Basic elements of differential geometry and topology by S.P. Novikov, A.T. Fomenko

By S.P. Novikov, A.T. Fomenko

One provider arithmetic has rendered the 'Et moi, ..., si j'avait su remark en revenir, je n'y serais element aile.' human race. It has positioned logic again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded n- sense'. The sequence is divergent; for that reason we are able to do whatever with it. Eric T. Bell O. Heaviside Matht"natics is a device for suggestion. A hugely worthy device in an international the place either suggestions and non linearities abound. equally, all types of elements of arithmetic seNe as instruments for different elements and for different sciences. using an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One carrier good judgment has rendered com puter technological know-how .. .'; 'One provider class idea has rendered arithmetic .. .'. All arguably actual. And all statements accessible this manner shape a part of the raison d'etre of this sequence.

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61) It means that Chern connection is almost compatible with metric g. 62) An Introduction to Finsler Geometry 34 Proof. 63) and k ufvf - unkVin =5ik-^Fyi. 62). jUas Proof. 53), we obtain the following - F2FyiyJykgks. 60) K% = -vtaF2Fyiyjykgkl. 64) we have K^uas = ysytgtiFyiyjykgkl - F2Fyly]ykgks = -ysFyiyJ F2Fyiyjykgks. 4 We can rewrite the formula for K^uas KijUa Proof. 18) we have 9iJFj = | . 67). 53), we obtain the following + FFyiyjxk)ui + (ysFyiyj + F \yk X \--p5nl{Gksn Fyiyjyrgrs) 1 - Fysxk + Fykxs) — -Ul (Gskn ~ Fykx, + Fy,xk)].

6) Denote the horizontal left of y by y. 10) Covariant Differentiation 41 Proof. 10). 2 smooth function. (4-11) Le£ c(t) 6e a geodesic on (M,F), Then, on (x,y), we have Fdf = — [f{c{i))]t=ou mod ua, and f : SM un where c(0) = y. Proof. 7) y(f) = Jt[f(c{t))}t=o. e. the Hilbert form, by " • ". For example, we have / := f\n and Hija := Hija\n, ect. Note that / : SM —» R is a function and Hija is a three order tensor. 3 Landsberg curvature Let Hiia be the Cartan tensor of {M,F)(ci. 57)) . Put —Hap7 and L := ^^ Laf3

Similarly, set •Ja •=z Va\n = Vet where r\ = ^,ar]aea denote the Cartan form. We call Jawa Landsberg curvature and denote it by J. 3 if J = 0. 4([Shen, 1994]) A Finsler manifold is of weak Landsberg type if and only if all projective spheres in the projective sphere bundle are minimal. Let Vol(x) denote the volume of the unit Finsler sphere ([Bao and Chern, 1996]) {y£TxM\F(x,y) = l}. For functions a1 — ai{x), we have where n I dV := y/det{gij) ^ {-\)k-1ykdy1 A • • • A dyk A • • • A dyn. 5([Bao and Chern, 1996]) manifold.

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