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,
References
| [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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_connections&oldid=18145
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
