Ivancevic_Applied-Diff-Geom

284 Applied Differential Geometry: A Modern Introductionvariational formula applied to these fields then inducesn−1∑i=1n−1∑¨L(0) ==i=1∫ l0∫ l{ (π0 l{(n − 1)) ( 2· cos2t · π )l( πl) ( 2· cos2t · π )l( − sec ( ˙γ, E i ) sin 2 t · π ) } dtl(− Ric ( ˙γ, ˙γ) sin 2 t · π ) } dt.lTherefore, if Ric( ˙γ, ˙γ) ≥ (n − 1) k 2 (this is the Ricci curvature of Sk n ), thenn−1∑i=1¨L(0) ≤ (n − 1)∫ l0= − (n − 1) 1 2l{ (πl) ( 2· cos2t · π )l(l 2 k 2 − π 2) ,( − k 2 sin 2 t · π ) } dtlwhich is negative when l > π · k −1 (the diameter of Sk n ). Thus at least oneof the contributions d2 L ids(0) must be negative as well, implying that the2geodesic cannot be a segment in this situation.3.10.2.2 Gauss–Bonnet FormulaIn 1926 Hopf proved that in fact there is a Gauss–Bonnet formula forall even–dimensional hypersurfaces H 2n ⊂ R 2n+1 . The idea is that thedeterminant of the differential of the Gauss map G : H 2n → S 2n is theGaussian curvature of the hypersurface. Moreover, this is an intrinsicallycomputable quantity. If we integrate this over the hypersurface, we get,1Vol S∫H2n det (DG) = deg (G) ,where deg (G) is the Brouwer degree of the Gauss map. Note that thiscan also be done for odd–dimensional surfaces, in particular curves, butin this case the degree of the Gauss map will depend on the embeddingor immersion of the hypersurface. Instead one gets the so–called windingnumber. Hopf then showed, as Dyck had earlier done for surfaces, thatdeg (G) is always half the Euler characteristic of H, thus yielding2Vol S∫H2n det (DG) = χ (H) . (3.135)Since the l.h.s of this formula is in fact intrinsic, it is natural to conjecturethat such a formula should hold for all manifolds.