Generally speaking, the indicatrix of a Finsler space can be a surface of a rather general form. The Finsler metric tensor induces a Riemannian metric on the indicatrix, converting it into a Riemannian space.
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For each fixed the Finsler metric tensor is Riemannian in the variables. The pair , where the are fixed and the are variable, is called the tangent Riemannian space at a Euclidean space in the case of Riemannian geometry ; the Riemannian curvature tensor of this space reduces to the expression. The indicatrix is a hypersurface that is imbedded in the tangent Riemannian space.
The most immediate example of a Finsler metric function is the -th root of a form of order. Let and be real scalar functions of class satisfying at each point the conditions or 2, and , and let be linearly independent real covariant vector fields of class ,. Then for. The determinant of the Finsler metric tensor is independent of if and only if , where. If a Finsler space is positive definite and the indicatrix is a convex surface, then.
The function is the only known example of a Finsler metric for which not counting the proper Riemannian case. One can select special types of Finsler spaces by postulating some special form of the characteristic Finsler tensors. If the base manifold admits a field of frames globally, and is the metric function of some Minkowski space, then one can introduce a Finsler metric on :.https://specdoncisiba.gq/3924.php
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In this case the Finsler space and the metric are called -form. The functions and are -form when and are constants.
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A Finsler space is called -reducible if it is not Riemannian, if and if the Cartan torsion tensor can be represented in the form. For example, the Lagrange function of an electric test charge in a gravitational or electromagnetic field is a Randers metric. The Finsler metric tensor corresponding to has signature , which makes it of interest in developing a Finslerian generalization of the general theory of relativity; this signature is also encountered in the case of the choice of metric tensors of the form. Such a generalization can be based on the concept of an oscillating Riemannian space to a Finsler space, according to which the Finsler metric tensor associates with each vector field the so-called osculating Riemannian metric tensor.
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Choosing tensor fields depending only on the , from which one constructs the Finsler metric according to , where is a scalar function, one can regard the as genuine gravitational field variables. The Finslerian geometrization of space-time also makes it possible to develop a theory of physical fields with various internal symmetries, relying on the concept of the group of transformations of the tangent vectors that leave the Finsler metric invariant.
Log in. For the mathematician this manifold is named after, see Paul Finsler. Categories : Finsler geometry Smooth manifolds.
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To a certain extent this was correct, but not altogether so, as we shall now indicate. The above-mentioned theories make use of a certain device which basically involves the consideration of a space whose elements are not the points of the underlying manifold, but the line-elements of the later, which form a 2n-1 -dimensional variety. This facilitates the introduction of what Cartan calls the "Euclidean connection", which , by means of certain postulates, may be derived uniquely from the fundamental metric function F x, dx.
The method also depends on the introduction of a so-called "element of support", namely, that at each point a previously assigned direction must be given, which then serves asdirectional argument in all functions depending on direction as well as position. Thus, for instance, the length of a vector and the vector obtained from it by an infinitesimal parallel displacement depend on the arbitrary choice of the element of support.
It is this device which led to the development of Finsler geometry in terms of direct generalisations of the methods of Riemannian geometry. It was felt, however, that the introduction of the element of support was undesirable from a geometrical point of view, while the natural link with the calculus of variations was seriously weakened. This view was expressed independently by several authors, in particular by Vagner , Busemann and the present writer.