Summary
Generalizations of principle axes are found for surfaces in E4. The singularities generalize umbilics. The generic indicies are computed. For these computations the Thom Transversality Theorem as applied by Feldman to geometry is used. Hower we « reduce the group » rendering the calculations more tractible. Also we show that a torus or sphere cannot be immersed in E4 with everywhere nonzero curvature of the normal bundle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. F. Baker,Principle of Geometry VI, Fredrick Ungar Publishing Company, New York, 1960.
H. Boerner,Representations of Groups, North Holland Publishing Company, Amsterdam, 1963, see pages 127–130.
S. S. Chern,On a theorem of algebra and its geometrical application, Journal of the Indian Mathematical Society, 8 (1944), 29–36.
S. S. Chern andE. Spanier,A theorem on orientable surfaces in four dimensional space, Commentarii Mathematici Helvetici, 25 (1951), 205–209.
L. P. Eisenhart,Minimal surfaces in Euclidean four space, American Journal of Mathematics, 34 (1912), 215–236.
E. A. Feldman,Geometry of immersions I, Transaction of the American Mathematical Society, 120 (1965), 185–224.
—— ——,Geometry of immersions II, Transaction of the American Mathematical Society, 125 (1966), 181–315.
—— ——,On parabolic and umbilic points of immersed hypersurfaces, Transactions of the American Mathematical Society, 127 (1967), 1–28.
K. Kommerell,Riemannsche Flächen in ebenen Raum von vier Dimensionen, Math. Ann., 60 (1905), 546–596.
E. P. Lane,Projective Differential Geometry of Curves and Surfaces, University of Chicago Press, Chicago, 1932, see page 274.
R. Lashof andS. Smale,On the immersion of manifolds in Euclidean space, Annals of Mathematics, 69 (1958), 562–582.
R. S. Palais,A global formulation of the Lie theory of transformation groups, American Mathematical Society, Memoirs, No. 22, 1957, page 35.
W. Pohl,Some integral formulas for space curves and their generalization, to appear in American Journal of Mathematics.
J. G. Semple andL. Roth,Introduction to Algebraic Geometry, Oxford University Press, Oxford, 1944, see pages 128–136.
H. Whitney,The self-intersections of a smooth n-manifold in 2n-space, Annals of Mathematics, 45 (1944), 220–246.
J. H. White,Self-linking and the Gauss integral in higher dimensions, Thesis, University of Minnesota, (1968), mimeographed.
C. L. E. Moore andE. B. Wilson,Differential geometry of two-dimensional surfaces in hyperspaces, Proceedings of the American Academy of Arts and Sciences, 52 (1916), 267–368.
Y. C. Wong,A new curvature theory for surfaces in a Euclidean 4-space, Commentarii Mathematii Helvetici, 26 (1952).
—— ——,Contributions to the theory of surfaces in a 4-space of constant curvature, Transactions of the American Mathematical Society, 59 (1946), 467–507.
—— ——,Fields of isocline tangent along a curve in a Euclidean 4-space, Tohoku Mathematical Journal, 3 (1951), 322–329.
Additional information
Entrata in Redazione il 19 novembre 1968.
Rights and permissions
About this article
Cite this article
Little, J.A. On singularities of submanifolds of higher dimensional Euclidean spaces. Annali di Matematica 83, 261–335 (1969). https://doi.org/10.1007/BF02411172
Issue Date:
DOI: https://doi.org/10.1007/BF02411172