Editing Differential geometry (section)

== Intrinsic versus extrinsic ==

From the beginning and through the middle of the 19th 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 [[Bernhard Riemann|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. However, there is a price to pay in technical complexity: the intrinsic definitions of curvature and [[connection (mathematics)|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]].<ref>{{cite book |first=David |last=Hestenes |chapter=The Shape of Differential Geometry in Geometric Calculus |chapter-url=https://davidhestenes.net/geocalc/pdf/Shape%20in%20GC-2012.pdf |title=Guide to Geometric Algebra in Practice |editor-first=L. |editor-last=Dorst |editor2-first=J. |editor2-last=Lasenby|editor2-link=Joan Lasenby |publisher=Springer Verlag |year=2011 |pages=393–410 }}</ref>