Abstract
Differential geometry is a field in which geometry is expressed in analysis, algebra, and calculations, and in which analysis and calculations are sometimes understood in intuitive steps that could be called geometric.
The author acknowledges with gratitude that part of this paper was written while he was a guest of IMPA in Rio de Janeiro in July 1979.
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References
T. F. Banchoff, Tightly embedded 2-dimensional polyhedral manifolds. Amer. J. Math. 87, 462 - 472 (1965).
T. F. Banchoff, Critical points and curvature for embedded polyhedra. J. Differential Geometry(1967).
T. F. Banchoff, The spherical two-piece property and tight surfaces in spheres. J. Differential Geometry 4, 193 - 205 (1970).
T. F. Banchoff, The two-piece property and tight n-manifolds-with-boundary in E“. Trans. Amer. Math. Soc 161, 233 - 267 (1971).
T. F. Banchoff, Tight polyhedral Klein bottles, projective planes and Moebius bands. Math. Ann. 207, 233 - 243 (1974).
K. Borsuk, Sur la courbure totale des courbes. Ann, de la Soc. Math. Pol. 20, 251 - 265 (1947).
S. Carter and A. West, Tight and taut immersions. Proc. London Math. Soc. 25701 - 720 (1972).
T. Cecil, Taut immersions of noncompact surfaces into a euclidean 3-space. J. Differential Geometry 11, 451 - 459 (1976).
T. E. Cecil and P. J. Ryan, Focal sets, taut embeddings and the cyclides of Dupin. Math. Ann. 236(1978), 177 - 190.
C. S. Chen and W. F. Pohl, On the classification of tight surfaces in a euclidean 4-space. Preprint.
C. S. Chen, On the tight isometric immersions of codimension two. Amer. J. Math. 94, 974 - 990 (1972).
S. S. Chern, La geometrie des sous-variétés d’un espace euclidien à plusieurs dimensions. Enseignement Math. 40, 5. 12 (1955).
S. S. Chern and R. K. Lashof, On the total curvature of immersed manifolds I, II. Amer. J. Math. 79, 306–313 (1957);
S. S. Chern and R. K. Lashof, Mich. Math. J. 5, 5. 12 (1958).
C. L. Chevalley, Theory of Lie groups. Princeton U. P., (1946).
R. D. Edwards, The topology of manifolds and cell-like maps. Preprint I.H.E.S., 1979, 28 pp.
J. Eells and N. H. Kuiper, Manifolds which are like projective planes, Publ. Math. LH.E.S. 14128 - 222 (1962).
S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology. Princeton U. P., 1952.
I. Fary, Sur la courbure totale d’une courbe gauche faisant un noeud. Bull. Soc. Math. France 77, 128 - 138 (1949).
W. Fenchel, Uber die Krummung und Windung geschlossener Raumkurven. Math. Ann. 101, 238–252 (1929).
D. Ferus, Uber die absolute Totalkrümmung höher-dimensionaler Knoten. Math. Ann. 171, 81 - 86 (1971).
D. Ferus, Totale Absolutkrúmmung in Differentialgeometrie und Topologie. Lecture Notes in Mathematics No. 66, Springer, Berlin, Heidelberg, New York, 1968.
D. Ferus, Symmetric submanifolds of euclidean space. preprint, 1979.
R. H. Fox, On the total curvature of some tame knots. Ann. Math. 52258 - 261 (1950).
H. Freudenthal, Zur ebenen Oktavengeometrie. Proc. Akad. Amsterdam A 56
H. Freudenthal, Indag. Math. 15, 195 - 200 (1953).
H. Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie. Preprint, Math. Inst. Univ. Utrecht, 1951.
P. Haupt and H. Künneth, Geometrische Ordnungen, Springer, Berlin, 1967.
Guy Hirsch, “La Géométrie projective et la topologie des espaces fibrés. In Topologie Algébrique. Colloque Int. CNRS 12, Paris, 1949.
Hurwitz, Uber die Komposition der quadratischen Formen. In Math. Werke II, p. 641; Math. Ann. 88 1–25 (1923).
S. Kobayashi, Imbeddings of homogenous spaces with minimum total curvature, Tohoku Math. J. 1963 - 70 (1967).
S. Kobayashi, Isometric imbeddings of compact spaces. Tohoku Math. J. 2021 - 25 (1968).
S. Kobayashi, and M. Takeuchi, Minimal imbeddings of R-spaces, J. Differential Geometry, 2, 230–215 (1968).
S. Kobayashi, and K. Nomizu, Foundations of Differential Geometry II. 1969, note 21, p. 361.
A. Kosinski, A theorem on Families of Acyclic Sets and its Applications. Pacific J. Math 12, 317 - 325 (1962).
W. Kühnel, Total curvature of manifolds with boundary in E“. J. London Math Soc. (2) 15, 173 - 182 (1977).
W. Kühnel, Tight and 0-tight polyhedral embeddings of surfaces. Invent. Math. 58, 161 - 177 (1980).
N. H. Kuiper, Immersions with minimal total absolute curvature. In Coll. de Géométrie Diff, Centre Belge de Recherches Math., Bruxelles, 1958, pp. 75–88.
