Why is $\Pi(X,Y)=\frac{\partial}{\partial t} \nabla_XY$?

Question

1. Let $X,Y$ be two vector fields. Then the book "Lectures on the Ricci Flow" says $\Pi(X,Y)=\frac{\partial}{\partial t} \nabla_XY$. I don't understand how this is the case. The second fundamental form is the normal part of $\tilde{\nabla}_XY$, where $\tilde{\nabla}$ is a connection on a bigger manifold.

2. Assuming the above formula, the author goes on to state that $\frac{\partial}{\partial t} \nabla_Y\nabla_X Z=\Pi(Y,\nabla_XZ) + \nabla_Y(\Pi(X,Z))$. How does he get this result?

Answer 1

(1) If $E_i$ is coordinate fields, then $X=X_iE_i$ Hence $\nabla_XY=\Gamma_{ij}^k X_iY_jE_k + X(Y_k)E_k$ so that $$ \frac{\partial }{\partial t}\nabla_XY= \bigg(\frac{\partial }{\partial t}\Gamma_{ij}^k\bigg) X_iY_jE_k :=\Pi (X,Y) $$

Note that $\Pi$ is a tensor.

(2) \begin{align*} \frac{\partial }{\partial t}\nabla_Y\nabla_X W &= [\nabla_Y( X(W_l)E_l + X_s W_t\Gamma_{st}^lE_l ) ]_t \\&= [ X(W_l) Y_u\Gamma_{ul}^q E_q + Y( X_s W_t\Gamma_{st}^q) E_q + X_s W_t\Gamma_{st}^l Y_u \Gamma_{ul}^q E_q ]_t\\ &= \Pi (Y,XW) +Y E_q^\ast\Pi (X,W)\ E_q + E_l^\ast\Pi(X,W)\ \nabla_YE_l +\Pi (Y,X_s W_t\Gamma_{st}^l E_l)\\ &= \Pi (Y,XW) +\nabla_Y ( \Pi (X,W) ) +\Pi (Y,X_s W_t\Gamma_{st}^l E_l) \\ &= \Pi (Y,\nabla_XW) +\nabla_Y ( \Pi (X,W) ) \end{align*}