Interior geometry

intrinsic geometry

The branch of geometry dealing with those properties of a surface and of figures on that surface that depend only on the lengths of the curves lying on the surface, and thus can be defined without recourse to their spatial environment. The interior geometry of regular surfaces involves concepts such as the angle between curves, the surface area of a domain, the Gaussian (or the complete) curvature of a surface, the geodesic curvature of a curve, and the Levi-Civita connection. The term "interior geometry" is also used in a more general situation to denote a structure (usually a metric or a connection) induced in a topological space by mapping it into another space which has a priori been assigned a similar structure.

The possibility of regarding the objects of interior geometry as the properties of the surface itself, regardless of the ambient space, led to the study of abstract spaces with an interior metric (cf. Internal metric), the properties of which are alike to the interior geometry of surfaces (cf. Riemannian space; Convex surface; Two-dimensional manifold of bounded curvature). Along with the interior approach, it is also possible to distinguish classes of immersed surfaces and submanifolds by their extrinsic geometric properties. A comparison of these two approaches constitutes the problem of isometric immersion and imbedding. In a number of important cases both methods yield identical classes of metrics. For instance, any interior geometry of a Riemannian metric (of class $C^r$, $r>3$) may be regarded as the interior geometry of some submanifold of a Euclidean space of sufficiently high dimension, while the geometry of any complete two-dimensional interior metric of non-negative curvature may be regarded as the interior geometry of a convex surface in $E^3$. A classical example of an opposite situation is the Hilbert theorem, according to which there is no regular isometric immersion of the Lobachevskii plane into $E^3$. The term "interior geometry" applied to abstract spaces of this kind is only meaningful as a juxtaposition with the extrinsic geometry in the framework of a definite theory. The clarification of the connections between the interior geometry of surfaces and their extrinsic geometry is one of the most difficult and interesting problems in geometry. It involves, in addition to the problem of isometric immersion, also problems such as the deformation of surfaces, infinitesimal deformation, unique determination of a surface by its metric, and the effect of the smoothness of a metric on the smoothness of the surface. The relations between extrinsic and intrinsic geometries were also studied in the context of superposition of immersions (curves on a surface, minimal submanifolds of spheres).

The fundamentals of interior geometry were laid by C.F. Gauss [1], were developed by B. Riemann [2] for the multi-dimensional case, and by A.D. Aleksandrov [3] for the irregular case.

References

[1]	C.F. Gauss, "Allgemeine Flächentheorie" , W. Engelmann , Leipzig (1900) (Translated from Latin)
[2]	B. Riemann, "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[3]	A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)

References

[a1]	W. Rinow, "Die innere Geometrie der metrischen Räume" , Springer (1961)
[a2]	M. Gromov, "Structures métriques des espaces Riemanniens" , F. Nathan (1981) (Translated from Russian)