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