Spherical curves and their analogues in affine differential geometry

Kreyszig, Erwin; Pendl, Alois

doi:10.1090/S0002-9939-1975-0365369-0

Spherical curves and their analogues in affine differential geometry
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by Erwin Kreyszig and Alois Pendl PDF

Proc. Amer. Math. Soc. 48 (1975), 423-428 Request permission

Abstract:

Necessary and sufficient conditions for curves, in Euclidean space to be spherical are derived in a fashion which can be generalized to affine differential geometry and analogues of those curves. This also includes a discussion of some geometrical aspects in recent papers by S. Breuer, D. Gottlieb, and Y.-C. Wong.

References

Shlomo Breuer and David Gottlieb, The reduction of linear ordinary differential equations to equations with constant coefficients, J. Math. Anal. Appl. 32 (1970), 62–76. MR 264143, DOI 10.1016/0022-247X(70)90318-5
Shlomo Breuer and David Gottlieb, Explicit characterization of spherical curves, Proc. Amer. Math. Soc. 27 (1971), 126–127. MR 270275, DOI 10.1090/S0002-9939-1971-0270275-2
Erwin Kreyszig, Differential geometry, Mathematical Expositions, No. 11, University of Toronto Press, Toronto, 1959. MR 0108795
E. R. Lorch, A curvature study of convex bodies in Banach spaces, Ann. Mat. Pura Appl. (4) 34 (1953), 105–112. MR 52679, DOI 10.1007/BF02415327
F. Nevanlinna and R. Nevanlinna, Absolute analysis, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 102, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959. MR 0121438

Affine Differentialgeometrie

L. A. Santaló, Quelques propriétés des courbes gauches dans la géométrie différentielle affine, Portugal. Math. 3 (1942), 63–68 (French). MR 5679
Yung-chow Wong, A global formulation of the condition for a curve to lie in a sphere, Monatsh. Math. 67 (1963), 363–365. MR 155237, DOI 10.1007/BF01299587
Yung Chow Wong, On an explicit characterization of spherical curves, Proc. Amer. Math. Soc. 34 (1972), 239–242. MR 295224, DOI 10.1090/S0002-9939-1972-0295224-3