Abstract:
An affinor structure is a generalization of the notion of an almost complex structure associated with a symplectic form on a manifold of even dimension for vector bundles of arbitrary rank. An affinor structure is the field of the automorphisms of the vector bundle preserving the exterior derivative of some $1$-form with radical of arbitrary dimension. The exterior derivative can be always defined on Lie algebroids, a special class of vector bundles. Therefore, the theory of affinor structures is considered on Lie algebroids. We show that the classical objects, such as a symplectic structure, a contact structure, and a Kähler structure, are particular cases of the general theory of affinor metric structures.
Keywords:affinor structure, radical of a $1$-form, Riemannian metric, Lie algebroid.
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