**23**

votes

**10**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 ...

**83**

votes

**16**answers

10k views

### What is torsion in differential geometry intuitively?

Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$
What is the geometric picture behind this ...

**41**

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 ...

**34**

votes

**5**answers

3k views

### Is there a “geometric” intuition underlying the notion of normal varieties?

I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot.
One thing that strikes me ...

**6**

votes

**2**answers

554 views

### Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems).
A ...

**2**

votes

**1**answer

540 views

### Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?

Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ?
Has it been done in the literature?
In textbooks, only the Banach case is treated, ...

**104**

votes

**16**answers

14k views

### How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...

**53**

votes

**6**answers

4k views

### Is there an analogue of curvature in algebraic geometry?

I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...

**27**

votes

**3**answers

3k views

### Can every manifold be given an analytic structure?

Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy ...

**11**

votes

**5**answers

2k views

### Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...

**10**

votes

**4**answers

2k views

### How to see the Phase Space of a Physical System as the Cotangent Bundle

Two things today motivated this question.
First, the professor said that in a lecture Thurston mentioned
Any manifold can be seen as the configuration space of some physical system.
Clearly we ...

**8**

votes

**8**answers

2k views

### Maxwells equations and differential forms

Hi,
is there a textbooks that explains the maxwell equations in differential form?
What I understood so far is, that the $E$ and $B$ fields can be assembled to
a differential 2 Form $F$, and the ...

**35**

votes

**3**answers

2k views

### $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$

I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded
in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem,
a claim repeated in this ...

**23**

votes

**3**answers

2k views

### Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...

**16**

votes

**2**answers

912 views

### Converse to Stokes' Theorem

Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : ...

**11**

votes

**3**answers

626 views

### Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...

**12**

votes

**1**answer

2k views

### Derivative of Exponential Map

Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector field along $\gamma$. Define $\varphi: (a,b) \to M$ by $t \mapsto \exp_{\gamma(t)}(E(t))$. Is there a ...

**9**

votes

**2**answers

1k views

### When does the tangent bundle of a manifold admit a flat connection?

Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...

**24**

votes

**1**answer

1k views

### Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...

**23**

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
...

**17**

votes

**2**answers

1k views

### Probing a manifold with geodesics

Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...

**12**

votes

**3**answers

899 views

### How transitive are the actions of symplectomorphism groups ?

This question is motivated by the classical fact from differential geometry :
Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ ...

**18**

votes

**5**answers

1k views

### Can one recover a metric from geodesics?

Assume there are two Riemannian metrics on a manifold ( open or closed) with the same set of all geodesics. Are they proportional by a constant? If not in general, what are the affirmative results in ...

**13**

votes

**3**answers

881 views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...

**11**

votes

**3**answers

583 views

### How to draw a Zoll surface?

I take into account that lots of questions on Zoll surfaces have already been asked on the forum. But I will stubbornly continue asking. Are there any chances to draw explicitely at least one Zoll ...

**9**

votes

**1**answer

450 views

### When is a given matrix of two forms a curvature form?

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of $1$-forms ...

**8**

votes

**2**answers

355 views

### Surface in 3D that realizes all pairs of principal curvatures

This is a question that Willie Wong raised in comments after he answered my question,
Surface analog of clothoid: curvatures covering $\mathbb{R}^3$. Willie's question is
more interesting (and ...

**4**

votes

**0**answers

233 views

### Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:
Theorem. All straight lines are extremals of the variational problem
$$
...

**2**

votes

**3**answers

352 views

### Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds

I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...

**3**

votes

**1**answer

229 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

**3**

votes

**1**answer

299 views

### Existence and Properties of 3D Curves with unusual 2D $(\kappa(s),\tau(s))$ Trajectories

This question is inspired by Surface in 3D that realizes all pairs of principal curvatures
While one can imagine, that a 3D surface could exist, that realizes all pairs of principal curvatures, ...

**3**

votes

**0**answers

465 views

### The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles

Consider $SO(n)$ bundles over smooth manifolds. Then using the fact that the Stiefel-Whitney classes are the modulo 2 reductions of the Chern classes, one can prove $w_{2i}^2(E) = p_i(E) \bmod 2$. Now ...

**2**

votes

**1**answer

139 views

### Inverse Problem for Pullback

Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...

**1**

vote

**1**answer

288 views

### Projectively equivalent connections

We can define projective structure on a manifold in two ways.
First we can define it as a maximal atlas of charts from the open subsets of manifold to the projective space , such that transitions maps ...

**1**

vote

**1**answer

295 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...

**1**

vote

**1**answer

307 views

### Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity

Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have ...

**58**

votes

**16**answers

8k views

### Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...

**60**

votes

**5**answers

4k views

### Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

**57**

votes

**11**answers

13k views

### why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...

**55**

votes

**16**answers

5k views

### What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...

**61**

votes

**6**answers

6k views

### Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$
Here is a somewhat more conceptual ...

**22**

votes

**14**answers

21k views

### Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...

**42**

votes

**2**answers

3k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

**65**

votes

**11**answers

9k views

### Is it possible to capture a sphere in a knot?

You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...

**29**

votes

**7**answers

6k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...

**45**

votes

**5**answers

2k views

### Random manifolds

In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its ...

**41**

votes

**10**answers

5k views

### Possibility of an Elementary Differential Geometry Course

I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.
I've found that in talking to professional physicists and ...

**24**

votes

**5**answers

3k 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 ...

**14**

votes

**4**answers

1k views

### What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...

**15**

votes

**5**answers

2k views

### A question on Generalized Gauss-Bonnet Theorem

Hello all,
I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is 'From Calculus to Cohomology' by Madsen&Tornehave. ...