**3**

votes

**1**answer

79 views

### The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...

**-1**

votes

**0**answers

114 views

### A metric on $S^{2}$ [on hold]

Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel z\parallel^{2}-\parallel w\parallel^{2},\;2z\bar{w})$.
Define a metric on $S^{2}$ as follows:
$$d(x,y)=Hd(p^{-1}(x), ...

**2**

votes

**0**answers

107 views

### Generalized metric on spacetimes

I read many articles about space-times. Most authors consider these spaces as warped product manifolds $I\times M$ where $I$ is an open connected interval of the real line and $M$ is a Riemannian ...

**3**

votes

**1**answer

103 views

### Has uniform ellipticity implications on the spectrum?

Let $X$ be a complete Riemann surface with a smooth metric, and $L$ a line bundle on it also equipped with a smooth metric; associated to this data there is a Laplace-Beltrami operator $D_L$ acting on ...

**4**

votes

**2**answers

475 views

### Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

**9**

votes

**3**answers

701 views

### Manifolds admitting flat connections

For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...

**3**

votes

**0**answers

35 views

### Does local reducibility imply global reducibility of universal covering?

Let $M$ be a locally reducible complete Riemannian manifold, that is, for any $p \in M$, we can find an open set $U$ around $p$ and two Riemannian manifolds $X$ and $Y$ such that $U$ is isometric to ...

**2**

votes

**0**answers

72 views

### Harmonic maps and centers of mass in Riemannian manifolds

Consider a smooth map $f : M \to N$ between two Riemannian manifolds $(M,g)$ and $(N,h)$. I would like to think of the tension field of $f$ and the harmonicity of $f$ in terms of centers of mass.
I ...

**4**

votes

**2**answers

141 views

### Reference for when a metric on a four-manifold is Kahler?

In a paper of Derdzinski (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...

**3**

votes

**1**answer

69 views

### Horizontal lift of differential operator

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
$X^{\mathrm{hor}}$ is a ...

**2**

votes

**0**answers

55 views

### Generalized Hawking Mass

This is a fairly general question. Let $(M^3,g)$ be a Riemannian 3-manifold. Let $\Sigma^2$ be a dimension-2 submanifold of $M$. The Hawking mass of $\Sigma^2$ is defined as
$m(\Sigma^2) := ...

**0**

votes

**1**answer

130 views

### Hilbert's Theorem relevance to positive curvature

In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative Gaussian curvature K immersed in $ R^3 $. This theorem answers the ...

**2**

votes

**0**answers

56 views

### Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow ...

**10**

votes

**2**answers

823 views

### Intuition for Levi-Civita connection?

Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or ...

**12**

votes

**1**answer

269 views

### If all balls around two points are isometric… — manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...

**0**

votes

**1**answer

150 views

### elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**1**

vote

**0**answers

108 views

### The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...

**23**

votes

**2**answers

622 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 ...

**2**

votes

**1**answer

229 views

### Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order ...

**4**

votes

**2**answers

178 views

### The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle

What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?

**1**

vote

**0**answers

82 views

### Riemann curvature of $S^1$-principal bundle

Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have,
$$T_pP \simeq T_pV\oplus\Gamma_p$$
Where $V$ ...

**4**

votes

**1**answer

155 views

### $C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a ...

**22**

votes

**9**answers

2k views

### Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...

**1**

vote

**1**answer

285 views

### Is it possible to make the principal bundle projection map totally geodesic?

Let $G$ be a compact (connected) Lie group. Suppose that a $G$-principal bundle $\pi:P\rightarrow Q$ is given.
Is it always possible to equip $P$ and $Q$ with Riemannian metrics, s.t. $\pi$ is ...

**7**

votes

**2**answers

321 views

### Examples of non isometric surfaces having the same curvature function

I think it is really natural to believe, after doing Riemannian geometry for a little time, that sectional curvature encodes the all local geometry of a Riemannian manifold. One of the first thing one ...

**2**

votes

**1**answer

135 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) ...

**7**

votes

**1**answer

84 views

### co-dimension one minimizing verifolds

It is known that a minimizing co-dimension verifolds within a manifold may need to be singular. I think a famous example first partially analyzed by Jim Simons is the cone on in the 8-ball of the ...

**5**

votes

**1**answer

806 views

### Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...

