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Differential Geometry of Manifolds

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Khokhlov S.V. Ignatochkina L.A.

Invariance of some classes of almost Hermitian structures concerning to the one-parameter group of diffeomorphisms generated by the Lie vector field

DOI
10.5922/0321-4796-2022-53-12
Pages
127-134
almost Hermitian structure almost complex structure Lie vector Lie form

Abstract

Finding the conditions for the invariance of geometric objects under the action of transformation groups is one of the main objects of geomet­ric research. Almost Hermitian structures and structures of the Gray — Her­vella classification on smooth manifolds are considered in this paper. All arguments are given using invariant Koszul’s calculus. Conditions for the invariance of the Kähler form in type structures are investigated and it is shown that the Kähler form is covariantly constant with respect to the Lie vector field. Conditions for the invariance of the Riemannian metric under the action of a one-parameter group of diffeomorphisms generated by a Lie vector field are studied. A criterion for the invariance of an al­most complex structure with respect to the local group of diffeomor­phisms generated by the Lie vector field in the class W4 is proved.

Conditions for the invariance of an operator of an almost complex structure, a tensor of a Riemannian metric, are proved. It is established that the invariance of the Riemannian structure g implies the invariance of the operator of an almost complex structure for some class of manifolds according to the Gray — Hervella classification, and conditions for the covariant constancy of the Lie form in certain classes of manifolds of dimensions above four were obtained. It is proved that the Lie form is covariantly constant in some classes of the type of dimensions above four.

Reference

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2. Ignatochkina, L. A.: Vaisman — Gray manifolds with J-invariant conformal curvature tensor. Sb. Math., 194:2, 61—72 (2003).

3. Kobayashi, Sh., Nomizu, K.: Foundations of Differential Geometry. Moscow (1981).

4. Aminova, A. V.: Projective transformations of pseudo-Riemannian manifolds. Moscow (2003).