Embedded Surfaces
When a surface M is embedded in E3, the Gauss map from M S2 extends to a SO(2)-equivariant map between the orthonormal frame bundles E SO(3). Indeed the triad consisting of the tangent frame and the normal vector gives an element of SO(3).
In 1956 Kobayashi proved that:
Under the extended Gauss map, the connection on SO(3) induces the connection on E.
This means that the forms ω, θ1 and θ2 on E are obtained by pulling back those on SO(3); and that lifting paths from M to E can be accomplished by mapping the path to the 2-sphere, lifting the path to SO(3) and then pulling back the lift to E. Thus for embedded surfaces, the 2-sphere with the principal connection on its frame bundle provides a "universal model", the prototype for the universal bundles discussed in Narasimhan & Ramanan (1965).
In more concrete terms this allows parallel transport to be described explicitly using the transport equation. Parallel transport along a curve c(t), with t taking values in, starting from a tangent from a tangent vector v0 also amounts to finding a map v(t) from to R3 such that
- v(t) is a tangent vector to M at c(t) with v(0) = v0.
- the velocity vector is normal to the surface at c(t), i.e. P(c(t))v(t)=0.
This always has a unique solution, called the parallel transport of v0 along c.
The existence of parallel transport can be deduced using the analytic method described for SO(3)/SO(2), which from a path into the rank two projections Q(t) starting at Q0 produced a path g(t) in SO(3) starting at I such that
g(t) is the unique solution of the transport equation
-
gtg−1 = ½ Ft F
with g(0) = I and F = 2Q − I. Applying this with Q(t) = P(c(t)), it follows that, given a tangent vector v0 in the tangent space to M at c(0), the vector v(t)=g(t)v0 lies in the tangent space to M at c(t) and satisfies the equation
It therefore is exactly the parallel transport of v along the curve c. In this case the length of the vector v(t) is constant. More generally if another initial tangent vector u0 is taken instead of v0, the inner product (v(t),u(t)) is constant. The tangent spaces along the curve c(t) are thus canonically identified as inner product spaces by parallel transport so that parallel transport gives an isometry between the tangent planes. The condition on the velocity vector may be rewritten in terms of the covariant derivative as
the defining equation for parallel transport.
The kinematic way of understanding parallel transport for the sphere applies equally well to any closed surface in E3 regarded as a rigid body in three dimensional space rolling without slipping or twisting on a horizontal plane. The point of contact will describe a curve in the plane and on the surface. As for the sphere, the usual curvature of the planar curve equals the geodesic curvature of the curve traced on the surface.
This geometric way of viewing parallel transport can also be directly expressed in the language of geometry. The envelope of the tangent planes to M along a curve c is a surface with vanishing Gaussian curvature, which by Minding's theorem, must be locally isometric to the Euclidean plane. This identification allows parallel transport to be defined, because in the Euclidean plane all tangent planes are identified with the space itself.
There is another simple way of constructing the connection form ω using the embedding of M in E3.
The tangent vectors e1 and e2 of a frame on M define smooth functions from E with values in R3, so each gives a 3-vector of functions and in particular de1 is a 3-vector of 1-forms on E.
The connection form is given by
taking the usual scalar product on 3-vectors.
Read more about this topic: Riemannian Connection On A Surface
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