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