By Hassan Akbar-Zadeh Doctorat d Etat en Mathématiques Pures June 1961 La Sorbonne Paris.

After a quick description of the evolution of pondering on Finslerian geometry ranging from Riemann, Finsler, Berwald and Elie Cartan, the booklet offers a transparent and particular remedy of this geometry. the 1st 3 chapters improve the elemental notions and strategies, brought by way of the writer, to arrive the worldwide difficulties in Finslerian Geometry. the following 5 chapters are self sustaining of one another, and care for between others the geometry of generalized Einstein manifolds, the category of Finslerian manifolds of continuing sectional curvatures. in addition they supply a remedy of isometric, affine, projective and conformal vector fields at the unitary tangent fibre bundle.

Key features

- thought of connections of vectors and instructions at the unitary tangent fibre package deal. - whole record of Bianchi identities for a standard conection of instructions. - Geometry of generalized Einstein manifolds. - category of Finslerian manifolds. - Affine, isometric, conformal and projective vector fields at the unitary tangent fibre package deal. - conception of connections of vectors and instructions at the unitary tangent fibre package deal. - whole checklist of Bianchi identities for a typical conection of instructions. - Geometry of generalized Einstein manifolds. - type of Finslerian manifolds. - Affine, isometric, conformal and projective vector fields at the unitary tangent fibre package deal.

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**Example text**

2) < = {Ci;(y))"1 < C'J(y) + ( C ? ( y ) ) " ^ 7 ( y ) , Ce O(n) We now show that to every Euclidean connection of directions is naturally associated a linear connection. Let us consider the connection defined with respect to a covering of M, endowed with orthonormal frames by the matrices n*, such that The linear connection thus defined is evidently independent of the covering chosen. 4) Vgij= - (coji + coy) = 0 It thus follows that the absolute differential of the metric tensor is zero. 4), it is clear that for a covering of M by the neighborhoods endowed with local sections of E(W, g) the matrices of the linear connection are skew-symmetric; on the other hand if 7i"' (UnV) we have > ClJ (y), where C" is an element of the group O(n); for the linear connection envisaged we have the condition of coherence It thus follows that the matrices (oy{!

1) ds = F(x,dx), F is by definition the arc element. 2) s(x o ,xi)= f F(x, x)du (*=|T) F being restricted of degree 1 the integral of the right hand side is independent of the parametric representation chosen for the path l(xo, Xi). 4) n = | £ 8x' and TTO and n\ correspond to the points xo and x\. We call the extremal of the problem of calculus of variations attached to F(x,x). It is a solution of the differential system of the second order. n). v ' where we have put F2(x, v) = 2L(x, v). n). 7) det(<%L)*0 This problem is called positively regular if <%L is a positive definite quadratic form.

8) by v1 and v* successively; in virtue of the homogeneity of F, we have gij (x, v x ) v V = ^ 8jF2 = 2L = F 2 (x, vx). We are thus led to the following definition : Definition . Let M be a differentiable manifold and V(M) the space of non-zero tangent vectors to M. The structure of a Finslerian manifold on M is defined by the data of a function F(x, Vx), positive, positively homogeneous of degree 1 on V(M) leading to a regular problem of the calculus of variationsL In the following we suppose that F leads to a positively regular problem.