Local differential geometry

The part of differential geometry that studies properties of geometrical forms, in particular curves and surfaces, "in the small" . In other words, the structure of a geometrical form is studied in a small neighbourhood of an arbitrary point of it.

Suppose that in three-dimensional Euclidean space $ E ^ {3} $ a curve $ \gamma $ is specified by its equation

$$ \mathbf r = \mathbf r ( t) . $$

The study of this curve reduces to the discovery of quantities that are invariant with respect to the group of motions of $ E ^ {3} $. The position vector $ \mathbf r $ of a point $ M $ on the curve is not invariant, but its derivatives

$$ \tag{* } \frac{d \mathbf r }{d t } ,\ \frac{d ^ {2} \mathbf r }{d t ^ {2} } \dots $$

are invariant. The differential neighbourhood of order $ n $ of a point $ M $ of a curve $ \gamma $ is the totality of all concepts and properties connected with the curve that can be expressed in terms of the first $ n $ vectors of the sequence (*). Thus, to the differential neighbourhood of order one belong the concepts of the tangent to the curve and its normal plane. To the differential neighbourhood of order two belong the concepts of curvature, the osculating plane, the Frénet trihedron, and the osculating circle of the curve. The concept of the torsion of the curve belongs to the differential neighbourhood of order three. The curvature and torsion of a curve form a complete system of invariants of it in the sense that any invariant of the curve is a function of curvature, torsion and their derivatives of some orders. The local theory of surfaces of $ E ^ {3} $ is constructed similarly. The local theory of curves and surfaces of $ E ^ {3} $ is the oldest part of local differential geometry, mainly created in the 18th century and 19th century. Already in the 19th century various generalizations of this theory had begun to appear. One of these generalizations is connected with the concept of a homogeneous space. In an arbitrary differential-geometric homogeneous space $ G / H $ one can construct a local theory of curves and surfaces of various dimensions, similar as above for $ E ^ {3} $; namely as the theory of invariants of the fundamental group $ G $. In this direction the greatest development occurred in affine differential geometry and projective differential geometry.

A generalization of the concept of the first fundamental form of a surface in $ E ^ {3} $ led to the theory of Riemannian spaces. The local theory of Riemannian spaces had already appeared in the middle of the 19th century and continues to be developed, finding numerous applications.

The concept of parallel displacement of a vector along a curve on a surface in space led to the theory of spaces of affine connection. In turn, this was the start of the development of the general theory of connections (cf. Connection).

References

[1]	J. Favard, "Cours de géométrie différentielle locale" , Gauthier-Villars (1957)

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)
[a3]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5