Metric connection

A linear connection in a vector bundle $\pi : X \rightarrow B$, equipped with a bilinear form in the fibres, for which parallel displacement along an arbitrary piecewise-smooth curve in $B$ preserves the form, that is, the scalar product of two vectors remains constant under parallel displacement. If the bilinear form is given by its components $g _ {\alpha \beta }$ and the linear connection by a matrix $1$- form $\omega _ \alpha ^ \beta$, then this connection is metric if

$$d g _ {\alpha \beta } = \ g _ {\gamma \beta } \omega _ \alpha ^ \gamma + g _ {\alpha \gamma } \omega _ \beta ^ \gamma .$$

In the case of a non-degenerate symmetric bilinear form, i.e. $g _ {\alpha \beta } = g _ {\beta \alpha }$ and $\mathop{\rm det} | g _ {\alpha \beta } | \neq 0$, the metric connection is called a Euclidean connection. In the case of a non-degenerate skew-symmetric bilinear form, the metric connection is called a symplectic connection in the vector bundle.

Under projectivization of a vector bundle, when the symmetric bilinear form generates some projective metric in each fibre (as in a projective space), the role of the metric connection is played by the projective-metric connection.

Comments

In the case of a positive-definite bilinear form, the metric connection is also called a Riemannian connection.

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1969)