ABSTRACT
So, R(X,Y) is a tensor and is of type (:) because the "contracted" product of R(X,Y) and a tensor of type ( ~) is a tensor of type (~).
By introducing the commutator
[Vx,Vy]==VXVy-VyVX' we denote:
R(X,Y) = [Vx' Vr]- V[x.Y]· (8-54)
4.1.2 Curvature tensor or Riemann-Christoffel tensor
Given any vector fields X,Y,Z E -%(M) we are going to define a new tensor called curvature tensor which operating on X,Y,Z leads to (~) tensor R(X,Y)Z. This curvature tensor is oftype (~). Make explicit the components of this tensor in a local chart. From
we deduce the following element of -%(M) :
In particular, make explicit
a a a R(-,'-j)-k =(V"Vj]Ok - V[O a10k'fu fu fu Pj
We successively have:
a a a R(-,'-j)-k = (VS j -VjVI-Vlo a I)Okox ox ox ,. j =Vo(Vook)-Vo (V~Ok)=vo(r~os)-Vo (~:O.)I J J v, I J' J
D The curvature tensor or Riemann-Christoffel tensor is the tensor of type (~) such that its components are
6V"' Ri.1l =air; - °jr;i +r; r;~ -~;rir. (8-55) The mapping
-%(M)x-%(M)x-%(M) ~-%(M): (X,Y,Z) H R(X,Y)Z is so well defined by
R(X Y)Z - XlyjZk R' ~ ,- k.1l ox'
and, in particular, by the following expressions where V[o/.ojIOk = 0:
R(o"Oj)Ok = [Vo/, Va) Ok - (V[a,.ojIOk) = R.i.1l a,.
