Riemannian connection on a surface

In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form.

The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds.

After the classical work of Gauss on the differential geometry of surfaces[1][2][3][4] and the subsequent emergence of the concept of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century, the geometric notion of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century represented a major advance in differential geometry.

The introduction of parallel transport, covariant derivatives and connection forms gave a more conceptual and uniform way of understanding curvature, allowing generalisations to higher-dimensional manifolds; this is now the standard approach in graduate-level textbooks.

[5][6][7] It also provided an important tool for defining new topological invariants called characteristic classes via the Chern–Weil homomorphism.

Covariant differentiation of tensors was given a geometric interpretation by Levi-Civita (1917) who introduced the notion of parallel transport on surfaces.

Cartan's approach was rephrased in the modern language of principal bundles by Ehresmann, after which the subject rapidly took its current form following contributions by Chern, Ambrose and Singer, Kobayashi, Nomizu, Lichnerowicz and others.

The monodromy of this equation defines parallel transport for the connection, a notion introduced in this context by Levi-Civita.

Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields.

Its existence can be proved directly for a general surface by noting that the four properties imply the Koszul formula so that

In turn the geodesic curvature determines how vectors in the tangent planes along the curve should rotate during parallel transport.

[24] This equation shows once more that parallel transport depends only on the metric structure so is an intrinsic invariant of the surface; it is another way of writing the ordinary differential equation involving the geodesic curvature of c. Parallel transport can be extended immediately to piecewise C1 curves.

[30] Objects appearing in the formulas of Gauss, such as the Christoffel symbols, can be given a natural geometric interpretation in this framework.

One of the key facts about F is that it is (absolutely or completely) parallelizable, i.e. for n = dim F, there are n vector fields on F which form a basis at each point.

Conversely the lift X* is uniquely characterised by the following properties: On the orthonormal frame bundle F of a surface M there are three canonical 1-forms: If π: F

These 1-forms satisfy the following structural equations, due in this formulation to Cartan:[41] where h1 and h2 are smooth functions on the frame bundle F and K is a smooth function on M. In the case of a Riemannian 2-manifold, the fundamental theorem of Riemannian geometry can be rephrased in terms of Cartan's canonical 1-forms: Theorem.

[43] 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.[44] Holonomy is exactly the phenomenon that occurs when a tangent vector (or orthonormal frame) is parallelly transported around a closed curve.

Since ω(Ui)=0, the flows αt and βt in F are lifts by parallel transport of their projections in M. Informally the idea is as follows.

The group SU(2) has a 3-dimensional Lie algebra consisting of complex skew-hermitian traceless 2 x 2 matrices, which is isomorphic to

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.

given by Their Lie brackets [X,Y] = 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.[51] 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

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.

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.

[55] 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.

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.

In 1956 Kobayashi proved that:[58] 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 (1961).

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.

One of the most comprehensive introductory surveys of the subject, charting the historical development from before Gauss to modern times, is by Berger (2004).

Accessible introductions to Cartan's approach to connections using moving frames can be found in Ivey & Landsberg (2003) and Sharpe (1997).