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Could you help me to solve the following problem?

Let $M$ be a parallelizable manifold of dimension $n$, $\{E_1,\dots, E_n\}$ a global frame of $M$. Let $X$, $Y$ be vector fields on $M$ with $Y= \sum_{i=1}^n Y^i E_i$. Let $$\nabla_X Y = \sum_{i=1}^n {X(Y^i) E_i}.$$ Compute the torsion and curvature of $\nabla$.

I tried to compute the curvature:

\begin{align*} &\ R(X,Y,Z)\\ =&\ R_{XY}Z\\ =&\ \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]} Z\\ =&\ \nabla_X (Y(Z^j)E_j) - \nabla_Y (X(Z^i)E_i) - [X,Y](Z^k)E_k\\ =&\ X(Y(Z^j))E_j + Y(Z^j) \nabla_X E_j - Y(X(Z^i))E_i - X(Z^i) \nabla_Y E_i - XY(Z^k)E_k +YX(Z^k)E_k\\ =&\ X(Y(Z^j))E_j - XY(Z^k)E_k - YX(Z^i)E_i + YX(Z^k)E_k + Y(Z^j) \nabla_X E_j - X(Z^i) \nabla_Y E_i\\ =&\ Y(Z^j) \nabla_X E_j - X(Z^i) \nabla_Y E_i\\ =&\ 0. \end{align*}

Is it correct? What about the torsion?

Thanks!

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For some reason you seem to expand $\nabla_X Y(Z^j)E_j$ as $X(Y(Z^j))E_j + Y(Z^j)\nabla_XE_j$, while according to the definition, $\nabla_X Y(Z^j)E_j = X(Y(Z^j))E_j$. It follows directly from the definition of $\nabla$ that $\nabla_XE_j = 0$, so you end up with the correct expression for $R(X, Y, Z)$, namely zero, but your computation is more complicated than it needs to be.

You can compute the torsion in the same way you computed the curvature. Note that

\begin{align*} \nabla_XY - \nabla_YX &= \nabla_X(Y^iE_i) - \nabla_Y(X^iE_i)\\ &= X(Y^i)E_i - Y(X^i)E_i\\ &= (X(Y^i)-Y(X^i))E_i\\ &= [X, Y] \end{align*}

so $T(X, Y) = \nabla_XY - \nabla_YX - [X, Y] = 0$.

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