Riemannian Connection On A Surface - Holonomy and Curvature

Holonomy and Curvature

Parallel transport in the frame bundle can be used to show that the Gaussian curvature of a surface M measures the amount of rotation obtained by translating vectors around small curves in M. Holonomy is exactly the phenomenon that occurs when a tangent vector (or orthonormal frame) is parallelly transported around a closed curve. The vector reached when the loop is closed will be a rotation of the original vector, i.e. it will correspond to an element of the rotaion group SO(2), in other words an angle modulo 2π. This is the holonomy of the loop, because the angle does not depend on the choice of starting vector.

This geometric interpretation of curvature relies on a similar geometric of the Lie bracket of two vector fields on E. Let U₁ and U₂ be vector fields on E with corresponding local flows α_t and β_t.

• Starting at a point A corresponding to x in E, travel along the integral curve for U₁ to the point B at .

• Travel from B by going along the integral curve for U₂ to the point C at .

• Travel from C by going along the integral curve for U₁ to the point D at .

• Travel from D by going along the integral curve for U₂ to the point E at .

In general the end point E will differ from the starting point A. As s 0, the end point E will trace out a curve through A. The Lie bracket at x is precisely the tangent vector to this curve at A.

To apply this theory, introduce vector fields U₁, U₂ and V on the frame bundle E which are dual to the 1-forms θ₁, θ₂ and ω at each point. Thus

Moreover V is invariant under K and U₁, U₂ transform according to the identity representation of K.

The structural equations of Cartan imply the following Lie bracket relations:

The geometrical interpretation of the Lie bracket can be applied to the last of these equations. Since ω(U_i)=0, the flows α_t and β_t in E are lifts by parallel transport of their projections in M.

Informally the idea is as follows. The starting point A and end point E essentially differ by an element of SO(2), that is an angle of rotation. The area enclosed by the projected path in M is approximately . So in the limit as s 0, the angle of rotation divided by this area tends to the coefficient of V, i.e. the curvature.

This reasoning is made precise in the following result.

Let f be a diffeomorphism of an open disc in the plane into M and let Δ be a triangle in this disc. Then the holonomy angle of the loop formed by the image under f of the perimeter of the triangle is given by the integral of the Gauss curvature of the image under f of the inside of the triangle.

In symbols, the holonomy angle mod 2π is given by

where the integral is with respect to the area form on M.

This result implies the relation between Gaussian curvature because as the triangle shrinks in size to a point, the ratio of this angle to the area tends to the Gaussian curvature at the point. The result can be proved by a combination of Stokes's theorem and Cartan's structural equations and can in turn be used to obtain a generalisation of Gauss's theorem on geodesics triangles to more general triangles.

One of the other standard approaches to curvature, through the covariant derivative, identifies the difference

as a field of endomorphisms of the tangent bundle, the Riemann curvature tensor. Since is induced by the lifted vector field X* on E, the use of the vector fields U_i and V and their Lie brackets is more or less equivalent to this approach. The vertical vector field W=A* corresponding to the canonical generator A of could also be added since it commutes with V and satisfies = U₂ and = —U₁.