On canonical forms on non-holonomic and semi-holonomic prolongations of principal fibre bundles
We deal with a -tensor field on the tangent bundle preserving vertical vectors and such that is a -tensor field on , where is the canonical almost tangent structure on . A connection on is constructed by . It is shown that if is a -almost complex structure on without torsion then is a unique linear symmetric connection such that and .
Two symplectic structures on a manifold determine a (1,1)-tensor field on . In this paper we study some properties of this field. Conversely, if is (1,1)-tensor field on a symplectic manifold then using the natural lift theory we find conditions under which , is symplectic.
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