**4**

votes

**2**answers

241 views

### What does it mean that the Hessian is proportional to the metric?

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential).
Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...

**10**

votes

**2**answers

887 views

### Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...

**3**

votes

**0**answers

75 views

### Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...

**3**

votes

**2**answers

141 views

### Triangles in rigid Riemann surfaces

Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows:
Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That ...

**17**

votes

**0**answers

150 views

### Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...

**3**

votes

**0**answers

55 views

### Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...

**2**

votes

**0**answers

68 views

### $Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here.
Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$
Then $\varphi $ is an automorphism of $O(2n)$, and ...

**3**

votes

**1**answer

124 views

### Non-flat totally geodesic surfaces

I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain ...

**3**

votes

**0**answers

40 views

### Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...

**2**

votes

**0**answers

69 views

### What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set ...

**0**

votes

**0**answers

46 views

### Toponogov comparison theorem for complex manifold

I want to know some reference for the Toponogov comparison theorem for complex manifold, in particular, for complex manifold with bounded holomorphic sectional curvature. As far as I know, the ...

**0**

votes

**1**answer

56 views

### Sobolev chain rule on non-compact manifolds

Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions ...

**5**

votes

**3**answers

584 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 ...

**4**

votes

**0**answers

96 views

### Difference between parallel transport composed with exponential maps along two different geodesics starting at the same point?

I asked this question on math.stackexchange too: it's not a homework problem, but something that came to my mind while thinking of commutation:
...

**11**

votes

**1**answer

295 views

### Flat manifolds and irreducible representations

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...

**4**

votes

**1**answer

161 views

### Prove that the Log-Euclidean distance is negative-definite

Let $\Bbb{S}_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices.
The Log-Euclidean distance between two points of ...

**3**

votes

**1**answer

224 views

### Volume of arithmetic quotients of symmetric spaces

Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...

**3**

votes

**0**answers

79 views

### Does null geodesic flow live on a natural compact bundle?

Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary).
A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$.
The geodesic flow can be seen as a ...

**5**

votes

**0**answers

90 views

### $C^1$ regularity of harmonic functions on Riemannian manifolds

Consider a smooth, connected and complete Riemannian manifold $M$. It is well known that harmonic functions defined on some open subset of $M$ are $C^\infty$.
I'm interested in knowing whether there ...

**3**

votes

**2**answers

368 views

### Iwaniec's conjecture

Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have ...

**4**

votes

**1**answer

62 views

### Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$

Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem?
The only thing I can find is ...

**1**

vote

**1**answer

105 views

### Diffeomorphism variation of the Christoffel symbol

Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative
$$
\delta g_{\mu\nu} = ({\mathcal L}_\xi G)_{\mu\nu}=\nabla_\mu \xi_\nu+\nabla_\nu \xi_\mu
$$
and under a change ...

**6**

votes

**0**answers

152 views

### Smooth morse theory of Riemannian distance functions

Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, ...

**1**

vote

**1**answer

97 views

### Decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...

**1**

vote

**1**answer

170 views

### Almost complex structure and nontrivial idempotents

Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of ...

**2**

votes

**0**answers

63 views

### Bounding distance between geodesics in manifolds with nonpositive curvature

This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...

**-2**

votes

**1**answer

138 views

### Finding Riemannian metric for this geodesic [closed]

In a $d$-dimensional manifold, given a geodesic equation $\gamma^i(t)=a^i\phi(tb^i),i\in 1\ldots d$, where $\phi:\mathbb{R}\rightarrow\mathbb{R}^+$ is an increasing function, $a^i>0,b^i$ are ...

**1**

vote

**0**answers

109 views

### Taylor expansions of Riemannian exponential map and Jacobi fields? [closed]

Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com,
...

**3**

votes

**1**answer

88 views

### Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$

Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on ...

**2**

votes

**1**answer

184 views

### existence of totally geodesic hypersurfaces

Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic ...

**6**

votes

**1**answer

1k 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, ...

**2**

votes

**1**answer

251 views

### Is the structure constant additive on connected components?

This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...

**6**

votes

**2**answers

117 views

### Geodesics on convex hypersufaces

Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior.
Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest ...

**5**

votes

**1**answer

351 views

### Intersections of open balls in manifolds

This question is motivated by the post Uncountable intersections of open balls in a separable metric space.
The general problem is the following: given a connected Riemannian manifold $M$, what are ...

**3**

votes

**1**answer

77 views

### Gauss-Bonnet formula for 2-dimensional Alexandrov spaces

EDIT: Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov.
Questions 1. Can one define a measure $K$ on $S$ (thought ...

**17**

votes

**11**answers

5k views

### Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.

**1**

vote

**0**answers

51 views

### Taylor expansion in Riemannian foliations

Take:
$M$ a Riemannian manifold, ${X_0}\in M$,
$N_{X_0}$ a submanifold of $M$ going through ${X_0}$,
and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.
At ${X_0} \in N_{X_0}$, we consider the ...

**0**

votes

**0**answers

58 views

### What is the relation between two Riemannian metrics with the same Riemannian curvatures?

Consider two flat Riemannian metrics on a manifold. What is the general relation between these two metrics if the manifold is not simply-connected? What is the answer if two Riemannian metrics have ...

**3**

votes

**1**answer

313 views

### What are Euler density and Weyl invariants?

I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...

**6**

votes

**2**answers

124 views

### Alexandrov spaces which are not limits of Riemannian manifolds

Are there important/ interesting/ natural examples of compact Alexandrov spaces with curvature bounded from below which are not Gromov-Hausdorff limits of smooth compact Riemannian manifolds with ...

**3**

votes

**2**answers

171 views

### Totally geodesic submanifold of a hyperbolic 3-manifold

If $M$ is a convex-cocompact hyperbolic 3-manifold, and $S$ is a closed surface with genus $\geq$ 2. Suppose $f:S\to M$ is a minimal immersion, and $f(S)$ is negatively curved. I know that all the ...

**190**

votes

**9**answers

16k views

### John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends.
Maybe this is an appropriate time to ask a ...

**5**

votes

**0**answers

109 views

### Nice applications of Liouville's theorem

I need a big list of nice-looking and simple applications of Liouville's theorem on geodesic flow in Riemannian geometry.
Please help.
Examples:
A Riemannian manifold with finite volume does not ...

**1**

vote

**1**answer

139 views

### Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?

I apologize if this is not a research level question (already tried asking http://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with ...

**0**

votes

**2**answers

146 views

### Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...

**6**

votes

**2**answers

259 views

### Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold

It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the ...

**3**

votes

**1**answer

160 views

### A surface on which all regular curves have nowhere vanishing curvature

Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that ...

**1**

vote

**0**answers

115 views

### Spherical cap is the only compact constant mean curvature surface bounded by a circle

I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap.
This is stated in the ...

**0**

votes

**0**answers

52 views

### Does anyone know how to describe the zero set of the Jacobian of injective harmonic maps in space?

For example consider the following question:
Let $\mathbb{B}^m$ be hyperbolic space and let $f
: \mathbb{B}^m \rightarrow \mathbb{B}^m$ be harmonic $K$-qc map.
Whether $f$ has critical points on ...

**0**

votes

**1**answer

128 views

### Riemann isometry vs Euclidean bi-Lipschitz mapping

Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...