**2**

votes

**1**answer

60 views

### The Chern connection on a Hermitian symmetric domain

There's a connection (the Chern connection) on the Tangent Bundle of a Kahler Manifold which is compatible with both the hermitan metric, and the holomorphic structure. In general, I guess there's no ...

**0**

votes

**0**answers

81 views

### How to prove this inequality of heat flow from Weitzenbock formula?

Let $(M,g), (N,h)$be a compact Riemannian manifolds, $m:=\dim M, n:=\dim N\geq 2$,
and $N$ is a non-positive curvature $K_N\leq 0$. All connections which appear below are the Levi-Civita connections. ...

**3**

votes

**1**answer

168 views

### Lifting a Diffeomorphism to the Cotangent Bundle

Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving.
...

**2**

votes

**1**answer

124 views

### How to find Darboux coordinates?

I would like to find local Darboux coordinates for symplectic structures on coadjoint orbits of some nilpotent Lie group. At first, I thought that this would be not very hard, and that it would be ...

**-1**

votes

**0**answers

43 views

### 3d kinematics of soft robots [on hold]

I am looking for literature where I can get the curvature and torsion information from the given 3d curve. Means I have 3d coordinates and I can generate a spline from it. I want a method to find out ...

**19**

votes

**1**answer

448 views

### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...

**-5**

votes

**1**answer

59 views

### Non Linear PDE's [on hold]

I want to solve two systems of questions being
$$\frac{C(r,y)''}{C(r,y)} + \frac{C(r,y)'^2}{C(r,y)^2}=0$$
and
$$A(r,y)'' + 2A(r,y)' \frac{C(r,y)'}{C(r,y)}=0$$
where ' is differentiating with respect ...

**0**

votes

**0**answers

62 views

### The limit of a sequence of embedded minimal disks in $\mathbb{R}^3$

Let $\Sigma_n,n\ge 1$ be a sequence of embedded minimal disks in $\mathbb{R}^3$ such that:
(1) $0\in\Sigma_n\subset B(0,r_n)$ with $r_n\to\infty$ as $n$ tend to $\infty$,
(2) ...

**2**

votes

**1**answer

185 views

### The points of half area of a triangle

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...

**4**

votes

**0**answers

89 views

### The geometric shape of domains of flows

Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow ...

**3**

votes

**1**answer

122 views

### Regular singularities and the infinitesimal site

Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$.
A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...

**0**

votes

**1**answer

53 views

### Calculate GPS coordinates at x meters [closed]

I want to calculate a pair of GPS coordinates(lat,long) that is at x meters N/S/E/W from a known point (lat_old,long_old).
I have found the Haversine formula
...

**1**

vote

**0**answers

57 views

### Structure of the zero set of analytic maps (Lojasiewicz’s Structure Theorem for Varieties)

The Lojasiewicz’s Structure Theorem for Varieties states that for $\Phi:\mathbb{R}^n\rightarrow \mathbb{R}$ real analytic with $\Phi(0)=0$ the set $\Phi^{-1}(0)$ is locally a union of subvarieties of ...

**1**

vote

**0**answers

69 views

### Computing the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$
and $\| v \| = \sqrt{|\langle v, v \rangle|}$.
Let ...

**2**

votes

**1**answer

172 views

### What is the difference between $\delta W^{\pm}=0$ and Einstein?

Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference ...

**1**

vote

**2**answers

199 views

### Schwarzian derivative of a diffeomorphism is zero iff Linear-fractionals?

I have found the following derivation of the Schwarzian derivatives in the book of Ovsienko and Tabachnikov:
For a diffeomorphism $\gamma$ which acts on 4 points $t_1,t_2,t_3,t_4 \in \mathbb{RP}_1$, ...

**0**

votes

**0**answers

78 views

### Does diff$(M)$ act transitively on the set of integrable $G$-structures?

Let $M$ be a compact manifold and diff$(M)$ its diffeomorphism group. Let $G$ be a Lie subgroup of $GL(n,\mathbb{R})$.
In general, the topology of a manifold may prevent it from having an integrable ...

**9**

votes

**1**answer

228 views

### Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a ...

**2**

votes

**1**answer

165 views

### The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof?

Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with ...

**4**

votes

**1**answer

156 views

### Analytic representatives for Kahler classes

If we are given compact complex manifold $X$ and a Kahler class $[\omega]$,
can we always find a positive definite representative $\omega \in [\omega]$ that is
real analytic?

**0**

votes

**0**answers

60 views

### Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|_{L^p} \leq ...

**1**

vote

**1**answer

150 views

### Universal bundles: construction of the map associated to a group homomorphism

For a Lie group $G$ let $EG \to BG$ denote the universal bundle. A Lie group homomorphism $\rho: G \to H$ determines a map $B \rho: BG \to BH$ as the classifying map for the principal $H$-bundle $EG ...

**2**

votes

**1**answer

76 views

### Projectively flat Hermitian curvature proportional to Kähler form?

Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to ...

