Adjoint connections

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Linear connections and such that for the corresponding operators of covariant differentiation and there holds

where and are arbitrary vector fields, is a quadratic form (i.e. a symmetric bilinear form), and is a -form (or covector field). One also says that and are adjoint with respect to . In coordinate form (where , , , ),

For the curvature operators and and torsion operators and of the connections and , respectively, the following relations hold:

In coordinate form,


[1] A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)


Instead of the name adjoint connections one also encounters conjugate connections.

Sometimes the -form is not mentioned in the notion of adjoint connections. Strictly speaking this notion of an "adjoint connection" should be called "adjoint with respect to B and w" .

How to Cite This Entry:
Adjoint connections. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098