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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
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Aug
25
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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
Jul
24
revised Complex structures on Riemann surfaces
edited tags
Jul
24
asked Complex structures on Riemann surfaces
Jul
24
comment Submersions and induced homomorphism on fundamental groups
I think you need a little more justification for the existence of the $\alpha_i$. You could use for example the regular value theorem (which maybe you implicitly are).