do Carmo gives a definition of sectional curvature as follows:
$K(x,y) = \frac{\langle R(x,y)x,y\rangle}{|x\times y|^2}$
where $x,y \in T_pM$ are linearly independent vectors.
My question: The curvature of a riemannian manifold is a correspondence that associates, for each vector fields $X,Y$, a linear map $R(X,Y)$, which takes vector fields in vector fields. In the definition above $x,y \in T_pM$ which means they are not vector fields, so how could one interpret $R(x,y)x$?