$|\nabla f|^2, |\nabla g|^2 \in W^{1,2}_{\mathrm{loc}}(M)$, then $\langle \nabla f ,\nabla g \rangle \in W^{1,2}_{\mathrm{loc}}(M)$?

Question

M is a Riemannian manifold, suppose $|\nabla f|^2, |\nabla g|^2 \in W^{1,2}_{\mathrm{loc}}(M)$, then $\langle \nabla f ,\nabla g \rangle \in W^{1,2}_{\mathrm{loc}}(M)$?

Answer 1

I do not know anything about Sobolev spaces on manifolds, but your assertion seems to be wrong even on $[-1,1]$:

Take $f(x) = |x|$ and $g(x) = x$. Then $\nabla f(x) = \mathrm{sgn}(x)$ and $\nabla g(x) = 1$. Hence, $|\nabla f(x)|^2 = 1$ and $|\nabla g(x)|^2 = 1$; and both functions belong to $H^1([-1,1])$. But $\nabla f \, \nabla g \not\in H^1(\Omega)$.