In our previous papers [7], [8],υ the notion of almost-analytic vector was introduced in certain almost-Hermitian spaces. In this paper we shall deal with tensors and obtain the notion of Φ-tensors which contains, as special cases, the one of analytic tensors and decomposable tensors.
In pseudo-Kahlerian manifolds, many interesting results concerning contravariant or covariant pseudo-analytic vectors are known.
2)Even though there were many papers about pseudo-Kahlerian manifolds, but were few about almost-Kahlerian ones. Recently, M. Apte generalized Bochner's theorem to compact almost-Kahlerian manifolds. His work seems to be very suggestive for me. In the present paper we shall generalize several theorems in pseudo-Kahlerian manifolds to almost-Kahlerian ones. The main results are integral formulas on vector fields in compact almost-Kahlerian manifolds.In §1 and §2 we shall prepare identities and lemmas and in §3 and §4 define almost-analytic vectors which are generalizations of pseudo-analytic vectors. As applications of integral formulas in §5, we shall obtain several theorems in §6. In §7, we shall give a decomposition theorem of the Lie algebra of contravariant almost-analytic vectors in a compact almost-Kahler-Einstein manifold. The canonical connection will be introduced in §8 and in the last section, to contravariant almost-analytic vectors, we shall generalize Apte's theorem.
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