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Graduate student in mathematics.


Dec
2
comment Curvature 2-form vs. Sectional Curvature
The definition of sectional curvature is in terms of the curvature 2-form (see e.g. en.wikipedia.org/wiki/Sectional_curvature)
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
Jul
25
comment Homeomorphism between SU(4) and SO(6)
The maps $SU(2) \to SO(3)$ and $SU(4) \to SO(6)$ are not homeomorphisms but are 2-to-1 Lie group covering maps (so local homeomorphisms).