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I have spent a lot of time on this problem and would appreciate some help. Please bear with me.

Let $M$ be a surface and $F\colon\mathbb{R}^3\rightarrow\mathbb{R}^3$ an isometry. Denote with $F_\ast$ the tangent map of $F$. Let $U$ be a unit normal vector field on $\mathcal{V}\subset M$ and let $\overline{U}=F_\ast U$ be the unit normal vector field on $F(\mathcal{V})$. Let $S$ be the shape operator on $M$ determined by $U$ and $\overline{S}$ the shape operator of $F(M)$ determined by $\overline{U}$. Let $p\in M$ and $v\in T_pM$. Show that $$F_\ast S_p(v_p)=\overline{S}_{F(p)}(F_\ast v_p).$$

Here is my attempt. First the left-hand side: the shape operator $S$ acting on a tangent vector $v_p$ is the covariant derivative of $U$ at $p$ in the direction of $v$. So if $\alpha$ is a curve such that $\alpha(0)=p$ and $\alpha'(0)=v_p$ then $$S_p(v_p) =-\nabla_{v_p}U = \frac{d}{dt}U(\alpha(t))\big|_{t=0}.$$ If we call $U'\colon\mathbb{R}^3\rightarrow\mathbb{R}^3$ the map that sends points to the vector part of $U$ (that is, ignoring the base point) then $$S_p(v_p) = -DU'(\alpha(0))\alpha'(0) ~\text{ at } p =J_{U'}(p)v ~\text{ at } p,$$ where $DU'$ is the derivative of $U'$ which is also the Jacobian matrix $J_{U'}$. Next, the tangent map $F_\ast$ sends tangent vectors $v_p$ to tangent vectors $J_F(p)v$ at $F(p)$. Then $$F_\ast S_p(v_p) = F_\ast\left(-J_{U'}(p)v ~\text{ at } p\right) = -J_F(p)J_{U'}(p)v ~\text{ at } F(p).$$ Now for the right-hand side: the shape operator $\overline{S}$ acting on a tangent vector $w_q$ is the covariant derivative of $\overline{U}$ at $q$ in the direction of $w$. Following our steps above, $$ \overline{S}_q(w_q) = -D\overline{U}'(q)w ~\text{ at } q.$$ Since $F_\ast v_p$ is $J_F(p)v$ at $F(p)$, we have $$ \overline{S}_{F(p)}(F_\ast v_p) = -D\overline{U}'(F(p))J_F(p)v ~\text{ at } F(p).$$ But what is $\overline{U}'$? By definition $\overline{U}=F_\ast U$, so at every point $p\in M$, $\overline{U}(p)=F_\ast U(p) = J_F(p)U'(p)$ at $F(p)$. So we can say $F_\ast U$ is the function $p\mapsto J_F(p)U'(p)$. Then $$\overline{S}_{F(p)}(F_\ast v_p) = -D\big(J_FU'\big)(F(p))J_F(p)v ~\text{ at } F(p).$$ From here, I don't know how to relate the two sides to each other. Part of my trouble is all the varying notation, which may have caused a mistake above. I would appreciate any guidance.

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