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So I'm trying to learn Riemannian geometry on my own... probably not a realistic goal! But anyway, for now I'm stuck on understanding part of this passage:

A vector field $X$ on a $C^{\infty}$ manifold $M$ is a smooth assignment of a tangent vector $X_p \in T_p M$ where smooth is defined to mean that for all $f \in C^{\infty}$, the function $Xf : M \to \mathbb{R}$ defined by

  1. $M \to \mathbb{R}$
  2. $p \mapsto (Xf)(p) := X_p(f)$

is infinitely differentiable.

This is from Isham, Modern Differential Geometry for Physicists 2ed p.97.

So I understand that $f$ is a smooth function defined on the manifold, that $X$ is a vector field assembled from the tangent space at each point, and probably smoothly changing as $p$ changes. The part I don't understand is this:

$p \mapsto (Xf)(p) := X_p(f)$

It's a mapping from $p$ to something, but can anyone explain in words what the right hand side means? What is $(Xf)$ ? And what is $X(f)$ ?

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$X_p$ is a tangent vector in $T_p(M)$ and $X_p(f)$ denotes the corresponding directional derivative. – Qiaochu Yuan Jan 5 '13 at 6:58
The symbol ‘$ X $’ can stand for both a vector field and a tangent vector at some point of a manifold. It is always good to watch out for such pitfalls in clarity. – Haskell Curry Jan 5 '13 at 7:20
Is $X_p(f)$ the directional derivative of the function $f(p)$ in the direction $X(p)$ ? The other alternative, that $X_p(f)$ would be a derivative of $X$ in the direction $f$, does not seem to make sense because $f$ is not a direction. – user55261 Jan 5 '13 at 8:45

Let $p\in M$, and let $T_pM$ be the tangent space of $M$ at $p$. There are several equivalent definitions of the tangent space, but it seems to me that the one intended here is this definition, where a tangent vector $X_p\in T_pM$ is a derivation on the space of smooth functions $f:M\to \mathbb{R}$. Specifically, $X_p$ eats smooth functions $f:M\to\mathbb{R}$ and outputs real numbers $X_p(f)$, it does so linearly in $f$, and finally it satisfies this formulation of the chain rule: $$X_p(fg)=X_p(f)g(p)+f(p)X_p(g).$$ The subscript $p$ on $X_p$ is purely a notational reminder that $X_p$ "lives at" $p$.

We can now define the notion of a vector field on $M$, denoted by a letter $X$, which consists of a choice of tangent vector $X_p$ at each point $p$, and this vector field $X$, i.e. this assignment $p\mapsto X_p$, is smooth when it satisfies the property you're asking about: for any smooth function $f:M\to\mathbb{R}$, the function "$Xf$" from $M$ to $\mathbb{R}$ defined by sending $p\in M$ to $X_p(f)\in\mathbb{R}$ must be smooth as well. Note that this explains the author's "in-line" definition of $Xf$, $$p\mapsto (Xf)(p):=X_p(f),$$ i.e. "$Xf$ is the function sending $p$ to $(Xf)(p)$, where $(Xf)(p)$ is the real number $X_p(f)$".

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