taken at all points on the manifold so that defines a smooth function on . Vector fields act (by definition) as differential operators on smooth functions. In local coordinates , the action reads where Einstein's summation convention is used. Formal definition An affine connection is called a Levi-Civita connection if it preserves the metric, i.e., . it is torsion-free, i.e., for any vector fields and we have , where is the Lie bracket of the vector fields and . Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text. Fundamental theorem of (pseudo) Riemannian Geometry Theorem Every pseudo Riemannian manifold has a unique Levi Civita connection . proof: If a Levi-Civita connection exists, it must be unique. To see this, unravel the definition of the action of a connection on tensors to find Hence we can write condition 1 as By the symmetry of the metric tensor we then find: By condition 2, the right hand side is therefore equal to and we find the Koszul formula Hence, if a Levi-Civita connection exists, it must be unique, because is arbitrary, is non degenerate, and the right hand side does not depend on . To prove existence, note that for given vector field and , the right hand side of the Koszul expression is function-linear in the vector field , not just real linear. Hence by the non degeneracy of , the right hand side uniquely defines some new vector field which we suggestively denote as in the left hand side. By substituting the Koszul formula, one now checks that for all vector fields , and all functions Hence the Koszul expression does, in fact, define a connection, and this connection is compatible with the metric and is torsion free, i.e. is a (hence the) Levi-Civita connection. Note that with minor variations the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion. Christoffel symbols Let be an affine connection on the tangent bundle. Choose local coordinates with coordinate basis vector fields and write for . The Christoffel symbols of with respect to these coordinates are defined as The Christoffel symbols conversely define the connection on the coordinate neighbourhood because that is, An affine connection is compatible with a metric iff i.e., if and only if An affine connection