Timeline for Relationship between sectional curvature, bisectional curvature and conjugate points

8 events

when toggle format	what		by	license	comment
May 12, 2017 at 3:19	comment	added	YangMills		I have not seen the expression "constant bisectional curvature" anywhere else, only in Tian's papers. Everybody else uses "constant holomorphic sectional curvature"
May 12, 2017 at 3:18	comment	added	YangMills		To me the only meaning that "constant bisectional curvature" could possibly have is that the bisectional curvature $R(X,JX,JY,Y)$ (or if you prefer $R_{i \bar{j} k \bar{\ell}}\xi^i \bar{\xi}^j \zeta^k \bar{\zeta}^\ell$) evaluated on pairs of unit vectors $X,Y$ (or unit $(1,0)$ vectors $\xi,\zeta$) is constant independent of the vectors and of the points, and this cannot happen unless $g$ is flat.
May 11, 2017 at 8:51	comment	added	diverietti		I don't think that Tian's book has a typo about that. He defines what he (but not only him, it is quite standard, isn't it?) means by "constant holomorphic bisectional curvature". The definition is modelled exactly on the curvature of complex space forms, that is $R_{i\bar j k\bar l}=\lambda(g_{i\bar j}g_{k\bar l}+g_{i\bar l}g_{k\bar j})$.
Sep 13, 2012 at 9:01	vote	accept	Reza
Sep 12, 2012 at 14:25	history	edited	YangMills	CC BY-SA 3.0	added 260 characters in body
Sep 12, 2012 at 14:23	comment	added	YangMills		Yes, of course, sorry about that. I will edit my answer accordingly.
Sep 12, 2012 at 8:39	comment	added	Benoît Kloeckner		Of course, it leaves flat spaces that do have constant bisectional curvature, right?
Sep 11, 2012 at 0:04	history	answered	YangMills	CC BY-SA 3.0