an inner product on each vector space fiber of E. The bundle metric allows one to define an orthonormal coordinate frame by the equation Given a vector bundle, it is always possible to define a bundle metric on it. Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity. However, to define the Hodge star, the Laplacian, the first Bianchi identity, and the Yang–Mills functional, one needs the bundle metric. The Hodge star additionally needs a choice of orientation, and produces the Hodge dual of its argument. Connection form Given a local bundle chart, the covariant derivative can be written in the form where A is the connection one-form. A bit of notational machinery is in order. Let denote the space of differentiable sections on E, let denote the space of p-forms on M, and let be the endomorphisms on E. The covariant derivative, as defined here, is a map One may express the connection form in terms of the connection coefficients as The point of the notation is to distinguish the indices j, k, which run over the n dimensions of the fiber, from the index i, which runs over the m-dimensional base space. For the case of a Riemann connection below, the vector space E is taken to be the tangent bundle TM, and . The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory. In mathematics, the notation is often used in place of A, as in the article on the connection form; unfortunately, the use of for the connection form collides with the use of to denote a generic alternating form on the vector bundle. Skew symmetry The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field , the matrix is skew-symmetric; equivalently, it is an element of the Lie algebra . This can be seen as follows. Let the fiber be n-dimensional, so that the bundle E can be given an orthonormal local frame with . One then has, by definition, that , so that: In addition, for each point of the