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The covariant differentiation with respect to An affine connection vU on an arbitrary open submanifold X,Y ((VU)X(Y))p = (vXY)p. Then tion on U. on (vU)a/ayl. U, X. U of and any M. In fact, for p E U, we put is a well-defined affine connec- VU In particular, if with local coordinate system of is called on M induces an affine connection v any two vector fields vX U is a coordinate neighborhood we write vi instead § 1. ozn If v. we obtain another set of functions U, vi () = E rl . 1) holds whenever ( two of these neighborhoods overlap.

F, 1. Differentiable Manifolds 18 w = In terms of a local coordinate system, if w1 . 1... ir du . then r 11 E dw = E i, <... 1 Adu.

U, X. U of and any M. In fact, for p E U, we put is a well-defined affine connec- VU In particular, if with local coordinate system of is called on M induces an affine connection v any two vector fields vX U is a coordinate neighborhood we write vi instead § 1. ozn If v. we obtain another set of functions U, vi () = E rl . 1) holds whenever ( two of these neighborhoods overlap. 1). 3) w f on M. of type (0,1) vX to arbitrary tensor fields. we define vXf as Xf. , a 1-form), we put (vXw) (Y) = VX(w(Y)) -w(vXY).

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Complex Analytic Differential Geometry by Demailly


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