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?
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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.