Hermitian Christoffel Symbols - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:28:15Z http://mathoverflow.net/feeds/question/100906 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100906/hermitian-christoffel-symbols Hermitian Christoffel Symbols Michael Albanese 2012-06-29T04:31:43Z 2012-07-04T23:37:58Z <blockquote> <p>Does anyone know of some good references for computing Christoffel symbols for Hermitian metrics?</p> </blockquote> <p>A quick Google search turns up <a href="http://arxiv.org/abs/1011.0207" rel="nofollow">this</a>. The following formula appears on page 4:</p> <p>$$\Gamma_{AB}^C = \frac{1}{2}h^{CE}\left(\frac{\partial h_{AE}}{\partial z^B} + \frac{\partial h_{BE}}{\partial z^A} - \frac{\partial h_{AB}}{\partial z^E}\right)$$</p> <p>where $A, B, C, E \in$ {$1, \dots, n, \bar{1}, \dots, \bar{n}$} and $z^{\bar{i}} = \bar{z}^i$. From this they get</p> <p>$$\Gamma_{\bar{i}j}^k = \frac{1}{2}h^{k\bar{l}}\left(\frac{\partial h_{j\bar{l}}}{\partial \bar{z}^i} - \frac{\partial h_{j\bar{i}}}{\partial \bar{z}^l}\right)$$</p> <p>How do they obtain this? Are they regarding $h$ as a map $(T^{1,0}M\oplus T^{0,1}M) \times (T^{1,0}M\oplus T^{0,1}M) \to \mathbb{C}$ where $h_{ab} = 0$, $h_{\bar{a}\bar{b}} = 0$, and $h_{\bar{a}b} = \overline{h_{b\bar{a}}}$? Even if they do, I don't see how they get the second term.</p> <p>Everything else I have found deals only with K&auml;hler metrics, in which case $\Gamma_{ab}^c$ and $\Gamma_{\bar{a}\bar{b}}^{\bar{c}}$ are the only non-trivial symbols.</p> <blockquote> <p>More generally, are there any treatments of Hermitian geometry which take this coordinate approach (as is common in Riemannian geometry texts)?</p> </blockquote> http://mathoverflow.net/questions/100906/hermitian-christoffel-symbols/100907#100907 Answer by Kevin Kordek for Hermitian Christoffel Symbols Kevin Kordek 2012-06-29T04:44:07Z 2012-06-29T04:44:07Z <p>You might want to try Complex Manifolds by Kodaira and Morrow. I seem to recall that a fair amount is done in local coordinates. </p> http://mathoverflow.net/questions/100906/hermitian-christoffel-symbols/100916#100916 Answer by diverietti for Hermitian Christoffel Symbols diverietti 2012-06-29T07:30:55Z 2012-06-29T07:30:55Z <p>Try to look at the book "Foundation of differential geometry" by Kobayashi and Numizu. You will find what you need (at least in the case of Kähler metrics) in Volume II, Chapter IX, Section 5.</p> http://mathoverflow.net/questions/100906/hermitian-christoffel-symbols/100967#100967 Answer by Daniel B. for Hermitian Christoffel Symbols Daniel B. 2012-06-29T17:55:10Z 2012-06-29T17:55:10Z <p>You might find something in "Lectures on Kahler Geometry" by Andrei Moroianu.</p> http://mathoverflow.net/questions/100906/hermitian-christoffel-symbols/101355#101355 Answer by S.A.A for Hermitian Christoffel Symbols S.A.A 2012-07-04T23:37:58Z 2012-07-04T23:37:58Z <p>Just another reference: the book of Bochner and Yano, curvature and Betti numbers, in the chapter where they deal with k\"ahlerian metrics, they do some calculation in co\"ordinates which you might find helpful.</p>