Curvature Computations in Finsler Geometry Using a Distinguished Class of Anisotropic Connections
- PDF / 493,290 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 188 Views
Curvature Computations in Finsler Geometry Using a Distinguished Class of Anisotropic Connections ´ Miguel Angel Javaloyes Abstract. We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections, and we find a family of anisotropic connections which are well suited to study the geometry of Finsler metrics. Mathematics Subject Classification. Primary 53C50, 53C60. Keywords. Anisotropic linear connections, Finsler geometry, Jacobi operator, Bianchi identities.
1. Introduction Traditionally, Finsler geometry is associated with lengthy computations in coordinates. This is due to the dependence on directions of all the elements, which allows for a large generality of the metrics, but sometimes makes it difficult to understand the geometric meaning of certain quantities. In order to overcome these difficulties, we will use affine connections ∇V , which are defined for every vector field V which is non-zero everywhere. The connections ∇V can be interpreted as osculating affine connections in the same way as one obtains the osculating metric gV of a Finsler metric by fixing at every point p ∈ M the direction of Vp in the fundamental tensor, namely (gV )p = gVp , where g is the fundamental tensor in (44). This approach was first considered in [9,11], later in [12, Sect. 7] and recently in [2,3,5]. Here we will go a step further. First, we consider anisotropic connections in a manifold M , which are not exactly connections on fiber bundles, but especially adapted to the dependence on the direction (see Definition 0123456789().: V,-vol
123
Page 2 of 21
´ Javaloyes M. A.
MJOM
2.2 and [4, Sect. 4.4] for the relationship with connections on the vertical bundle). Then we will use the anisotropic tensor calculus developed in [4], and the formulas (6), (9) and (13), wherein the derivative of a tensor and the curvature are computed using ∇V . To take advantage of this approach, we make a fundamental observation in Proposition 2.13: that there is a privileged choice of the extension V which allows one to compute the derivative of a tensor. This choice has the property that at a fixed point p, the vector field V is parallel in all directions, namely, (∇VX V )p = 0 for all vector fields X. This simplifies dramatically the computations involving curvature tensors and derivatives. Indeed, it reduces, for example, the proof of the Bianchi identities to the classical case of an affine connection in a manifold, Sect. 2.4. It also allows us to relate the curvature tensors of two different anisotropic connections using the difference tensor, Sect. 2.5. In particular, this relation will lead us to distinguish a family of connections which are well suited for studying Finsler metrics, Sect. 3. Amongst these
Data Loading...
