Riemannian Connection On A Surface - Orthonormal Frame Bundle

Orthonormal Frame Bundle

See also: Connection (principal bundle)

Let M be a surface embedded in E3. The orientation on the surface means that an "outward pointing" normal unit vector n is defined at each point of the surface and hence a determinant can be defined on tangent vectors v and w at that point:

using the usual scalar triple product on E3 (itself a determinant).

An ordered basis or frame v, w in the tangent space is said to be oriented if det(v, w) is positive.

  • The tangent bundle of M consists of pairs (p, v) in M x E3 such that v lies in the tangent plane to M at p.
  • The frame bundle E of M consists of triples (p, e1, e2) with an e1, e2 an oriented orthonormal basis of the tangent plane at p.
  • The circle bundle of M consists of pairs (p, v) with ||v|| = 1. It is identical to the frame bundle because for each unit tangent vector v there is a unique unique tangent vector w with det(v, w) = 1.

Since the group of rotations in the plane SO(2) acts simply transitively on oriented orthonormal frames in the plane, it follows that it also acts on the frame or circle bundles of M. The definitions of the tangent bundle, the unit tangent bundle and the (oriented orthonormal) frame bundle E can be extended to arbitrary surfaces in the usual way. There is a similar identification between the latter two which again become principal SO(2)-bundles. In other words:

The frame bundle is a principal bundle with structure group SO(2).

There is also a corresponding notion of parallel transport in the setting of frame bundles:

Every continuously differentiable curve in M can be lifted to a curve in E in such a way that the tangent vector field of the lifted curve is the lift of the tangent vector field of the original curve.

This statement means that any frame on a curve can be parallelly transported along the curve. This is precisely the idea of "moving frames". Since any unit tangent vector can be completed uniquely to an oriented frame, parallel transport of tangent vectors implies (and is equivalent to) parallel transport of frames. The lift of a geodesic in M turns out to be a geodesic in E for the Sasaki metric (see below). Moreover the Gauss map of M into S2 induces a natural map between the associated frame bundles which is equivariant for the actions of SO(2).

Cartan's idea of introducing the frame bundle as a central object was the natural culmination of the theory of moving frames, developed in France by Darboux and Goursat. It also echoed parallel developments in Albert Einstein's theory of relativity. Objects appearing in the formulas of Gauss, such as the Christoffel symbols, can be given a natural geometric interpretation in this framework. Unlike the more intuitive normal bundle, easily visualised as a tubular neighbourhood of an embedded surface in E3, the frame bundle is an intrinsic invariant that can be defined independently of an embedding. When there is an embedding, it can also be visualised as a subbundle of the Euclidean frame bundle E3 x SO(3), itself a submanifold of E3 x M3(R).

Read more about this topic:  Riemannian Connection On A Surface

Famous quotes containing the words frame and/or bundle:

    Writing a novel is not merely going on a shopping expedition across the border to an unreal land: it is hours and years spent in the factories, the streets, the cathedrals of the imagination.
    —Janet Frame (b. 1924)

    In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.
    Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)