**5**

votes

**1**answer

244 views

### A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let ...

**1**

vote

**0**answers

131 views

### Computation with the Legendre Transform

Let $M$ be a manifold and fix a Lagrangian $L\in C^\infty(T M )$. Let $x_1,\dots x_n$ be local coordinates for $M$ and equip the tangent bundle and cotangent bundle with standard coordinates ...

**1**

vote

**0**answers

185 views

### The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**6**

votes

**1**answer

86 views

### Integrating representations of Lie algebroids

If $A \to M$ is a Lie algebroid over a smooth manifold $M$ then a representation of $A$ is a vector bundle $E \to M$ with a flat $A$-connection
$$
\nabla : \Gamma(E) \to \Gamma(E\otimes A^*).
$$
If ...

**11**

votes

**1**answer

210 views

### Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...

**5**

votes

**1**answer

116 views

### Curvature and Failure to return to starting point

Assume I have a geodesic polygon $P$ in a Riemannian manifold $M$ that is given by the image of a piecewise geodesic closed curve $\gamma(t)$ (parametrized by arclength), with vertices $x_i = ...

**6**

votes

**0**answers

92 views

### Bundles over Function Spaces

Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable ...

**3**

votes

**3**answers

208 views

### Is the group of isometries of a homogeneous Riemannian manifold maximal?

I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
Iso is a proper subgroup of ...

**2**

votes

**0**answers

103 views

### Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...

**11**

votes

**3**answers

464 views

### Is there a “unique” homogeneous contact structure on odd-dimensional spheres?

Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in ...

**5**

votes

**1**answer

248 views

### Equivalence of exterior forms

Let us start with the following definition.
Let $1\leqslant k\leqslant n$ and let $\omega_1,\omega_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega_1$, $\omega_2$ are equivalent, if there exists ...

**0**

votes

**0**answers

96 views

### Gromov-Floer compactness for C^0 convergence of complex structure/ C^1 convergence of Hamiltonian

Let $M$ be a compact symplectic manifold, $J$ a possibly surface dependent complex structure, and $H$ a Hamiltonian on $M$. I am interested in a variant of Gromov-Floer convergence for solutions of ...

**5**

votes

**2**answers

181 views

### Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...

**7**

votes

**1**answer

126 views

### When are quotients of the diffeomorphism group Fréchet manifolds?

Let $M$ be a compact manifold and $\text{diff}(M)$ its diffeomorphism group. Various quotients of $\text{diff}(M)$ appear in the literature, oftentimes with geometric significance. A well-known ...

**0**

votes

**0**answers

141 views

### An inequality on closed manifolds

Excuse me again, I am not sure what kind of question the following is. Any suggestion is appreciated!
Let $(M,g)$ be a closed Riemannian manifold, $g$ is real analytic. Let $u,v$ be two nonnegative ...

**2**

votes

**1**answer

151 views

### When do curves exist in infinite-dimensional submanifolds?

Let me explain the motivation for my question by talking about the finite-dimensional situation. Let's say we have a $d$-dimensional $C^\infty$ manifold $M$ embedded smoothly in $\mathbb R^n$. We fix ...

**0**

votes

**1**answer

63 views

### Condition for infinite dimensional complex manifold to be Kähler by pullback form

For a finite dimensional Kähler manifold $M$, there is the condition that if $N\xrightarrow{f}M$ is a holomorphic map with maximal rank at each point, then $N$ is Kähler with the pullback form induced ...

**7**

votes

**2**answers

267 views

### A cohomology group which depends on the connection

Warning: I am not a differential geometer, so some of the following might not make sense.
Background:
Let $w: (T\Omega)^k \to \mathbb{R}$ be a $k$-tensor on $\Omega$, an open subset of ...

**0**

votes

**0**answers

51 views

### Does the number of zero eigenvalues correspond to the dimension of equilibrium manifold for nonlinear system?

Consider nonlinear dynamical system with $m \times m$ Jacobian matrix $J(x)$ that has $k$ zero eigenvalues for all $x$. The rest $m-k$ eigenvalues have negative real part for all $x$. Is that true ...

**17**

votes

**5**answers

3k views

### Why differential forms are important?

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...

**1**

vote

**1**answer

185 views

### Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function
$h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is:
...

**2**

votes

**0**answers

51 views

### isoperimetric problems on Alexandrov spaces

For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by
$$
...

**3**

votes

**2**answers

118 views

### Differentiable structure on the Gauge group of a principal bundle?

I am currently reading this paper in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most ...

**2**

votes

**1**answer

254 views

### A question on Schrodinger operator

I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated!
Let $(M,g)$ be ...

**1**

vote

**1**answer

99 views

### Volume bounds of balls in Riemannian manifolds

Let $(M,g)$ be a complete Riemannian manifold and suppose $\mathrm{Ric}(g) \geq -k$ for some $k>0$. Suppose we know that $\mathrm{vol}_g (B_1^g (x_0)) \geq \nu$ for some particular $x_0 \in M$ and ...

**4**

votes

**0**answers

77 views

### Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...

**3**

votes

**1**answer

180 views

### is the stress tensor (elasticity theory) actually a pseudo tensor? [closed]

Please, is the stress tensor (elasticity theory) actually a pseudo tensor? It seems to me it must change its sign when coordinate system changes its orientation.