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 \ …

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

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 $t_f$ …

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 …

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 …

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. …

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, …

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 …

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_{\ …

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 $f$ …

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. …

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 …

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 …

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 …

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 …