page_title: Riemannian connection on a surface
section_title: Covariant derivative
hierarchical_section_title: Riemannian connection on a surface / Covariant derivative
caption_attribution_description: English: Combing a hairy doughnut flat. (I, anonymous author of this work, claim no author's rights.)
context_page_description: 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.
context_section_description: 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 nₙ at p, uniquely defined up to a sign, and v in E³, the projection is given by P(p)(v) = v - (v · nₚ) nₚ.
  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
   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.
  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.
  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 u an isometric embedding of M 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 X, Y and Z set to 's; in that case all the Lie brackets commute.
caption: A vector field on the torus
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