MathDB

Let

\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)

be a real vector bundle of finite rank, and let

\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)

be the tangent bundle of

E

, where

V\xi=\mathrm{Ker}\, d\pi

is the vertical subbundle of

\tau_E

. Let us denote the projection operators corresponding to the splitting

(*)

v

and

h

. Construct a linear connection

\nabla

V\xi

such that

\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]

(

X

and

Y

are vector fields on

E

[.,\, .]

is the Lie bracket, and all data are of class

\mathcal C^\infty

. [J. Szilasi]

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