By Michel Coste (auth.), J. -D. Boissonnat, J. -P. Laumond (eds.)
Read or Download Geometry and Robotics: Workshop, Toulouse, France, May 26–28, 1988 Proceedings PDF
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Extra resources for Geometry and Robotics: Workshop, Toulouse, France, May 26–28, 1988 Proceedings
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Hicks, Linear perturbations of connections, Mich. Math. J. 12 (1965), 389-397. [KOB-N] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I, II, Interscience Publis hers (1962, 1969). [KOW] O. Kowalski, A classification of Riemannian 3-manifolds with constant principal Ricci curva tures p\ = pi ^ p3, Nagoya Math. J. 132 (1993), 1-36. [L-S-V-W] H. L. Liu, U. Simon, L. Verstraelen and C. P. Wang, The third fundamental hypersurfaces in nonflat space forms, J. of Geometry, to appear.
Corollary. Let M 2 (7) be a closed, orientable surface of genus 7 > 1. p(n) = 1. Proof. Consider the principal curvatures fcj with eigenvectors Ei of S. 1) IC(Ei, Ej) = 1 + hkj. 3) there is a point p € M such that 0 = (detS)(p) = *i(p)fc2(p)fc3(p), thus at least one of the principal curvatures vanishes at p € M. 1) then gives the assertion. 4. Four-dimensional manifolds. 1. Lemma. Let x,x" : M 4 -> S 5 (l) be a polar pair. (i) -Ufa. = - £ « is a polarization invariant; 44 (ii) at p € M , K" = 1 if and only if K = 1.
2, since A = 0 if and only if / " = 0 and since / ' = b = const is the unique solution of (18) for the given initial conditions. 1 we can assume ||T||2 ^ 0. From 0 =/= const = A = nf" we get /'(t) = n\t + c. Because of K = 0 and (8) this surface would be regular. The linear / ' leads to a contradiction in (18). ii \\C\\2 = const is equivalent to r' = 0 or w = 3 / " = const, which in turn is equivalent to K = 0. 3 Singular Tchebychev surfaces Singularity leads to the second order ODE r = \f3-vf (19) for /'.