By H. Busemann (auth.), E. Bompiani (eds.)
H. Busemann: the unreal method of Finsler areas within the large.- E.T. Davies: Vedute generali sugli spazi variazionali.- D. Laugwitz: Geometrical tools within the differential geometry of Finsler spaces.- V.V. Wagner: Geometria del calcolo delle variazioni.
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Extra info for Geometria del calcolo delle variazioni
7) theorem. q 1, then the me- tric is Minkowskian. The proofs of all these theorems are too long to be given here. 5) for spaces of the elliptic type which is easy to prove. Convexity of spheres must be rephrased. ::J are quadrics. 7) Theorem. Let R be a space of the 8111pt10 type with geodesios ot length 2 J and dimension at least 3 • It the spheres X(p. f g <. ~ ) with han order 2 then the space 1s elllptlc. J Order 2 means, of course, that a geodes10 does not intersect the sphere more than twice.
Thus G(a ', b') passes + sb = ab • through s • It 0 is any other point of L then}unless c = s, let c lie on the same side of Gas b and say (scb) . Then ab = a lb', ac = ab - ac give b'c' so that LRG = G(a' Ib' ) = alb ' - a ' c' ac = a'c' and = be. Therefore also c'€ G(a',b l ) and the map of L on L' isometric. If H (I G = B and H = B(p,q) then H' = HRG = B( p' ,q' )( P1gu- re 7) . To prove this we ohoose t so close to son H that G(q,t) and G(p,t) intersect G. 3) RG = Yh, moreover . Prom here the proof proceeds as tallows : we want to show that RG also maps a line H whioh does not intersect G isometricallyon a line H'.
A supporting line of K(p, J' ). The point x was arbitrary on Z and the lines G(z,x) are all parallel (z6K(:p». l1e set Z I Blasch- kels Theorem yields now that X(p'J' ) is an ellipsoid. If we make differentiability hypothesis we can conclude that the "1nt1nltesimal" spheres are ellipsoids, and that the metric is Riemannian. Beltramits Theorem shows then that the metric has constant curvature and is therefore elliptic. The same can be shown without differentiability hypotheses by using projective geo- metry.
Geometria del calcolo delle variazioni by H. Busemann (auth.), E. Bompiani (eds.)