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Vaisman-Gray manifold of pointwise holomorphic sectional conharmonic tensor. (English) Zbl 1422.53056
The authors discuss geometrical properties of Vaisman-Gray manifolds (VG manifolds).
Recall that in the adjoint $$G$$-structure space, an almost Hermitian manifold $$(M,J,g)$$ is called a Vaisman-Gray manifold if $$B^{abc}=-B^{bac}$$ and $$B_c^{ab}=\theta^{[a}\delta_c^{b]}$$, where $$B^{abc}=\frac i2 J^a_{[\hat b,\hat c]}$$, $$B_c^{ab}=\frac i2 J^a_{\hat b,c}$$ and $$\theta$$ is the Lee form.
In particular the authors give necessary and sufficient conditions for a VG manifold to admit a pointwise holomorphic sectional conharmonic (PHT)-tensor, that is, a tensor satisfying $$\langle T(X,JX,X,JX,)\rangle=h\|X\|^4$$. Moreover, conditions assuring that these manifolds are Einstein manifolds are established.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53B35 Local differential geometry of Hermitian and Kählerian structures
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