...

**3**

votes

**0**answers

143 views

### Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$.
...

**-1**

votes

**1**answer

137 views

### $S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$

**6**

votes

**2**answers

377 views

### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...

**2**

votes

**0**answers

103 views

### On Corlette's paper Flat G-bundles with canonical metrics

I am now reading Kevin Corlette's paper:
Flat G-bundles with canonical metrics, JDG, 1988
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214442469
I ...

**0**

votes

**0**answers

130 views

### Integral of Square of Mean Curvature

Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When ...

**1**

vote

**1**answer

129 views

### Right Invariant Randers metrics

I'm hoping to determine the geodesic equation for a right invariant Randers metric $F(x) = \sqrt{a(x,x)} + b(x)$ on $SU(N)$. In my special case the navigation data $(h,W)$ for the Randers metric are ...

**2**

votes

**1**answer

155 views

### Closed surface with uncountably many conic points?

Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere?
We call $p\in M$ a ...

**3**

votes

**1**answer

150 views

### equivariant whitney product formula

Let $G$ be a compact Lie group and $B$ be a closed smooth $G$-manifold. If we have two smooth oriented $G$-vector bundles:
$$E\longrightarrow B$$
and
$$F\longrightarrow B$$
then is there exists an ...

**8**

votes

**3**answers

357 views

### Number of disjoint simple closed geodesics

According to Jairo comment on the first version of this question I revise the question as follwos;
Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that:
There are at most ...

**2**

votes

**0**answers

147 views

### Surface with bounded torsion propery [closed]

Let $S$ be a surface in $\mathbb{R}^{3}$ with the following property:
There is a uniform constant $M$ such that for every Frenet curve $\gamma(t)$, contained in $S$, $| \tau(t) | \leq M$, for all ...

**6**

votes

**2**answers

468 views

### Trying to Understand Lefschetz Pencils

All:
I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) below better, though I would appreciate insights on condition i), and in general. Please forgive if the presentation ...

**3**

votes

**0**answers

110 views

### Orthonormal frame bundle orthogonal to a curve

This is a duplicate of this question on math.stackexchange, since I got there not a single answer.
Let $M$ be a $n$-dimensional smooth riemannian manifold and ...

**5**

votes

**1**answer

172 views

### Holonomy group of Enriques surface

I expect that the holonomy group of an Enriques surface $S$ is $SU(2)\times C_2$. I think this can be proven by the fact that its double cover, which is a K3 surface, has the full $SU(2)$ holonomy, ...

**1**

vote

**0**answers

147 views

### Characterization of Differentiability via Lie Derivatives [closed]

As the starting point of a project I intend to propose the proof of the statement below, but I want to make sure that it is not already proved somewhere else before.
Let $G$ be a Lie group, and $f$ a ...

**2**

votes

**1**answer

220 views

### Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies.
The situation is this:
Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...

**6**

votes

**2**answers

218 views

### How to compute the normals to Costa's minimal surface?

I am trying to draw Costa's minimal surface in high resolution using the PovRay raytracer. For this I need to compute points on the surface as well as the normals. It is relatively easy to compute the ...

**4**

votes

**2**answers

103 views

### Normalized packing number

Suppose $M$ is a $n$-dimensional closed Riemannian manifold. Let the packing number $N(t)$ be the maximum number of balls with radius $t$ in $M$ that are disjoint. I am wondering whether the following ...

**1**

vote

**0**answers

94 views

### How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]

myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in ...

**4**

votes

**3**answers

219 views

### On discrete version of curve shortening flow

One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...

**-2**

votes

**2**answers

281 views

### Are there some websites for self learining of advanced mathematics? [closed]

Are there some websites for self learining of advanced mathematics?
For example, some great lecture vedio of differential geometry, Lie group , Lie algebra, algebraic topology and so on. Thanks

**2**

votes

**0**answers

79 views

### Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action

My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...