By Ngaiming Mok
This monograph stories the matter of characterizing canonical metrics on Hermitian in the community symmetric manifolds X of non-compact/compact forms by way of curvature stipulations. The proofs of those metric stress theorems are utilized to the examine of holomorphic mappings among manifolds X of an identical variety. furthermore, a twin model of the generalized Frankel Conjecture on characterizing compact Kähler manifolds also are formulated.
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Additional info for Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds (Series in Pure Mathematics, V. 6)
Example text
Seminegative). If (X,g) is furthermore irreducible then it is of constant positive (negative) Ricci curvature. 2 under the assumption that (X,g) is irreducible. ) we have R(A,B;B,A) = (R(A,B)B, A) = (—([A,BJ,B], A). When (X,g) is irreducible and of compact type the Riemannian inner product on T0(X) m is given by where Bg is the Killing form and c is a positive constant since both and BgIm are invariant under K and K acts irreducibly on m. On the other hand from the invariance of B9 under inner automorphisms we have B9([u,vJ,w) = —B9(v,[u,wj).
Tensor A = Im(E defines denoting Hermitian inner the Hermitian metric g is given Also associated to the Hermitian metric g is the real dz' ø dii). It follows from the Hermitian property of dz' A an that A is skew—symmetric. It can be identified with w = alternating (1,1)—form. We call w the Hermitian form of (X,g). By partition of unity any complex manifold can be endowed with Hermitian metrics. Of special interest among Hermitian metrics is the dass of Kähler metrics. We give the following geometric definition of Kähler manifolds.
As asserted in the proposition. 1 we obtain — VBVAC(o) 46 PROPOSITION 2 Let (X,g) be a Riemannian symmetric manifold of compact (resp. non—compact) type. Then, the sectional curvature of (X,g) is semipositive (resp. seminegative). If (X,g) is furthermore irreducible then it is of constant positive (negative) Ricci curvature. 2 under the assumption that (X,g) is irreducible. ) we have R(A,B;B,A) = (R(A,B)B, A) = (—([A,BJ,B], A). When (X,g) is irreducible and of compact type the Riemannian inner product on T0(X) m is given by where Bg is the Killing form and c is a positive constant since both and BgIm are invariant under K and K acts irreducibly on m.