When I read books of geometry, I sometimes find the term "intrinsic" and "extrinsic" but don't understand them precisely.
What are definitions of them? Are intrinsic properties more important than extrinsic properties?
$\begingroup$
$\endgroup$
-
1$\begingroup$ could you give an example of these books' usage? $\endgroup$ – ziggurism Nov 17 '17 at 13:03
-
$\begingroup$ Can you explain what you didn't understand or found unsatisfying when you searched for the terms in the context of differential geometry? For example, I think that the beginning of Wikipedia's page of differential geometry of surfaces has a good first explanation (at the time of this comment). $\endgroup$ – Mark S. Nov 17 '17 at 13:20
-
$\begingroup$ Gaussian curvature is intrinsic one. $\endgroup$ – marimo Nov 17 '17 at 13:20
-
$\begingroup$ @marimo: But for surfaces in $\Bbb R^3$, Gaussian curvature is defined extrinsically. So it's one of Gauss's deep results that it in fact does turn out to be intrinsic (i.e., dependent only on the metric and not on the embedding). $\endgroup$ – Ted Shifrin Nov 17 '17 at 16:54
$\begingroup$
$\endgroup$
I will try to address the question in the context of differential geometry of manifolds. A property of a manifold is termed intrinsic if it depends only on distances as measured within the manifold. In a more mathematical language, intrinsic properties depends on the metric, or the first fundamental form. On the other hand, a property is termed extrinsic if it depends on the way you embed your manifold in an higher dimensional Euclidean space. An example of extrinsic 'property' is the second fundamental from from which you can deduce the shape operator.
Another important example that may use the term intrinsic is the curvature. Gauss's Theorema Egregium asserts that the curvature is an intrinsic property of a manifold - because it can be deduced from the metric.
