Riemannian Connection On A Surface - Example: The 2-sphere

Example: The 2-sphere

See also: Nonholonomic system and Geometric phase

The differential geometry of the 2-sphere can be approached from three different points of view:

• analytic geometry, since the 2-sphere is a submanifold of E3;
• group theory, since the compact matrix group SO(3) acts transitively on the 2-sphere as a continuous group of symmetries;
• classical mechanics, since a rigid 2-sphere can roll on a plane.

S2 can be identified with the unit sphere in E3

Its tangent bundle T, unit tangent bundle U and oriented orthonormal frame bundle E are given by

The map sending (a,v) to (a, v, a x v) allows U and E to be identified.

Let

be the orthogonal projection onto the normal vector at a, so that

is the orthogonal projection onto the tangent space at a.

The group G = SO(3) acts by rotation on E3 leaving S2 invariant. The stabilizer subgroup K of the vector (1,0,0) in E₃ may be identified with SO(2) and hence

S2 may be identified with SO(3)/SO(2).

This action extends to an action on T, U and E by making G act on each component. G acts transitively on S2 and simply transitively on U and E.

The action of SO(3) on E commutes with the action of SO(2) on E that rotates frames

Thus E becomes a principal bundle with structure group K. Taking the G-orbit of the point ((1,0,0),(0,1,0),(0,0,1)), the space E may be identified with G. Under this identification the actions of G and K on E become left and right translation. In other words:

The oriented orthonormal frame bundle of S2 may be identified with SO(3).

The Lie algebra of SO(3) consists of all skew-symmetric real 3 x 3 matrices. the adjoint action of G by conjugation on reproduces the action of G on E3. The group SU(2) has a 3-dimensional Lie algebra consisting of complex skew-hermitian traceless 2 x 2 matrices, which is isomorphic to . The adjoint action of SU(2) factors through its centre, the matrices ± I. Under these identifications, SU(2) is exhibited as a double cover of SO(3), so that SO(3) = SU(2) / ± I. On the other hand SU(2) is diffeomorphic to the 3-sphere and under this identification the standard Riemannian metric on the 3-sphere becomes the essentially unique biinvariant Riemannian metric on SU(2). Under the quotient by ± I, SO(3) can be identified with the real projective space of dimension 3 and itself has an essentially unique biinvariant Riemannian metric. The geometric exponential map for this metric at I coincides with the usual exponential function on matrices and thus the geodesics through I are have the form exp Xt where X is a skew-symmetric matrix. In this case the Sasaki metric agrees with this biinvariant metric on SO(3).

The actions of G on itself, and hence on C∞(G) by left and right translation induce infinitesimal actions of on C∞(G) by vector fields

The right and left invariant vector fields are related by the formula

The vector fields λ(X) and ρ(X) commute with right and left translation and give all right and left invariant vector fields on G. Since C∞(S2) = C∞(G/K) can be identified with C∞(G)K, the function invariant under right translation by K, the operators λ(X) also induces vector fields Π(X) on S2.

Let A, B, C be the standard basis of given by

Their Lie brackets = XY – YX are given by

The vector fields λ(A), λ(B), λ(C) form a basis of the tangent space at each point of G.

Similarly the left invariant vector fields ρ(A), ρ(B), ρ(C) form a basis of the tangent space at each point of G. Let α, β, γ be the corresponding dual basis of left invariant 1-forms on G. The Lie bracket relations imply the Maurer-Cartan equations

These are also the corresponding components of the Maurer-Cartan form

a left invariant matrix-valued 1-form on G, which satisfies the relation

The inner product on defined by

is invariant under the adjoint action. Let π be the orthogonal projection onto the subspace generated by A, i.e. onto, the Lie algebra of K. For X in, the lift of the vector field Π(X) from C∞(G/K) to C∞(G) is given by the formula

This lift is G-equivariant on vector fields of the form Π(X) and has a unique extension to more general vector fields on G / K.

The left invariant 1-form α is the connection form ω on G corresponding to this lift. The other two 1-forms in the Cartan structural equations are given by θ₁ = β and θ₂ = γ. The structural equations themselves are just the Maurer-Cartan equations. In other words;

The Cartan structural equations for SO(3)/SO(2) reduce to the Maurer-Cartan equations for the left invariant 1-forms on SO(3).

Since α is the connection form,

• vertical vector fields on G are those of the form f · λ(A) with f in C∞(G);
• horizontal vector fields on G are those of the form f₁ · λ(B) + f₂ · λ(C) with f_i in C∞(G).

The existence of the basis vector fields λ(A), λ(B), λ(C) shows that SO(3) is parallelizable. This is not true for SO(3)/SO(2) by the hairy ball theorem: S2 does not admit any nowhere vanishing vector fields.

Parallel transport in the frame bundle amounts to lifting a path from SO(3)/SO(2) to SO(3). It can be accomplished by directly solving a matrix-valued ordinary differential equation ("transport equation") of the form g_t = A · g where A(t) is skew-symmetric and g takes values in SO(3).

In fact it is equivalent and more convenient to lift a path from SO(3)/O(2) to SO(3). Note that O(2) is the normaliser of SO(2) in SO(3) and the quotient group O(2)/SO(2), the so-called Weyl group, is a group of order 2 which acts on SO(3)/SO(2) = S2 as the antipodal map. The quotient SO(3)/O(2) is the real projective plane. It can be identified with space of rank one or rank two projections Q in M₃(R). Taking Q to be a rank 2 projection and setting F = 2Q − I, a model of the surface SO(3)/O(2) is given by matrices F satisfying F2 = I, F = FT and Tr F = 1. Taking F₀= diag (–1,1,1) as base point, every F can be written in the form g F₀ g−1.

Given a path F(t), the ordinary differential equation, with initial condition, has a unique C1 solution g(t) with values in G, giving the lift by parallel transport of F.

If Q(t) is the corresponding path of rank 2 projections, the conditions for parallel transport are

Set A = ½F_t F. Since F2 = I and F is symmetric, A is skew-symmetric and satisfies QAQ = 0.

The unique solution g(t) of the ordinary differential equation

with initial condition g(0) = I guaranteed by the Picard–Lindelöf theorem, must have gTg constant and therefore I, since

Moreover

since g−1Fg has derivative 0:

Hence Q = g Q₀ g−1. The condition QAQ=0 implies Q g_t g−1 Q = 0 and hence that Q₀ g−1 g_t Q₀ =0.

There is another kinematic way of understanding parallel transport and geodesic curvature in terms of "rolling without slipping or twisting". Although well known to differential geometers since the early part of the twentieth century, it has also been applied to problems in engineering and robotics. Consider the 2-sphere 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. At each point of contact the different tangent planes of the sphere can be identified with the horizontal plane itself and hence with one another.

• The usual curvature of the planar curve is the geodesic curvature of the curve traced on the sphere.
• This identification of the tangent planes along the curve corresponds to parallel transport.

This is particularly easy to visualize for a sphere: it is exactly the way a marble can be rolled along a perfectly flat table top.

The roles of the plane and the sphere can be reversed to provide an alternative but equivalent point of view. The sphere is regarded as fixed and the plane has to roll without slipping or twisting along the given curve on the sphere.