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.

 

[blue_raver22]

Yes, there is a standard way to write the Riemann and Ricci curvature tensors without using coordinate indices. This approach is known as the abstract index notation, where tensors are expressed in terms of abstract indices that do not depend on any specific coordinate system.

In this notation, the Riemann curvature tensor is denoted as R_{abcd} and the Ricci curvature tensor is denoted as R_{ab}. The abstract indices a, b, c, and d represent the abstract components of the tensors, which can be transformed to any coordinate system without changing the underlying mathematical structure.

This notation is widely used in modern differential geometry and general relativity, as it allows for a coordinate-free description of tensors and their properties. It also simplifies calculations and makes it easier to generalize results to different coordinate systems.

In summary, the Riemann and Ricci curvature tensors can be denoted in a coordinate-free way using the abstract index notation, providing a more elegant and versatile approach to expressing these important mathematical objects.