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., Chapter XII.
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 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 , later generalized by , 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. This approach, together with its higher-dimensional generalisations, is discussed in great detail in Chapters 1 and 2.
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
The space of all vector fields (M) forms a module over C∞(M), closed under the Lie bracket
with a C∞(M)-valued inner product (X,Y), which encodes the Riemannian metric on M.
Since (M) is a submodule of C∞(M, E3)=C∞(M) E3, the operator X I is defined on (M), taking values in C∞(M, E3).
Let P be the smooth map from M into M3(R) such that P(p) is the orthogonal projection of E3 onto the tangent space at p. Thus for the unit normal vector np at p, uniquely defined up to a sign, and v in E3, the projection is given by (p)(v) = v - (v · np) np.
Pointwise multiplication by P gives a C∞(M)-module map of C∞(M, E3) onto (M) . The assignment
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defines an operator on (M) called the covariant derivative, satisfying the following properties
is C∞(M)-linear in X (Leibniz rule for derivation of a module) (compatibility with the metric) (symmetry property).
The first three properties state that is an affine connection compatible with the metric, sometimes also called a hermitian or metric connection. The last symmetry property says that the torsion tensor
vanishes identically, so that the affine connection is torsion-free.
Although the Riemannian connection was defined using an embedding in Euclidean space, this uniqueness property means that it is in fact an intrinsic invariant of the surface.
Its existence can be proved directly for a general surface by noting that the four properties imply the Koszul formula
so that depends only on the metric and is unique. On the other hand, if this is used as a definition of , it is readily checked that the four properties above are satisfied.
For an isometric embedding of in E3, the tangent vectors and yield a matrix It is a positive-definite matrix. Its inverse is also positive-definite symmetric, with matrix . The inverse also has a unique positive-definite square root, with matrix . It is routine to check that form an orthonormal basis of the tangent space. In this case, the projection onto the tangent space is given by so that
Thus , where
Since , this gives another way to derive the Christoffel symbols:
Formulas for covariant derivative can be also be derived from local coordinates (x,y) without the use of isometric embeddings. Taking and ' as vector fields, the connection can be expressed purely in terms of the metric using the Christoffel symbols:
To derive the formula, the Koszul formula can be applied with , and set to 's; in that case all the Lie brackets commute.
Curvature operator
The Riemann curvature tensor can be defined by covariant derivatives using the curvature operator:
Since the assignment is C∞(M)-linear in each variable, it follows that R(x,Y)p is an endomorphism at p. For X and Y linearly independent tangent vectors at ,
is independent of the choice of basis and is called the Gaussian curvature at . The Riemann curvature tensor is given by
To check independence of it suffices to note that it does not change under elementary transformations sending (,) to (,), (,) and ( + ,). That in turn relies on the fact that the operator is skew-adjoint. Skew-adjointness entails that = 0 for all , which follows because
Parallel transport
Given a curve in the Euclidean plane and a vector at the starting point, the vector can be transported along the curve by requiring the moving vector to remain parallel to the original one and of the same length, i.e. it should remain constant along the curve. If the curve is closed, the vector will be unchanged when the starting point is reached again. This is well known not to be possible on a general surface, the sphere being the most familiar case. In fact it is not usually possible to identify simultaneously or "parallelize" all the tangent planes of such a surface: the only parallelizable closed surfaces are those homeomorphic to a torus.
Parallel transport can always be defined along curves on a surface using only the metric on the surface. Thus tangent planes along a curve can be identified using the intrinsic geometry, even when the surface itself is not parallelizable.
The Euler equations for a geodesic c(f) can be written more compactly as
Parallel transport along geodesics, the "straight lines" of the surface, is easy to define. A vector in the tangent plane is transported along a geodesic as the unique vector field with constant length and making a constant angle with the velocity vector of the geodesic.
For a general curve, its geodesic curvature measures how far the curve departs from being a geodesics; it is defined as the rate at which the curve's velocity vector rotates in the surface. In turn the geodesic curvature determines how vectors in the tangent planes along the curve should rotate during parallel transport.
A vector field v(t) along a unit speed curve c(t), with geodesic curvature kg(t), is said to be parallel along the curve if
it has constant length the angle θ(t) that it makes with the velocity vector satisfies
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This yields the previous rule for parallel transport along a geodesic, because in that case kg = 0, so the angle θ(t) should remain constant. The existence of parallel transport follows from standard existence theorems for ordinary differential equations. The above differential equation can be rewritten in terms of the covariant derivative as
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This equation shows once more that parallel transport depends only on the metric structure so is an intrinsic invariant of the surface. Parallel transport can be extended immediately to piecewise C1 curves.
When M is a surface embedded in E3, this last condition can be written in terms of the projection-valued function P as
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or in other words:A fuller discussion is given in the section on embedded surfaces.