Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
Questions tagged [dg.differential-geometry]
8666 questions
63
votes
12 answers
How much of differential geometry can be developed entirely without atlases?
We may define a topological manifold to be a second-countable Hausdorff space such that every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^n$. We can further define a smooth manifold to be a topological manifold…
Harry Gindi
- 19,374
52
votes
1 answer
Atiyah's May 2018 paper on the 6-sphere
A couple years ago Atiyah published a claimed proof that $S^6$ has no complex structure. I've heard murmurs and rumors that there are problems with the argument, but just a couple months ago he apparently published a follow-up which fleshes out the…
Paul Siegel
- 28,772
50
votes
5 answers
Usefulness of Nash embedding theorem
Nash embedding theorem states that every smooth Riemannian manifold can be smoothly isometrically embedded into some Euclidean space $E^N$. This result is of fundamental importance, for it unifies the intrinsic and extrinsic points of view of…
Ettore Minguzzi
- 1,341
46
votes
7 answers
Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness
I'm soon giving an introductory talk on de Rham cohomology to a wide postgraduate audience. I'm hoping to get to arrive at the idea of de Rham cohomology for a smooth manifold, building up from vector fields and one-forms on Euclidean space.…
Janos Erdmann
- 1,383
38
votes
3 answers
Smoothness of distance function in Riemannian Manifolds
Let $(\mathcal{M},g)$ be a $C^{\infty}$-Riemannian manifold. A basic fact is that $g$ endows the manifold $\mathcal{M}$ with a metric space structure, that is, we can define a distance function…
Mauricio
- 1,415
- 1
- 13
- 19
32
votes
6 answers
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 this direction?
Axel
- 1,297
31
votes
2 answers
Compact surfaces of negative curvature
John Hubbard recently told me that he has been asking people if there are compact surfaces of negative curvature in $\mathbb{R}^4$ without getting any definite answers. I had assumed it was possible, but couldn't come up with an easy example off…
Matt Noonan
- 3,984
28
votes
7 answers
Why is the Leibniz rule a definition for derivations?
In differential geometry, the tangent space is defined as a generalization of directional derivatives, which in turn are defined as functionals following Leibniz's product rule.
I understand all the proofs, but what is the intuition behind choosing…
R S
- 985
24
votes
1 answer
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 "nice" expression for $\varphi'(t)$?
This…
24
votes
7 answers
Nash embedding theorem for 2D manifolds
The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$. What can we say in the special case of 2-manifolds? For example, can we always embed a 2-manifold in $R^3$?
user39719
24
votes
8 answers
Geodesics on spheres are great circles
How does one prove that on $S^n$ (with the standard connection) any geodesic between two fixed points is part of a great circle?
For the special case of $S^2$ I tried an naive approach of just writing down the geodesic equations (by writing the…
Anirbit
- 3,453
22
votes
3 answers
The rank of a symmetric space
I would like to work a theorem on a article who deals with the rank one symmetric spaces.
i looked up the definition of symmetric spaces of rank one, but I did not find a satisfactory definition then what is the meaning of rank, intuitively and…
22
votes
2 answers
Flows of vector fields and diffeomorphisms isotopic to the identity
Let $M$ be a compact manifold and $\varphi : M \longrightarrow M$ be a diffeomorphism which is isotopic to the identity. Does there exist a vector field $ X $ on $M$ such that $\varphi$ is the flow at time $1$ of $X$? If that is not always the case,…
Selim G
- 2,626
22
votes
1 answer
Can an Einstein metric have the same Levi-Civita connection with a non-Einstein one?
We say that two metrics are affinely equivalent if their Levi-Civita connections coincide. Is it possible that an Einstein (=Ricci tensor is proporional to the metric) is affinely equivalent to a metric which is not Einstein?
Of course, since…
Vladimir S Matveev
- 4,712
- 20
- 30
20
votes
1 answer
Integral formula for Euler class
The tangent bundles of closed hyperbolic surfaces have flat $PSL(2,\mathbb{R})$ connections showing that there can be no integral formula for the Euler class such connections. This contrasts the situation for the Orthogonal group where the Pfaffian…
michael freedman
- 201