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Topics in Direct Differential Geometry

Published online by Cambridge University Press:  20 November 2018

Ralph Park
Ralph Park*
Affiliation:
University of Calgary, Calgary, Alberta
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In the theory of curves, one often makes differentiability assumptions in order that analytic methods can be used. Then one tries to weaken these assumptions as much as possible. The theory of curves which is presented here uses geometric methods, such as central projection, rather than analysis. In this way, no analytic assumptions are needed and a purely geometric theory results. Since this theory is not so well known as the analytic one, I have tried to make the treatment as self-contained as possible. It is hoped that this paper will form a quick introduction for a reader who has had no previous acquaintance with the subject.

We assume that our curves satisfy a condition, which we call direct differentiability. Roughly this condition is that, at each point of the curve, all the osculating spaces exist.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 98 - 148
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Barner, M., Über die Mindestzahl stationärer Schmiegebenen bei geschlossenen streng-konvexen Raumkurven, Abh. Math. Sem. Univ. Hamburg 20 (1956), 196–215.Google Scholar
2. Barner, M. and Flohr, F., Der Vierscheitelsatz und seine Verallgemeinerungen, Der. Math. Unterr. (1958), 43-73.Google Scholar
3. Derry, D., The duality theorem for curves of order n in n-space, Can. J. Math. 3 (1951), 159–163.Google Scholar
4. Derry, D., On closed differentiable curves of order n in n-space, Pacific J. Math. 5 (1955), 675–686.Google Scholar
5. Haupt, O., Streng-konvexe Bogen und Kurven in der direkten Infinitesimalgeometrie, Arch. Math. 9 (1958), 110–116.Google Scholar
6. Haupt, O., Über einige Grundeigenschaften der Bogen ohne (n — 2, k)-Sekanten im projektiven Pn(n ≦ k). Math. Ann. 139 (1959), 151–170.Google Scholar
7. Haupt, O. and Künneth, H., Geometrische Ordnungen (Springer-Verlag, Berlin-Heidelberg, 1967).Google Scholar
8. Rudin, W., Principles of Mathematical Analysis (McGraw-Hill, New York, 1953).Google Scholar
9. Scherk, P., Über differenzierbare Kurven und Bögen I. Zum Begriff der Charakteristik, Časopis pěst. mat. a fys. 66 (1937), 165–171.Google Scholar
10. Scherk, P., Über differenzierbare Kurven und Bogen II. Elementarbogen und Kurve n-ter Ordnung im Rn, Časopis pěst. mat. a fys. 66 (1937), 172–191.Google Scholar
11. Scherk, P., Dually differentiable points on plane arcs, Trans. Roy. Soc. Canada 48 (1954), 43–48.Google Scholar
12. Scherk, P., Elementary points on plane arcs, Trans. Roy. Soc. Canada 48 (1954), 49–53.Google Scholar
13. Turgeon, J., On the rank numbers of an arc, Ph.D. Thesis, University of Toronto, Toronto, 1968.Google Scholar