# Parallelity of the pullback of 2nd fundamental form

I've got the following problem with the pullback of the second fundamentalform: Let $\tau \rightarrow Gr$ be the tautological bundle over a Grassmannian (the fibre at a point is just the point itself) Then one can prove easily that the 2nd fundamentalform $H \in \Omega^1(Gr,Hom(\tau,\tau^c))$ of $\tau \subset \underline{\mathbb{C} ^N}$ is parallel, i.e. we have a product rule

$\nabla_X^{\tau^c} H_Y\sigma=H_{\nabla_XY}\sigma+H_Y\nabla^{\tau}_X\sigma$

where $\nabla$ is the Levi-Civita Connection of $Gr$. Now I want to pullback this, i.e. I look at a map $\bar f:M \times I\rightarrow Gr$ and pullback $H$, i.e. by definition

$(\bar f^*H)_Xs:=H_{\bar f_*X}s \in \Gamma(\bar f^*\tau^c)$

for $s \in \Gamma(\bar f^*\tau)$ and $X \in \Gamma(TM \times TI)$. For simplicity I denote the time vectorfield on $M \times I$ just by $\frac{\partial}{\partial t}$ meaning $(0,\frac{\partial}{\partial t})\in TM \times TI$. Then I can derive $(\bar f^*H)_Xs$ with the pullback connection $\nabla^{\bar f^*\tau^c}$:

$\nabla^{\bar f^*\tau^c}_{\frac{\partial}{\partial t}}(\bar f^*H)_Xs$

Now my question is: Do I have the same product rule as above? Which connection do I have to use in the vector argument of $\bar f^*H$? Is the pullback of the second fundamental form parallel w.r.t. toe pullback connection?

I tried to compute this stuff in local coordinates and I think I got the same product rule as above with the pulled back Levi-Civita connection of $Gr$ in the vector argument of $\bar f^*H$ instead of the Levi Civita connection but I'm not sure...anyway are there more global arguments if it is true what I said?

I'm grateful for any sort of help or advice.

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