Indefinite K�hler manifolds

Abstract

IntroductionLet M be a complex-n-dimensional indefinite Kfihler manifold, that means M is endowed with an almost complex structure J and with an indefinite Riemannian metric g, which is J-Hermitian, i.e., for all mEM g (Ju, Jv)=g(u,v) for all u,v~T"M, and VJ=O,where 17 denotes the Levi-Civita connection of g. It follows then, that J is integrable and the index of g is an even number 2s with 0 < s-< n. (The manifolds, maps .... will be understood to be of class CO~ Given these data, we shall use all the well-known conceigts for complex resp. for indefinite Riemannian manifolds. In particular the curvature tensor R of M satisfies :Ruv is a G-linear skew-adjoint operator of (T,,M, Or,), Ruv= -R~u and R~w+ Rvwu+ Rwuv=O (1.2) for all meM and u, v, w~T,,M. For brevity we will write u* for Ju, u^v for spanR(u, v) and g for g,.. A plane (this will allways mean a real-2-dimensional vector subspace) P of T,,M is nondegenerate (with respect to g) if and only if P has a basis {u, v} with A(u, v)= g(u, u)g(v, v)-g(u, 0 2 :~ 0 (1.3) (and we sometimes denote such a P by P+, PS, P+ if gIP • P is positive definite, negative definite or indefinite respectively). The sectional curvature function K is defined for a nondegenerate plane P of T,.M as usual by [-see (1.3)] K(P)=g(RuvV, U)/A(u,v), if P=uAv.(1.4)The restriction of K to the nondegenerate holomorphic resp. totally real planes is called the holomorphic resp. totally real sectional curvature (function) of M.A holomorphic plane is nondegenerate if and only if it contains some u with g(u, u)~ 0 and we write then Hu for K(u ^ u*).

Real hypersurfaces of indefinite Kaehler manifolds

“…Consider the semi-Euclidean space R("+x) with the indefinite Kahlerian structure (cf. Barros-Romero [1] PROOF. By direct calculations taking account of (1.5), (2.13), (2.14) and (4. [7].…”

Quasi-biharmonic Lagrangian surfaces in Lorentzian complex space forms

“…Barros and Romero [2] showed that locally any complex space formM n s (4ǫ) is isometric holomorphically to C n s , CP n s (4ǫ) CH n s (4ǫ) according to ǫ = 0, ǫ > 0 or ǫ < 0.…”

Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms

“…For more details, we refer the reader to [2]. LetM n s (4c) be an indefinite complex space form of complex dimension n and complex index s. The complex index is defined as the (complex) dimension of the largest complex negative definite vector subspace of the tangent space.…”

“…We refer to [2] for the construction of the standard models of indefinite complex space forms CP n s (4c), whenc > 0, CH n s (4c), whenc < 0 and C n s . For our purposes it is sufficient to know that there exist pseudo-Riemannian submersions, called Hopf fibrations,…”

“…In [2] it is shown that locally any indefinite complex space form is holomorphically isometric to either C …”