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4 votes
Accepted

Can two different spin structures on a manifold induce the same spin$^c$ structure?

Spin structures naturally form a torsor over $H^1(M, \mathbb{Z}/2)$; in particular this means that the "difference" of two spin structures is an element of $H^1(M, \mathbb{Z}/2)$ on the nose....
Qiaochu Yuan's user avatar
3 votes
Accepted

Equivalent forms of second Bianchi identity on $TM$

See here (and the various sublinks) for any background about the exterior covariant derivative and how to do computations with it. Fix a local coordinate chart $(U,x=(x^1,\dots, x^n))$ on $M$, and ...
peek-a-boo's user avatar
  • 58.3k
3 votes

Total space of the normal bundle

$\newcommand\R{\mathbb{R}}$Note that there is a canonical spitting of the tangent bundle of $\R^n$, $$ T_*\R^n = \R^n\times\R^n. $$ Let $F: M \rightarrow \R^n$ be an embedding. The Jacobian (i.e, ...
Deane's user avatar
  • 8,492
3 votes
Accepted

How to show that a sheaf is itself a sheaf of modules?

Let $X$ be a space and $\mathcal{O}_X$ a sheaf of rings (thought of as continuous, smooth, holomorphic, or algebraic functions). A sheaf of Abelian groups $\mathcal{F}$ is a sheaf of $\mathcal{O}_X$-...
Alekos Robotis's user avatar
3 votes

Fundamental equations of a Riemannian immersion

Our goal is to understand the ambient curvature in terms of the curvatures of submanifolds. You always need 2 tangent vectors (in the correct slots); this is because curvature is an endomorphism-...
peek-a-boo's user avatar
  • 58.3k
2 votes
Accepted

Scheme-theoretic construction of tensor product of vector bundles on a scheme.

The total space of a vector bundle does have a universal property. Let $p: E \to X$ be the total space of the vector bundle $\mathcal E$ on $X$. The universal property of $E$ is: a map $f: Y \to E$ is ...
Joshua Mundinger's user avatar
1 vote
Accepted

Extension of a vector bundle over the smooth compactification of an algebraic variety

Let me assume $A$ is quasi-projective. Let $D$ be an ample divisor on $A$. Then for $n \gg 0$ the twist $E(nD)$ is generated by a finite-dimensional vector space $V$ of global sections, i.e., there is ...
Sasha's user avatar
  • 18k
1 vote
Accepted

Sign discrepancy in covariant derivative

For $\eta\in\Omega^k(\mathrm{End}(E))$ and $\mu\in\Omega^l(\mathrm{End}(E))$ the form $[\eta,\mu]\in\Omega^{k+l}(\mathrm{End}(E))$ is defined by $$ [\eta,\mu](X_1,\dots,X_{k+l})=\frac1{k!l!}\sum_{\...
Claire's user avatar
  • 4,809

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