I am currently researching gradient flows on Riemannian Hilbert manifolds and in this paper, Trombe considers the Hilbert space $((H^1([0,1]; \mathbb{R}^3), \langle \cdot, \cdot \rangle_{H})$ with inner product $\langle u,v\rangle_H := \langle u,v\rangle_{L^2} + \langle u', v'\rangle_{L^2}$ as a Hilbert manifold and I want to know why for $p \in H(I;S^2)$ the dot product he uses
$$ \langle \cdot, \cdot \rangle: T_pH(I;S^2) \times T_pH(I;S^2), (u,v) \mapsto \langle \frac{Du}{\partial t}, \frac{Dv}{\partial t} \rangle $$
defines a Riemannian metric on $H(I;S^2$). I know that I may of course use the by the identity embedding $i: H(I;S^2) \hookrightarrow H(I;\mathbb{R}^3)$ induced pullback metric
$$g_{H(I;S^2)_p} (u,v)(p) := \langle di_p(u), di_p(v) \rangle_{T_{i(p)}H \simeq H} = \langle u, v \rangle_H$$ for $u,v \in T_w H(I;S^2)$.
Any help would be greatly appreciated.