TITLE: Why is first fundamental form considered intrinsic
QUESTION [9 upvotes]: I am reading Kuhnel's differential geometry book, and in chapter 4, it says that "intrinsic geometry of a surface" can be considered to be things that can be determined solely from the first fundamental form. But I am unsure on why the first fundamental form itself can be considered to be something intrinsic to the surface. Kuhnel defines the first fundamental form to be the inner product induced from that of $\mathbb{R}^3$ restricted to $T_pM$. So isn't the larger $\mathbb{R}^3$ used?
So my question is: Supposed I have a smooth 2-dimensional manifold (e.g. in the sense defined by Lee's Introduction to Smooth Manifolds), but I didn't put it in any ambient space. Is there a way for me to define the first fundamental form? $T_pM$ is intrinsically defined (as the space of derivations in Lee's book), and so what is the inner product that I should put on $T_pM$?

REPLY [5 votes]: 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.