# Tagged Questions

**13**

votes

**0**answers

287 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**22**

votes

**4**answers

1k views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**13**

votes

**2**answers

724 views

### Why Donaldson's Four-Six Conjecture?

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times ...

**2**

votes

**0**answers

297 views

### weak hopf conjecture

Hi:
I am thinking of the following problem which is related to weak Hopf conjecture:
Let $E$ be the total space of a vector bundle over a compact nonnegatively curved manifold $B$. Let k=the rank ...

**15**

votes

**3**answers

2k views

### How to Tackle the Smooth Poincare Conjecture

The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare Conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. ...

**25**

votes

**0**answers

1k views

### Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...

**29**

votes

**1**answer

924 views

### Example of a compact Kähler manifold with non-finitely generated canonical ring?

A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...

**3**

votes

**0**answers

360 views

### Compact surfaces smoothly immerse in: $\mathbb{R}^4$ or $\mathbb{R}^5$?

I wonder if someone can clarify whether it is known
that
every closed, orientable surface (2-manifold) has a
smooth isometric immersion in $\mathbb{R}^4$?
This topic has been discussed rather ...

**7**

votes

**1**answer

391 views

### Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.
My question is as ...

**15**

votes

**9**answers

4k 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.

**14**

votes

**1**answer

797 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 ...

**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 ...

**17**

votes

**1**answer

1k views

### Coiling Rope in a Box

What is the longest rope length L of radius r that can fit into a box?
The rope is a smooth curve with a tubular
neighborhood of radius r, such that the rope does not
self-penetrate. For an open ...

**15**

votes

**2**answers

1k views

### Are there Ricci-flat riemannian manifolds with generic holonomy?

This may well be an open problem, I'm not sure.
In Berger's classification (refined by Simons, Alekseevsky, Bryant,...) of the holonomy representations of irreducible non-symmetric complete ...

**10**

votes

**2**answers

1k views

### A generalization of Cauchy's mean value theorem.

The following simple theorem is known as Cauchy's mean value theorem. Let $\gamma$ be an immersion of the segment $[0,1]$ into the plane such that $\gamma(0) \ne \gamma(1)$. Then there exists a point ...

**14**

votes

**0**answers

654 views

### Milnor's cartography problem

Let $\Omega$ be a round disc of radius $\alpha<\pi/2$ on the standard sphere.
It is easy to construct a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map from $\Omega$ to the plane.
Is it true ...

**17**

votes

**3**answers

1k views

### Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space
and $r$ is the distance from $\Sigma$ to the center of the ball.
Is it true that
$$\mathop{\rm area} \Sigma\ge ...

**6**

votes

**1**answer

531 views

### Infinite dimensional Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes.
I heard that this statement is not true in infinite dimensions, ...

**10**

votes

**2**answers

446 views

### Is a smooth closed surface in Euclidean 3-space rigid?

Classical theorem of Cohn-Vossen: A closed convex surface in Euclidean 3-space cannot be deformed isometrically.
Robert Connelly found an example of a polyhedral surface that can be deformed ...

**30**

votes

**6**answers

5k views

### Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...

**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 ...