Adjoint connections

Linear connections $ \Gamma $ and $ \widetilde \Gamma $ such that for the corresponding operators of covariant differentiation $ \nabla $ and $ \widetilde \nabla $ there holds

$$ Z B ( X , Y ) = B ( \nabla _ {Z} X , Y ) + B ( X , {\widetilde \nabla } _ {Z} Y ) + 2 \omega (Z) B ( X , Y ) , $$

where $ X , Y $ and $ Z $ are arbitrary vector fields, $ B ( \cdot , \cdot ) $ is a quadratic form (i.e. a symmetric bilinear form), and $ \omega ( \cdot ) $ is a $ 1 $- form (or covector field). One also says that $ \nabla $ and $ \widetilde \nabla $ are adjoint with respect to $ B $. In coordinate form (where $ X , Y , Z \Rightarrow \partial _ {i} $, $ B \Rightarrow b _ {ij} $, $ \omega \Rightarrow \omega _ {i} $, $ \nabla \Rightarrow \Gamma _ {ij} ^ {k} $),

$$ \partial _ {k} b _ {ij} - \Gamma _ {ki} ^ {s} b _ {sj} - {\widetilde \Gamma } _ {kj} ^ {s} b _ {is} = \ 2 \omega _ {k} b _ {ij} . $$

For the curvature operators $ R $ and $ \widetilde{R} $ and torsion operators $ T $ and $ \widetilde{T} $ of the connections $ \nabla $ and $ \widetilde \nabla $, respectively, the following relations hold:

$$ B ( R ( U , Z ) X , Y ) + B ( X , \widetilde{R} ( U , Z ) Y ) = $$

$$ = \ 2 \{ \omega ( [ U , Z ] ) - U \omega (Z) + Z \omega (U) \} B ( X , Y ) , $$

$$ B ( Z , \Delta T ( X , Y ) ) - B ( \Delta T ( Z , Y ) X ) = $$

$$ = \ B ( \Delta T ( Z , X ) , Y ) ,\ \Delta T = \widetilde{T} - T . $$

In coordinate form,

$$ R _ {rsj} ^ {m} b _ {im} + {\widetilde{R} } _ {rsi} ^ {m} b _ {jm} = \ - 2 ( \partial _ {r} \omega _ {s} - \partial _ {s} \omega _ {r} ) b _ {ij} , $$

$$ \Delta T _ {ij} ^ {s} b _ {sk} - \Delta T _ {kj} ^ {s} b _ {si} - \Delta T _ {ki} ^ {s} b _ {sj} = 0 . $$

References

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

Comments

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

Sometimes the $ 1 $- form $ \omega $ 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" .