Tagged Questions

3
votes
3answers
95 views

Does the Baker-Campbell-Hausdorff formula hold for vector fields on a (compact) manifold?

Consider a compact manifold M. For a vector field X on M, let $\phi_X$ denote the diffeomorphism of M given by the time 1 flow of X. If X and Y are two vector fields, is $\phi_X \ …
-1
votes
0answers
80 views

tangent bundle and orientation [closed]

Let M is a smooth manifold, TM is tangent bundle. We know TM is Manifold too. How to make TM orientable?
3
votes
3answers
138 views

Existence of Fermi coordinates on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ …
6
votes
1answer
192 views

Seiberg-Witten theory on 4-manifolds with boundary

What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist? I would be especially interested in theories which "behave good" under gluing along the bounda …
2
votes
2answers
208 views

Equivalent singular chains and differential forms, as functionals on forms, on compact Riemannian manifolds

On a compact Riemannian oriented manifold $M$,for each singular $k$-chain $\sigma$ (with real coefficients), $\sigma$ induces a linear functional on the $\mathbb{R}$-vector space o …
8
votes
2answers
234 views

What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy?

This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf. …
8
votes
3answers
175 views

Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex?

Perhaps this is basic knowledge in Riemannian geometry, but I can't seem to figure out the answer. Here is the precise statement of my question. Let $M$ be a Riemannian manifold, …
9
votes
2answers
192 views

Can we decompose Diff(MxN)?

If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there …
2
votes
1answer
207 views

Deriving symmetries of a Gauge theory

Hello, I don't know if this is a good place for exposing my problem but I'll try... I have a gauge theory with action: $S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\ …
2
votes
1answer
257 views

Definition of a complex structure on a vector bundle

Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$. The complex conjugation $f$ …
0
votes
3answers
266 views

Following curves on S^n

Suppose $V$ is a no-where zero vector field on $S^n$ ($n$ odd). Let $p \in S^n$. Let $\gamma_p$ be the unique curve on $S^n$ through $p$ and tangential to $V$ everywhere along it. …
4
votes
6answers
318 views

Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commente …
1
vote
1answer
132 views

Power series for meromorphic differentials on compact Riemann surfaces

Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/\Gamma$ where I do know $\Gamma$ explicit. Is there a way to write down the power series of meromorphic f …
10
votes
2answers
308 views

Sheaves and bundles in differential geometry

Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as …
1
vote
2answers
220 views

Orientation of a smooth manifold using sheaves

Is there any way to define the orientation of an orientable smooth manifold using sheaves (when our smooth manifold is viewed as a locally ringed space) without our definition bein …

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