Actions

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)

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" .