Abstract

AbstractWe provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.

Highlights

  • A Sasakian manifold M is a contact metric manifold that satis es a normality condition, encoding the integrability of a canonical almost complex structure on the product M × R

  • Sasakian manifolds may be considered as an odd-dimensional analogue of nearly Kähler manifolds

  • The prototypical example of nearly Sasakian manifold is the -sphere as totally umbilical hypersurface of S, endowed with the almost contact metric structure induced by the well-known nearly Kähler structure of S

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Summary

Introduction

A Sasakian manifold M is a contact metric manifold that satis es a normality condition, encoding the integrability of a canonical almost complex structure on the product M × R. In particular one can show that an almost contact metric structure (g, φ, ξ , η) is Sasakian if and only if the covariant derivative of the endomorphism φ satis es (∇X φ)Y − g(X, Y)ξ + η(Y)X = ,. Sasakian manifolds may be considered as an odd-dimensional analogue of nearly Kähler manifolds. The prototypical example of nearly Sasakian manifold is the -sphere as totally umbilical hypersurface of S , endowed with the almost contact metric structure induced by the well-known nearly Kähler structure of S. A peculiarity of nearly Sasakian ve dimensional manifolds which are not Sasakian is that upon rescaling the metric one can de ne a Sasaki-Einstein structure on them. The theory of nearly Sasakian non-Sasakian manifolds is essentially equivalent to the one of Sasaki-Einstein manifolds. Concerning other dimensions, there have been many attempts of nding explicit examples of nearly Sasakian non-Sasakian manifolds until the recent result obtained in [4] showing that every nearly Sasakian

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Curvature properties of nearly Sasakian manifolds
We will use the following notation for curvature tensors
Covariant derivative of φ
As a result we get
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