I was wondering if anyone can give me an insight of what is meant with "Intrinsics Geometry" and "Extrinsics Geometry". At the beginning I thought this was like a distinction between differentiation and integration (in order to distiguish local analysis from global analysis), however I have the feeling I might have got this completely wrong.
So I looked this up on wikipedia and I quote what I've found:
From the beginning and through the middle of the 18th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium, to the effect that Gaussian curvature is an intrinsic invariant.
The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be "outside" of it?). However, there is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive.
These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator.
I still don't think like I get it, can anyone clarify? From the quote above what I'm getting is that intrinsics geometry assumes that the geometry analyzed is embedded in a bigger space (for example in differential geometry of curve and surface when we define the first fundamental form we rely on the euclidean metric of $\mathbb{R}^3$ to provide a definition, while we don't do this in Riemannian geometry where we define the Riemann tensor).
I'm not sure again though I fully understand the difference.
