Riemann & Ricci Curvature Tensors: No Coord. Indices?

[Matterwave]

For you math people who like to express objects in a coordinate-free way, how would you denote the Riemann and the Ricci curvature tensors? They are both usually denoted R but with different indices to show which one is which. Is there a standard way to write them without the coordinate indices?

 

[phyzguy]

I strongly recommend Doran and Lasenby's "Geometric Algebra for Physicists". They distinguish the two by showing their arguments. The Riemann tensor is a bivector-valued function of a bivector argument, so it can designated R(v1^v2), where ^ is the wedge-product. The Ricci tensor is a vector valued function of a vector argument, so it can be designated R(v).

 

[quasar987]

Yes, I believe writing R and Ric to denote the respective tensors is standard. R is tricky though because sometimes it is interpreted as a trilinear map R: TM x TM x TM --> TM. In which case we write not R(X,Y,Z) but R(X,Y)Z or R_Z(X,Y) and call R the riemann curvature endomorphism because given (X,Y), R(X,Y) is an endomorphism TM-->TM: Z-->R(X,Y)Z. Other times, R is interpreted as a 4-linear map R:TM x TM x TM x T*M-->R. This is related of course to the curvature endomorphism by means of the musical isomorphism (aka raising/lowering the last index).

 

[Matterwave]

Since I'm always working on a manifold with metric, I think I'll just use R and Ric. The raising and lowering of indices is trivial.