Riemannian Connection On A Surface - Covariant Derivative

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.

Pointwise multiplication by P gives a C∞(M)-module map of C∞(M, E3) onto (M) . The assignment

defines an operator on (M) called the covariant derivative, satisfying the following properties

1. is C∞(M)-linear in X
2. (Leibniz rule for derivation of a module)
3. (compatibility with the metric)
4. (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.

The assignment is uniquely determined by these four conditions and is called the Riemannian connection or Levi-Civita connection.

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.

It existence can be proved directly for a general surface by noting that the four properties imply

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.

Equivalently, in local coordinates (x,y) with basis tangent vectors e₁= and e₂ =, the connection can be expressed purely in terms of the metric using the Christoffel symbols:

If c(t) is a path in M, then the Euler equations for c to be a geodesic can be written more compactly as