# Is Riemann curvature tensor defined for the Riemannian connection?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

I refer to Section 22.3, Section 22.4, Section 22.6 and Section 22.7.

Question: Is Riemann curvature tensor, in Section 22.6, supposed to be, in this particular section of this book, defined for the Riemannian connection?

Reasons why I would think Riemann curvature tensor is not defined for Riemannian connection:

1. I'm fairly certain that there were only 2 conventions in this book where the Riemannian connection is the default connection $$\nabla$$ for a Riemannian manifold. The first is for geodesics, as stated in Remark 14.2, and the second is for parallel translation, as stated in Section 14.7.

2. Section 22.6 could instead begin with "If $$R(X,Y)$$ is the curvature endomorphism on a Riemannian manifold, then we define the Riemann curvature..."

3. Theorem 22.9 could omit "if a connection $$\nabla$$ is compatible with the metric, then".

4. Is Ricci curvature defined for the Riemannian connection?

## 1 Answer

According to the introduction paragraph of section 22.6. that you linked, they say that $$R(X,Y)Z$$ is the curvature of the given connection $$\nabla$$ which can by any connection.

• Thanks but I forgot to include a link to my other question on Ricci curvature, which seems to be based on Riemann curvature tensor and which seems to assume Riemannian connection. Edited. Given that Ricci curvature seems to be based on Riemann curvature tensor. Is Riemann curvature tensor indeed defined for arbitrary connections ? – user636532 Oct 7 '19 at 6:52
• @SeleneAuckland I saw the other question and that paragraph seemed confusing to me too. It looks like they define Ricci curvature for an arbitrary connection, then prove something for the Levi-Civita connection and then claim the result for an arbitrary connection. – quarague Oct 7 '19 at 6:54
• Thanks quarague! – user636532 Oct 7 '19 at 7:59
• You can answer the other question if you want. – user636532 Oct 9 '19 at 9:29