 a tensorial derivative , called covariant derivative is constructed. For each vector field and each tangent vector it provides another unique vector field. On the components of vector fields and linear forms it is defined by
We have adopted the notational convention used by Schouten [300, 310, 389]. Eisenhart and others [121, 234] change the order of indices of the components of the connection:symmetric, this does not make any difference as For both kinds of derivatives we have: 22.
A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special role: With regard to it the connection transforms as a tensor (cf. Section 2.1.5).
For a vector density (cf. Section 2.1.5), the covariant derivative of contains one more term:
A smooth vector field is said to be parallely transported along a parametrised curve with tangent vector if for its components holds along the curve. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point23:
A transformation mapping autoparallels to autoparallels is given by:projective structure on [404, 403].
The particular set of connections.
In Part II of this article, we shall find the set of transformations playing a role in versions of Einstein’s unified field theory.
From the connection further connections may be constructed by adding an arbitrary tensor field to its symmetrised part24:antisymmetric part of the connection, i.e., torsion; it is a tensor field. The trace of the torsion tensor is called torsion vector; it connects to the two traces of the affine connection ; , as
The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
The curvature tensor (22) satisfies two algebraic identities:
From both affine curvature tensors we may form two different tensorial traces each. In the first case , and is called homothetic curvature, while is the first of the two affine generalisations from and of the Ricci tensor in Riemannian geometry. We get2627. In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor (cf., however, ).
For a symmetric affine connection, the preceding results reduce considerably due to . From Equations (29,30,32) we obtain the identities:independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor holds. This equation will be important for the physical interpretation of affine geometry.
In affine geometry, the simplest way to define a fundamental tensor is to set , or . It may be desirable to derive the metric from a Lagrangian; then the simplest scalar density that could be used as such is given by 28.
As a final result in this section, we give the curvature tensor calculated from the connection (cf. Equation (20)), expressed by the curvature tensor of and by the tensor :, p. 141).
From the symmetric part of the first fundamental form , a connection may be constructed, often called after Levi-Civita ,Riemannian curvature tensor defined as in Equation (22) with (cf. Section 2.1.3); is called the Christoffel symbol. Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. In the remaining part of this section we will deal with a symmetric fundamental form only, and denote it by . With the help of the symmetric affine connection, we may define the tensor of non-metricity by29
Then the following identity holds:contorsion tensor , a linear combination of torsion , is defined by30
The inner product of two tangent vectors is not conserved under parallel transport of the vectors along if the non-metricity tensor does not vanish:
A connection for which the non-metricity tensor vanishes, i.e.,metric-compatible31. J. M. Thomas introduced a combination of the terms appearing in and to define a covariant derivative for the metric (, p. 188), . We will have to deal with Equation 