Geometry on Connected Manifolds

The differential geometry on manifolds is considered in an abstract setting without resorting to the concept of a metric. To this end, vectors and covectors are distinguished according to their transformation behavior upon changes of coordinates. Since partial derivatives of vectors and covectors do not transform like tensors, the concept of the covariant derivative of tensors, obeying proper tensor transformation behavior, is motivated. This is achieved by introducing the connection, a third-order non-tensorial object, as one of the most important objects of differential geometry. The covariant derivative also serves to identify what is considered as the parallel transport of tensors. Moreover, the (right) skew symmetric contribution to the connection is denoted the torsion, a third-order tensor that discriminates symmetric from non-symmetric manifolds. Along these lines also the anholonomic object is introduced as a third-order tensor that is related to the concept of the dislocation density in the sequel. Finally, the fourth-order curvature tensor is derived from considering the parallel transport of vectors and covectors along infinitesimal circuits in the manifold. Various aspects of the curvature tensor, in particular the so-called Bianchi identities and the Ricci tensors, are carefully discussed. For the sake of transparency the exposition follows mainly an index notation, however, in order to relate to more modern representations, the main concepts are also given in a coordinate-free invariant formulation and in terms of elements of exterior calculus.