bio | website | calvino.polito.it/~salamon |
---|---|---|
location | Turin | |
age | ||
visits | member for | 4 years, 10 months |
seen | Jan 14 '13 at 13:47 | |
stats | profile views | 552 |
Jun 25 |
awarded | Revival |
Nov 15 |
awarded | Enlightened |
Nov 14 |
awarded | Nice Answer |
Feb 13 |
awarded | Yearling |
Jan 3 |
awarded | Necromancer |
Nov 8 |
awarded | Necromancer |
Feb 13 |
awarded | Yearling |
Dec 4 |
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. |
Sep 24 |
awarded | Nice Answer |
Aug 6 |
awarded | Critic |
Aug 2 |
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 |