bundle chart, the local frame is orthonormal: It follows that, for every vector , that That is, is skew-symmetric. This is arrived at by explicitly using the bundle metric; without making use of this, and using only the pairing , one can only relate the connection form A on E to its dual A on E, as This follows from the definition of the dual connection as Curvature There are several notations in use for the curvature of a connection, including a modern one using F to denote the field strength tensor, a classical one using R as the curvature tensor, and the classical notation for the Riemann curvature tensor, most of which can be extended naturally to the case of vector bundles. None of these definitions require either a metric tensor, or a bundle metric, and can be defined quite concretely without reference to these. The definitions do, however, require a clear idea of the endomorphisms of E, as described above. Compact style The most compact definition of the curvature F is to define it as the 2-form taking values in , given by the amount by which the connection fails to be exact; that is, as which is an element of or equivalently, To relate this to other common definitions and notations, let be a section on E. Inserting into the above and expanding, one finds or equivalently, dropping the section as a terse definition. Component style In terms of components, let where is the standard one-form coordinate bases on the cotangent bundle T*M. Inserting into the above, and expanding, one obtains (using the summation convention): Keep in mind that for an n-dimensional vector space, each is an n×n matrix, the indices of which have been suppressed, whereas the indices i and j run over 1,...,m, with m being the dimension of the underlying manifold. Both of these indices can be made simultaneously manifest, as shown in the next section. The notation presented here is that which is commonly used in physics; for example, it can be immediately recognizable as the gluon field strength tensor. For the abelian case, n=1, and the vector bundle is one-dimensional; the commutator vanishes, and the above can then be recognized as the electromagnetic tensor in more or less standard physics notation. Relativity style All of the indices can be made explicit by providing a smooth frame , on . A given section