Why is first fundamental form considered intrinsic

Solution 1:

What you're looking for is known as a Riemannian metric. One way of phrasing it is as follows. (Lee talks about these later in a slightly different way of phrasing later in his book.)

A Riemannian metric on a smooth manifold $M$ is a choice of inner product $g_p$ on each tangent space $T_pM$ that varies smoothly, in the sense that given any two smooth vector fields $X$ and $Y$ on $M$, the function $g(X,Y) = p \mapsto g_p(X_p,Y_p)$ is smooth.

Now let's start with an embedding $f: M \hookrightarrow \Bbb R^n$. The first fundamental form gives us a Riemannian metric on $M$ by setting $g_p(x,y) = Df(x) \cdot Df(y)$. To see that this is smoothly varying as defined above, consider the maps $TM \to \Bbb R^n \times \Bbb R^n \to \Bbb R^n, (p,v) \mapsto (f(p), Df(v)) \mapsto Df(v)$. Write a smooth vector field $X$ as $p \mapsto (p,X(p))$. Then the map $M \to \Bbb R^n, p \mapsto Df(X(p))$ is smooth, and so is the inner product map $\Bbb R^n \times \Bbb R^n \to \Bbb R, (x,y) \mapsto x\cdot y$. It is readily seen that $g(X,Y) = Df(X(p)) \cdot Df(Y(p))$ and that this latter map is smooth, as desired.

I should also say that a smooth manifold does not automatically come with a Riemannian metric, which is much more structure.

Solution 2:

Like you said, a property is intrinsic if it can be computed with knowledge of the first fundamental form (metric). I take it as meaning a property that a 'being' living on the surface could calculate. For example, Gaussian curvature is intrinsic. A being on a sphere, with only their metric, could see their world was 'curved' by finding a triangle whose interior angles sum up to greater than 180.