Читать книгу Algebra and Applications 2 - Группа авторов - Страница 41

Let M be a differentiable manifold, and let ▽ be the covariant derivation operator associated with a connection on the tangent bundle TM. The covariant derivation is a bilinear operator on vector fields (i.e. two sections of the tangent bundle): (X, Y) ↦ ▽XY, such that the following axioms are fulfilled:

The torsion of the connection is defined by:

[1.91]

and the curvature tensor is defined by:

[1.92]

The connection is flat if the curvature R vanishes identically, and torsion-free if . The following crucial observation by Matsushima (1968, Lemma 1) is an immediate consequence of equation [1.43]:

PROPOSITION 1.14.– For any smooth manifold M endowed with aflat torsion-free connection ▽, the space χ(M) of vector fields is a left pre-Lie algebra, with pre-Lie product given by:

[1.93]

Note that on M = ℝⁿ, endowed with its canonical flat torsion-free connection, the pre-Lie product is given by:

[1.94]