**1**

vote

**1**answer

87 views

### Nonpositive curvature of Stein manifolds

It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I am curious about what kinds of additional ...

**2**

votes

**0**answers

62 views

### Normal-like coordinates for weakly differentiable metrics

Let $(M,g)$ be a Riemannian $W^{2,p}$ metric, with $p>n/2$. Thus $g$ is at least continuous. At any point $P\in M$, do there exist local coordinates $x^i$ such that $g$ can be decomposed as $g_{ij} ...

**10**

votes

**1**answer

275 views

### Simple, closed geodesics in $\mathbb{S}^3$ manifold

Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., ...

**2**

votes

**2**answers

310 views

### Approximation theorem for Anti-Self-Dual Metrics

Rounge's Theorem states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on ...

**4**

votes

**2**answers

145 views

### The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold

There is a theorem :
1) 2-dim (pseudo-)Riemannian manifold must be local conformal flat;
2) 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.
3) n-dim (n>3) ...

**1**

vote

**2**answers

120 views

### How is the metric tensor related to the Hessian of the first fundamental form?

I know that the metric tensor can not always be formulated as a Hessian, but sometimes it can. Can you help me to understand what the special conditions are under which the metric tensor is a Hessian ...

**5**

votes

**3**answers

842 views

### Totally Geodesic Submanifolds

Suppose that $N$ is a totally geodesic submanifold of a complete Riemannian manifold $(M,g)$. Is it the case that a geodesic segment that minimizes length in the submanifold $N$ also minimizes length ...

**4**

votes

**2**answers

202 views

### A question on certain elliptic PDE

Consider the elliptic PDE "CR"
$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are similar to the Cauchi ...

**9**

votes

**2**answers

280 views

### Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$.
On the other ...

**7**

votes

**1**answer

99 views

### Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...

**-1**

votes

**1**answer

46 views

### Parallel transport along a reparametrized geodesic

Let $M$ denote a Riemannian manifold, $\gamma$ a geodesic of $M$ defined on $\mathbb{R}$. Let $t_{0} \in \mathbb{R}$ and $(\alpha,\beta) \in \mathbb{R}^{2}$. I define the reparametrized geodesic ...

**2**

votes

**2**answers

87 views

### Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...

**0**

votes

**0**answers

127 views

### Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...

**3**

votes

**0**answers

97 views

### Focal points for the exponential map and Jacobi fields

It is known that in a Riemannian manifold $(M,g)$, if there is a closed geodesic and a non-zero, periodic, non-constant Jacobi field along it, then M has a focal point. Is the converse true? That is ...

**3**

votes

**2**answers

403 views

### riemannian length of an element of the fundamental group of a manifold

It is a stupid question i guess but like they say if you ask you are stupid for 5 minutes and if you don't ask you are stupid forever . here is the question given a closed manifold $(M,g)$ and ...

**5**

votes

**1**answer

409 views

### Is it true that the geodesics on SO(n) and SU(n) are closed?

I mean for the bi-invariant metric (but actually any metric would work). In this metric geodesics are translates of 1-parameter subgroups so we need only to show that $exp(t X)$ for any X in the lie ...

**1**

vote

**0**answers

156 views

### connections and curvature

Let $(M, g)$ be a Riemannian manifold. Is it possible to construct two different affine (or metric) connections, say $\nabla$ and $\nabla'$, which induce the SAME curvature tensor, i.e. $R(X, ...

**3**

votes

**1**answer

72 views

### Local geodesics in uniquely geodesic spaces

A while ago I asked this
question in Math Stackexchange. Since I didn't receive an answer so far, I thought I'd ask it here.
Suppose $Y$ is a proper length space, where every pair of points $x,y\in ...

**10**

votes

**7**answers

3k views

### Roadmap to learning about Ricci Flow?

Hello,
I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...

**6**

votes

**1**answer

170 views

### Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...

**1**

vote

**1**answer

93 views

### Smoothness of the exponential map at the origin

Let $(M, g)$ be a smooth Riemannian manifold, $p \in M$, and $\exp_P$ the exponential map at the point $P$:
$\exp_P: T(P) \to M$
It seems clear to me that $\exp_P$ is smooth on $U \setminus \{0\}$, ...

**5**

votes

**1**answer

116 views

### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function,
...