**197**

votes

**9**answers

17k 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 ...

**63**

votes

**4**answers

8k views

### When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion free connection $\nabla_g$, the Levi-Civita connection, that is compatible witht metric.
I was wondering if one can ...

**46**

votes

**7**answers

5k views

### Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...

**46**

votes

**2**answers

2k views

### “Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...

**33**

votes

**5**answers

2k views

### Is the Laplacian on a manifold the limit of graph Laplacians?

Here's the sort of thing I have in mind. Let $M$ be a Riemannian manifold, compact if it helps, and let $\Delta_M$ be the Laplace-Beltrami operator. Choose a sequence of triangulations of $M$ so ...

**33**

votes

**0**answers

2k views

### Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...

**32**

votes

**4**answers

3k views

### What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...

**32**

votes

**1**answer

2k views

### Many flat totally geodesic surfaces ⇒ flat?

Let $M$ be a 4-dimensional Riemannian manifold.
Assume there is a huge number (say 100) of flat totally geodesic 2-dimensional surfaces
passing through a point $p\in M$ and assume that their tangent ...

**31**

votes

**5**answers

2k views

### Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...

**30**

votes

**5**answers

3k views

### $G_2$ and Geometry

In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...

**30**

votes

**2**answers

4k views

### Does the curvature determine the metric?

Hello,
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...

**26**

votes

**10**answers

3k 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 ...

**26**

votes

**5**answers

4k views

### A geometric interpretation of the Levi-Civita connection?

Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the ...

**24**

votes

**12**answers

10k views

### Introductory text on Riemannian geometry

I have studied differential geometry, and am looking for basic introductory texts on Riemannian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, ...

**24**

votes

**1**answer

560 views

### Do free higher homotopy classes of compact Riemannian manifolds have preferred representatives?

A well known theorem of Cartan states that every free homotopy class of closed paths in a compact Riemannian manifold is represented by a closed geodesic (theorem 2.2 of Do Carmo, chapter 12, for ...

**23**

votes

**9**answers

2k views

### Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...

**23**

votes

**2**answers

793 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 ...

**23**

votes

**2**answers

2k views

### The Origin of the Musical Isomorphisms

In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are sometimes referred to as ...

**23**

votes

**2**answers

1k views

### Is there a Chern-Gauss-Bonnet theorem for orbifolds?

There's a Gauss-Bonnet theorem for compact 2-orbifolds(due to Satake, I think), which gives a relation between the curvature of a Riemannian orbifold and the orbifold topology(i.e. taking into account ...

**23**

votes

**0**answers

232 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 ...

**23**

votes

**0**answers

1k views

**22**

votes

**9**answers

4k views

### Some questions about scalar curvature

Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
...

**20**

votes

**2**answers

1k views

### When is a closed differential form harmonic relative to some metric?

Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$.
Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is ...

**20**

votes

**2**answers

2k views

### Curves of constant curvature on S^2

Most probably this is a well known question.
Consider $S^2$ with a Riemannian metric. I would like to ask what is known about the structure of the set of simple (without self-intersections) closed ...

**19**

votes

**2**answers

524 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 ...

**19**

votes

**2**answers

1k views

### Complex manifolds in which the exponential map is holomorphic

Let $X$ be a complex manifold and $g$ a hermitian metric on $X$. Consider the Riemannian exponential $\exp_p: T_p X \to X$.
If $\exp_p$ is holomorphic for every $p \in X$, then $(\exp_p)^{-1}$, ...

**19**

votes

**3**answers

1k views

### Algebraic (semi-) Riemannian geometry ?

I hope these are not to vague questions for MO.
Is there an analog of the concept of a Riemannian metric, in algebraic geometry?
Of course, transporting things literally from the differential ...

**19**

votes

**2**answers

1k views

### Area of distance sphere in manifold with Ricci $\ge 0$.

Let $M$ be a open complete manifold with Ricci curvature $\ge 0$.
By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear.
I am wondering whether the following statement is ...

**18**

votes

**2**answers

915 views

### Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?

The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...

**18**

votes

**1**answer

641 views

### Is the following a sufficient condition for asphericity?

I recently came across the following question while working on some problems on manifolds with lower Ricci curvature bounds.
Given $n$ does there exist a large $R>0$ with the following property:
...

**17**

votes

**11**answers

6k 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.

**17**

votes

**4**answers

2k views

### Largest hyperbolic disk embeddable in Euclidean 3-space?

Hilbert proved that there's no complete regular (C^k for sufficiently large k) isometric embedding of the hyperbolic plane into R^3. On the other hand, the pseudosphere is locally isometric to the ...

**17**

votes

**2**answers

956 views

### If there is a dense geodesic, are almost all geodesics equidistributed? Dense?

Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian ...

**17**

votes

**3**answers

413 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 ...

**17**

votes

**4**answers

2k views

### Why is it important that partial derivatives commute?

I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...

**17**

votes

**2**answers

649 views

### Manifold with all geodesics of Morse index zero but no negatively curved metric?

A closed oriented Riemannian manifold with negative sectional curvatures has the property that all its geodesics have Morse index zero.
Is there a known counterexample to the "converse": if (M,g) is ...

**17**

votes

**1**answer

426 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...

**17**

votes

**0**answers

642 views

### Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...

**16**

votes

**3**answers

2k views

### Why are they called isothermal coordinates?

On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:
$$g_{ij} = e^{f} \delta_{ij}$$
My question ...

**16**

votes

**1**answer

548 views

### Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood

Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...

**16**

votes

**5**answers

3k views

### Curvature and Parallel Transport

Here is an updated formulation of the question, which is more precise and I think completely correct:
Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be a neighborhood of ...

**16**

votes

**4**answers

1k views

### When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?

Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...

**16**

votes

**1**answer

1k views

### Nonnegative to Positive Curvature.

This questions asks for your intuition and insight as I'm surprised by how little is known about the difference between nonnegative and positive curvature. I don't want to be completely vague, so I ...

**16**

votes

**2**answers

3k views

### A question about the proof of Mostow rigidity

I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...

**16**

votes

**1**answer

394 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 ...

**16**

votes

**0**answers

341 views

### Negative Einstein manifolds

In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values.
For ...

**16**

votes

**2**answers

2k views

### Conjugate points in Lie groups with left-invariant metrics

For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...

**15**

votes

**3**answers

2k views

### Riemann's formula for the metric in a normal neighborhood

I would love to understand the famous formula $g_{ij}(x) = \delta_{ij} + \frac{1}{3}R_{kijl}x^kx^l +O(||x||^3)$, which is valid in Riemannian normal coordinates and possibly more general situations.
...

**15**

votes

**4**answers

1k views

### Equations satisfied by the Riemann curvature tensor

It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
...

**15**

votes

**5**answers

1k views

### Point singularity of a Riemannian manifold with bounded curvature

Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold ...