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I'm beginning with tensorial calculus and I have some questions. Let $(M,g)$ a riemannian manifold with $nabla$ his Levi Civita connection. The curvature tensor $R$ is defined as

beginalign*
R : mathfrakX(M) times mathfrakX(M) times mathfrakX(M) &to mathfrakX(M)\
(X,Y,Z) &mapsto R(X,Y)Z
endalign*

where

$$R(X,Y)Z = nabla_X nabla_Y Z - nabla_Y nabla_X Z - nabla_[X,Y]Z.$$

In all books of differential geometry, they said that $R$ is a $(1,3)$ tensor but I don't know why, because a $(1,3)$ tensor is a multilinear map

$$Omega^1(M) times mathfrakX(M) times mathfrakX(M)timesmathfrakX(M) to C^infty(M).$$

I think that I don't understand any concept or because for example, let $X in mathfrakX(M)$, then $nabla X : TM to TM$, $(nabla X)(v_p) = nabla_v_pX$, is a $(1,1)$ tensor field and I don't know why.

$textbfRemark$: $Omega^1(M)$ is the set of all 1-forms, $alpha : M to TM^*$.

A $(p, q)$-tensor field on a smooth manifold $M$ is a $C^infty(M)$-multilinear map $T : Omega^1(M)^ptimesmathfrakX(M)^q to C^infty(M)$.

Given a $C^infty(M)$-multilinear map $S : Omega^1(M)^ptimesmathfrakX(M)^q to mathfrakX(M)$, there is an associated $(p + 1, q)$-tensor field $T : Omega^1(M)^p+1timesmathfrakX(M)^q to C^infty(M)$ defined by

$$T(beta, alpha^1, dots, alpha^p, X_1, dots, X_q) := beta(S(alpha^1, dots, alpha^p, X_1, dots, X_q)).$$

Likewise, given a $C^infty(M)$-multilinear map $S : Omega^1(M)^ptimesmathfrakX(M)^q to Omega^1(M)$, there is an associated $(p, q + 1)$-tensor field $T : Omega^1(M)^ptimesmathfrakX(M)^q+1 to C^infty(M)$ defined by

$$T(alpha^1, dots, alpha^p, Y, X_1, dots, X_q) := (S(alpha^1, dots, alpha^p, X_1, dots, X_q))(Y).$$