I am reading John Lee's $\textit{Introduction to Riemannian Manifolds}$ and I am having trouble understanding where he defines the gradient of a smooth function $f:M\to \mathbb{R}$. Here is the setup: $(M,g)$ is a Riemannian manifold and $f\in C^\infty(M)$. We define grad $f$ as the vector field obtained from $df$ by raising an index. That is, if $df=w$, then grad $f$ has local basis expression $(g^{ij}E_i f) E_j$.
Now the part that I don't understand. The author asserts that grad $f$ is characterized by the fact that
$$
df_p(w)=\langle \text{grad } f|_p, w\rangle \; \; \text{ for all } p\in M, w\in T_pM.
$$
Could someone explain why this is true?
Best Answer
That's the definition of "index-raising", as you say. The vector field $X$ comes from raising an index in the 1-form $\omega$ if and only if $\langle X, v\rangle = \omega(v)$ for all vectors $v$. This can actually be taken as the definition of index-raising, this is the coordinate-free one.