Kähler submanifolds and the Umehara algebra

Abstract

We show that the indefinite complex space form C r,s is not a relative to the indefinite complex space form CP N l (b) or CH N m (b). We further study whether two Fubini-Study spaces are relatives or not.2010 Mathematics Subject Classification. 32H02, 32Q40, 53B35.

Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space forms

“…The Umehara algebra (see [29]) is defined to be the R-algebra Λ p generated by the elements of the form h k+ hk, for h, k ∈ O p . Umehara algebra has been an important tool in the study of relatives Kähler manifolds (see [11,27,28,29,8]).…”

CR relatives Kaehler manifolds

“…Further, in [13] it has been proven that Euclidean spaces and Hermitian symmetric spaces of noncompact type are not relatives, and the same result for Hermitian symmetric spaces of compact type follows by Umehara's result and Nakagawa-Takagi embedding of Hermitian symmetric spaces of compact type into the complex projective space. The relativity problem has been investigated also in [9], where it has been reformulated in terms of the Umehara's algebra, and in [18,19,21].…”

Strongly not relatives Kähler manifolds

“…Umehara's work has been generalized in the recent paper by X. Cheng and A. J. Di Scala [4], where the authors state necessary and sufficient conditions for complex space forms of finite dimension and different curvatures to not be relative to each others. In [6] A. J.…”

“…Observe that in general there are not reasons for a Kähler manifold which is not relative to another Kähler manifold to remain so when its metric is rescaled. For example, consider that the complex projective space (CP 2 , c g F S ) where g F S is the Fubini-Study metric, for c = 2 3 is not relative to (CP 2 , g F S ), while for positive integer values of c it is (see [4] for a proof).…”