By A. T. Fomenko

ISBN-10: 1904868320

ISBN-13: 9781904868323

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Let 7(i) be an integral curve of G. 1) (xl(t),yl(t)) 2/8W + 2Gi(x(i), 2/(<)) = 0. Let a(t) :— n("f(t)) be the projection of -j(t) under ir. Then the coordinates 51 52 Geodesies (x'(t)) of a(t) satisfy a^t) + 2Gi(a(t),&(t)J = 0. 2) Here we identify a(t) and &(t) = cfl{t)-^r\a(t) with their coordinates (crl(i)) and (**(<))• Conversely, given a curve a =

The curvature forms flj* are defined by Slji:=duji-ujk/\ujki. 29), one obtains d2^ = dwj A w / - wj A duj> = {u,™ A ^ } Ac^ " ^ A {0/+u,/" A < } Since d2u>1 = 0, one obtains the following identity: UJJ A ft/ = 0. The above identity is called the first Bianchi identity, ft/ can be expressed in terms of ujl ALJJ, ujlAwn+j and u>n+% Aojn+:>. 34) R/ki+R/ik=O, i i i Ri kl+Rk iJ+Ri ik = O, (2-35) (2-36) P/u = PkV Let fi* := dun+i - LJn+j A u>. 31), we obtain SV = dun+i - wn+i A w / = d2^* + dy* A w / + y»dw/ - un+j A w/ = [un+i - ymwl} A w/ + 2/ {fi/ + ^ m A < } - W"+^ A w/ = i/*jy.

The Finsler metric F can also be defined in the following way. First, define $* and V* on T*M by $*(*,£):= sup J ^ L , V(O:=£(f*), £erx*M. , F ^^)= SU P J^TY The proof is left to the reader. 2 Let $ = $(x,y) be a Finsler metric on an n-dimensional manifold M and V = V%{x)-g~ be an arbitrary vector field on M with $(x, —Vx) < 1, x e M. 31). Then the Finsler volume forms of F and $ are equal, dVF = dV*. 33) Proof. Fix a basis {b4} for TXM and let Vx := t/bj. ,*) €R n | $ ( x , y i b i ) < l } , UF : = {(i/') G R" |F(x, 2 / i b,)

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A Short Course in Differential Geometry and Topology by A. T. Fomenko


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