all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients (also known as Christoffel symbols) specifying what happens to components of basis vectors under infinitesimal parallel transport: Despite their appearance, the connection coefficients are not the components of a tensor. Generally speaking, there are independent connection coefficients at each point of spacetime. The connection is called symmetric or torsion-free, if . A symmetric connection has at most unique coefficients. For any curve and two points and on this curve, an affine connection gives rise to a map of vectors in the tangent space at into vectors in the tangent space at : and can be computed component-wise by solving the differential equation where is the vector tangent to the curve at the point . An important affine connection in general relativity is the Levi-Civita connection, which is a symmetric connection obtained from parallel transporting a tangent vector along a curve whilst keeping the inner product of that vector constant along the curve. The resulting connection coefficients (Christoffel symbols) can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection. The covariant derivative Let be a point, a vector located at , and a vector field. The idea of differentiating at along the direction of in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve such that and . The formula for a covariant derivative of along associated with connection turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative. It can be expressed using connection coefficients: The expression in brackets, called a covariant derivative of (with respect to the connection) and denoted by , is more often used in calculations: A covariant derivative of can thus be viewed as a differential operator acting on a vector field sending it to a type tensor (increasing the covariant index by 1) and can be generalised to act on type tensor fields sending them to type tensor fields. Notions of parallel transport can then be defined similarly as for the case of vector fields. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field. In the literature, there are three common methods of denoting covariant differentiation: Many standard