Intrinsic Versus Extrinsic
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?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay.
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.)
Read more about this topic: Differential Geometry
Famous quotes containing the words intrinsic and/or extrinsic:
“Three elements go to make up an idea. The first is its intrinsic quality as a feeling. The second is the energy with which it affects other ideas, an energy which is infinite in the here-and-nowness of immediate sensation, finite and relative in the recency of the past. The third element is the tendency of an idea to bring along other ideas with it.”
—Charles Sanders Peirce (1839–1914)
“Authors communicate with the people by some special extrinsic mark; I am the first to do so by my entire being, as Michel de Montaigne.”
—Michel de Montaigne (1533–1592)