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bio website calvino.polito.it/~salamon
location Turin
age
visits member for 4 years, 10 months
seen Jan 14 '13 at 13:47

Jun
25
awarded  Revival
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15
awarded  Enlightened
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14
awarded  Nice Answer
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awarded  Yearling
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awarded  Necromancer
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awarded  Yearling
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comment Spin structures on the Grassmannians
One can compute the first Chern class of the complex Grassmannian of $k$-planes in $C^N$ as follows. For the purpose of computing its Chern character, the holomorphic tangent bundle $T$ is a product of the tautological rank $k$ bundle $V$ and (formally) $N−V$. This leads to the formula $c_1(T) = (N−2k)v$ where $v=c_1(V)$ is a generator of $H^2$. So it looks like the Grassmannian is spin iff $N$ is even.
Dec
4
comment Is the cotangent bundle to a Kahler manifold hyperkahler?
This answer is pretty complete, but it is worth reading the paper of Calabi in Ann. Ec. Norm. Sup. 12 (1979) for an explicit construction of the HK metric on the cotangent bundle of complex projective space. The precise form of the metric is not obvious, and his approach (subsequently generalized to other HSS's) was to find the Kaehler potential. As in applications of Yau's theorem in the compact case, the HK metric is indeed compatible with the underlying holomorphic symplectic structure.
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awarded  Nice Answer
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6
awarded  Critic
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comment A geometric interpretation of the Levi-Civita connection?
Yes, but any such section $s$ that passes through $p\in P$ is unique to first order. If we set $a_{ijk} = \Gamma_{ij}^r g_{rk}$ then $s$ is tangent to $P$ at $p$ iff $a_{ijk}+a_{ikj}=0$, which forces the Christoffel symbols to vanish at the point in question.
Aug
1
answered A geometric interpretation of the Levi-Civita connection?
Jul
29
answered Do hyperKahler manifolds live in quaternionic-Kahler families?
Jul
29
awarded  Supporter
Jul
27
answered Diffeomorphism group of the unit sphere of complex n-space
Jul
27
revised projection of the co-derivative == co-derivative of the projection ?
added 3 words
Jul
27
answered projection of the co-derivative == co-derivative of the projection ?
Jul
25
awarded  Editor
Jul
25
revised Which journals publish expository work?
added last sentence