Q:

Definition of Riemannian Curvature Tensor on Wikipedia

The definition of the Riemannian curvature tensor on wikipedia expresses it as inner products of tangent vectors:
https://en.wikipedia.org/wiki/Sectional_curvature
Now, my question is are these inner products just the Euclidean inner products, or are they the inner products induced by the metric tensor at the point on the manifold where the two tangent vectors are lying?

A:

It's the metric. Note that it need not make sense to apply a Euclidean inner product to tangent vectors. If you see $\langle\cdot,\cdot\rangle$ in the context of a Riemannian manifold, it's the metric.

