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. These concepts were put in their current form with principal bundles only in the 1950s. 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. This account is intended as an introduction to the theory of connections.
Historical overview
After the classical work of Gauss on the differential geometry of surfaces 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. It also provided an important tool for defining new topological invariants called characteristic classes via the Chern–Weil homomorphism. Although Gauss was the first to study the differential geometry of surfaces in Euclidean space E3, it was not until Riemann's Habilitationsschrift of 1854 that the notion of a Riemannian space was introduced. Christoffel introduced his eponymous symbols in 1869. Tensor calculus was developed by Ricci, who published a systematic treatment with Levi-Civita in 1901. Covariant differentiation of tensors was given a geometric interpretation by Levi-Civita (1917) who introduced the notion of parallel transport on surfaces. His discovery prompted Weyl and Cartan to introduce various notions of connection, including in particular that of affine connection. 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. Connections on a surface can be defined in a variety of ways. The Riemannian connection or Levi-Civita connection is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in terms of orthogonal projection. Indeed, the vector bundles associated with the frame bundle are all sub-bundles of trivial bundles that extend to the ambient Euclidean space; a first order differential operator can always be applied to a section of a trivial bundle, in particular to a section of the original sub-bundle, although the resulting section might no longer be a section of the sub-bundle. This can be corrected by projecting orthogonally. The Riemannian connection can also be characterized abstractly, independently of an embedding. The equations of geodesics are easy to write in terms of the Riemannian connection, which can be locally expressed in terms of the Christoffel symbols. Along a curve in the surface, the connection defines a first order differential equation in the frame bundle. The monodromy of this equation defines parallel transport for the connection, a notion introduced in this context by Levi-Civita. This gives an equivalent, more geometric way of describing the connection as lifting paths in the manifold to paths in the frame bundle. This formalises the classical theory of the "moving frame", favoured by French authors. Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields. The approach of Cartan, using connection 1-forms on the frame bundle of M, gives a third way to understand the Riemannian connection, which is particularly easy to describe for an embedded surface. Thanks to a result of Kobayashi (1956), later generalized by Narasimhan & Ramanan (1961), the Riemannian connection on a surface embedded in Euclidean space E3 is just the pullback under the Gauss map of the Riemannian connection on S2. Using the identification of S2 with the homogeneous space SO(3)/SO(2), the connection 1-form is just a component of the Maurer–Cartan 1-form on SO(3). In other words, everything reduces to understanding the 2-sphere properly.
Covariant derivative
For a surface M embedded in E3 (or more generally a higher-dimensional Euclidean space), there are several equivalent definitions of a vector field X on M:
a smooth map of M into E3 taking values in the tangent space at each point; the velocity vector of a local flow on M; a first order differential operator without constant term in any local chart on M; a derivation of C∞(M). The last condition means that the assignment f ↦ Xf on C∞(M) satisfies the Leibniz rule
X ( f g ) = ( X f ) g + f ( X g ) . {\displaystyle X(fg)=(Xf)g+f(Xg).}
The space of all vector fields X {\displaystyle {\mathcal {X}}} (M) forms a module over C∞(M), closed under the Lie bracket
[ X , Y ] f = X ( Y f ) − Y ( X f ) {\displaystyle [X,Y]f=X(Yf)-Y(Xf)}
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