Eric O. Korman
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 Mar 16 answered Principal bundles, connection forms and fundamental vector fields Mar 16 answered Exterior derivatives involving representations Nov 30 comment Differential forms on $S^1$ @self-learner the cohomology groups are the kernel divided by the image (since $d^2 = 0$, the image is contained in the kernel). Nov 23 comment Prove that any map $f\colon S^1 \to S^1$ mapping antipodal point to antipodal point has $\deg_2(f)=1$ by a direct computation. What definition of degree are you using? Nov 19 reviewed Approve mathematical model Nov 19 comment Differential forms on $S^1$ The star means the dual space, so $T^*M$ is the space of dual vectors and sections of its exterior product are differential forms. Any $n$-form on an n-dimensional manifold looks like $\mu = f dx_1 \wedge \cdots \wedge dx_n$ in coordinates. Therefore $d\mu = \sum_j \partial_j f dx_j \wedge dx_1 \wedge \cdots \wedge dx_n = 0$. Nov 19 comment Differential forms on $S^1$ This is because $\mu$ is an $n$-form on an $n$-dimensional manifold. Thus $d\mu$ is an $n+1$-form and so must be zero since $\Lambda^{n+1} T^*M = 0$. Nov 18 comment Differential forms on $S^1$ @orion: Not every form can be written that way. Nov 18 answered Differential forms on $S^1$ Nov 18 comment Complex structure on a real vector bundle $J$ is the complex structure. The $e_i$ and $\sigma_i$ are local sections of $E$ and so are acted on by $J$. Having a trivialization (i.e. iso from $E$ restricted to $U$ to $U\times \mathbb R^{2n}$) is equivalent to having $2n$ pointwise independent sections of $E$ restricted to $U$. Nov 18 answered Complex structure on a real vector bundle Nov 17 comment Complex structure on a real vector bundle Are you familiar with how a metric on a vector bundle reduces the structure group to the orthogonal group? Oct 27 comment Lie Bracket, Hopf Fibration, independence of choice A horizontal lift is not unique unless you fix a connection on the fibration. Oct 20 awarded Notable Question Sep 24 awarded Autobiographer Aug 25 awarded Enlightened Aug 25 awarded Nice Answer Aug 22 answered Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$? Aug 14 comment Is this a manifold? So you're looking at the sphere with like a ring around it thats touching it? This won't be locally homeomorphic to $\mathbb R^2$. Jul 26 revised Complex structures on Riemann surfaces added 171 characters in body