**2**

votes

**1**answer

132 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}...

**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

**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: ...

**0**

votes

**0**answers

40 views

### Limit of argmin of sum

Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...

**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 $...

**1**

vote

**0**answers

117 views

### The Yamabe problem and $\phi^4$ scalar field theory?

The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap
In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the ...

**7**

votes

**4**answers

493 views

### General Relativity and Differential Geometry intuitions of Second Bianchi Identity

In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-
$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$
It is said that ...

**5**

votes

**1**answer

141 views

### Comparison of angles in Alexandrov space

Let $X$ be a finite dimensional Alexandrov space with curvature bounded below. Let $p\in X$ be a fixed point.
Is it true that for any $\varepsilon >0$ there exists $\delta>0$ such that for any $...

**0**

votes

**0**answers

59 views

### Linearization of specific plane vector field

I have a vector field $v = (f(x,y), \alpha y)$, such that $f(0, 0) = 0$ and $df (0, 0) = (1, 0)$. When is smooth linearization possible and when is it not?
I only see obstacles in the form like this:...

**2**

votes

**1**answer

233 views

### Relation between Harmonic vector field and Harmonic 1-form

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^...

**5**

votes

**2**answers

113 views

### References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...

**2**

votes

**0**answers

77 views

### Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group.
Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...

**0**

votes

**0**answers

79 views

### How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields.
I dont know ...

**2**

votes

**2**answers

149 views

### Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...

**9**

votes

**1**answer

311 views

### A careful roadtrip from locally symmetric spaces to algebra

I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...

**5**

votes

**0**answers

59 views

### Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...

**11**

votes

**4**answers

327 views

### Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...

**4**

votes

**1**answer

147 views

### Compact manifolds locally bi-Lipschitz to Euclidean space

I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...

**0**

votes

**0**answers

39 views

### Nonstandard support function for the Busemann function

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold.
Assume that $M$ contains a ray $\gamma : [0, \infty) \to \mathbb{R}$. Let $b_\gamma$ be the associated Busemann function, i.e.,
$$
b_\...

**2**

votes

**0**answers

58 views

### Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$.
My question concerns solutions to $\triangle_g u =0$ that are say ...

**20**

votes

**2**answers

555 views

### Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity?
Of course, such a manifold must not admit a diffeomorphism ...

**4**

votes

**1**answer

76 views

### How isometric action on Riemannian manifold acts on cut locus

Assume that $M$ is a simply connected closed Riemannian manifold with no boundary and nonnegative sectional curvaure Assume that ${\bf Z}_n=(g),\ n\geq 3$ acts on $M$ isometrically. Then if $gx=x$, i....

**5**

votes

**1**answer

109 views

### Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....

**-1**

votes

**1**answer

70 views

### Zariski open set in orthogonal grassmanian [closed]

I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...

**4**

votes

**1**answer

142 views

### how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...

**1**

vote

**0**answers

48 views

### Is Fano Kahler surface with reverse orientation also Kahler?

In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?

**9**

votes

**1**answer

110 views

### Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...

**17**

votes

**3**answers

474 views

### Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...

**3**

votes

**1**answer

93 views

### coisotropic action on $TS^{2n+1}$

Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(...

**2**

votes

**0**answers

64 views

### A problem of defining addition in a Quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...

**5**

votes

**2**answers

152 views

### Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?

(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the $...

**6**

votes

**2**answers

239 views

### Index of Modified Dirac Operator

Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...

**2**

votes

**1**answer

138 views

### isometric action on the $n$-sphere

Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely
$$ \Psi \colon S^1 \times ...

**2**

votes

**0**answers

89 views

### Willmore functional

Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$.
We know that $(\...

**2**

votes

**1**answer

65 views

### Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$

Consider the group $GL_n(\mathbb{R})$ with its standard topology.
It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's ...

**4**

votes

**1**answer

139 views

### Distance comparison in submanifold versus in the underlying manifold

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$.
QUESTION I:
The above ...

**0**

votes

**1**answer

83 views

### When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold? [duplicate]

Also asked here: http://math.stackexchange.com/questions/1725787/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz
A reference on totally geodesic submanifold (TGS):
...

**1**

vote

**1**answer

112 views

### Riemannian metric on a level set of a smooth function on a manifold

Also asked here: http://math.stackexchange.com/questions/1725491/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold
Let $(M,g)$ be a finite or infinite dimensional Riemannian ...

**16**

votes

**1**answer

422 views

### Just how close can two manifolds be in the Gromov-Hausdorff distance?

Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...

**6**

votes

**2**answers

173 views

### Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...

**4**

votes

**1**answer

274 views

### On the complexification of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...

**2**

votes

**2**answers

136 views

### Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...

**3**

votes

**0**answers

67 views

### Invariant Lagrangians of a connection and its derivatives: how do they look like?

Let
$$
L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma)
$$
be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...

**2**

votes

**0**answers

109 views

### Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...

**0**

votes

**1**answer

125 views

### Projection of geodesic is geodesic

Background : If a compact Riemannian manifold $M$ with a no curvature condition has disjoint two submanifolds $N_i$, then the distance between them is attained by some minimizing geodesic $c$.
If $c'...

**2**

votes

**1**answer

71 views

### Some manifold which is not totally geodesic in a compact manifold

(1) If $N^k$ is a submanifold in a compact Riemannian manifold $M^{k+m},\ m\geq 1$ s.t. each $p\in N$ has the following property : There exists independent set $\{ X_i\}_{i=1}^k$ tangent to $T_pN$ s.t....

**0**

votes

**0**answers

47 views

### Foliation by Umbilic Surfaces

Suppose $(M,g)$ is a simply connected 3 dimensional Riemannian Manifold which is a foliation by Umbilic surfaces.
Can I make the claim that there exists a coordinate system $(x_1,x_2,x_3)$ in which ...

**-3**

votes

**1**answer

252 views

### Well-known name for a certain connection

Have $X \subset \mathbb{R}^3$ be a smoothly embedded surface. Then we try to define a connection on the tangent bundle $TX$ as follows. The tangent space $T\mathbb{R}^3$ is naturally a trivial $\...

**1**

vote

**1**answer

151 views

### Applying Cheeger and Colding segment inequality

The question turns out quite long and maybe a bit vague, I apologize in advance for that.
I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I ...

**1**

vote

**0**answers

62 views

### Possible directions of saddle connections

Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...