N. H. Kuiper, Sur les immersions a courbure totale minimale, In Seminaire de Topologie et Géométrie Différentielle Dirigé par Ch. Ehresmann, Faculte des Sciences, Paris, 1959, pp. 1–5.
N. H. Kuiper, La courbure d’indice ket les applications convexes. In Séminaire de Topologie et Géométrie Différentielle Dirigé par Ch. Ehresmann, Faculté des Sciences, Paris, 1960, pp. 1–15.
N. H. Kuiper, On surfaces in euclidean three space. Bull Soc. Math. Belg. 12, 5 - 12 (1960).
N. H. Kuiper, Convex immersions of closed surfaces in E3, Comm. Math. Helv. 35, 85 - 92 (1961).
N. H. Kuiper, On convex maps. Nieuw Archief voor Wisk 10, 147 - 164 (1962).
N. H. Kuiper, Der Satz von Gauss Bonnet fir Abbildungen im E N . Jahr. Ber. DMV. 69, 77 - 88 (1967).
N. H. Kuiper, Minimal total absolute curvature for immersions. Invent. Math. 10, 209 - 238 (1970).
N. H. Kuiper, Morse relations for curvature and tightness. In Proc. Liverpool Singularities Symp. II (C. T. C Wall, Ed.) Springer Lecture Notes in Mathematics 209, 1971, pp. 77–89.
N. H. Kuiper, Tight topological embeddings of the moebius band. J. Diff. Geometry 6, 271 - 283 (1972).
N. H. Kuiper, Stable surfaces in euclidean three space. Math. Scand. 36, 83 - 96 (1975).
N. H. Kuiper, and W. F. Pohl, Tight topological embeddings of the real projective plane in E5. Invent. Math. 42, 177 - 199 (1977).
N. H. Kuiper, Curvature measures for surfaces in E N. Lobachevski Colloquium. Kazan, USSR (1976).
R. C. Lacher, Cell-like maprings and their generalizations. Bull. AMS 83, 495 - 552 (1977).
R. Langevin and H. Rosenberg, On curvature integrals and knots. Topology 15, 405 - 416 (1976).
W. Lastufka, on TPP immersions. Thesis, Univ. of Minnesota, 1979.
J. A. Little and W. F. Pohl, On tight immersions of maximal codimension. Invent. Math. 13, 179 - 204 (1971).
B. Mazur, A note on some contractible 4-maniflods. Ann. of Math. (2) 73, 221–228 (1961).
W. Meeks, Lectures on Plateau’s problem(July 1978), Escola de Geometria Differencial DO CEARA Brasil, 1979, 53 pp.
J. W. Milnor, On the total curvature of knots. Ann. of Math. 52(1950).
J. W. Milnor, On the total curvature of closed space curves. Math. Scand. 1, 289 - 296 (1953).
J. W. Milnor, Morse Theory, Ann. Math. Stud. 51, Princeton, 1963.
J. D. Moore, Codimension two submanifolds of positive curvature. Proc. AMS 70, 72–78 (1978).
H. R. Morton, A criterion for an embedded surface in R 3 to be unkotted. preprint, Liverpool, 1976; in Proc. Conf. on Low Dimensional Topology, Sussex, 1977, to appear.
V. Poenaru, Les Décompositions de l’hypercube en produit topologique. Bull. Soc. Math. France113–129 (1960).
M. Retberg, Komplexe Mannigfaltigkeiten in einen euklidischen Raum. Diplomarbeit, Univ. Bielefeld, 1978.
G. Ringel, Map colour theorem. In Grundlehre der Math. Wiss., Band 209, Springer 1974.
I. L. Rodriguez, The two piece property and convexity for surfaces with boundary. J. Diff. Geometry 11, 235 - 250 (1976).
L. C. Siebenmann, Approximating cellular maps by homeomorphisms. Topology II, 271 - 294 (1973).
E. Spanier, Algebraic Topology. McGraw-Hill, 1966.
S. S. Tai, On the minimum imbeddings of compact symmetric spaces of rank one. J. Diff. Geom. 2, 55 - 66 (1968).
Eberhard Teufel, Total krummung und totale Absolutkriimmung in der spharischen Differentialgeometrie und Differentialtopologie. Thesis, Stuttgart, 1979.
A. Weinstein, Positively curved n-manifolds in Rn+2. J. Diff. Geom. 4, l-4 (1970).
T. J. Willmore, Tight immersions and total absolute curvature. Bull. London Math. Soc. 3, 129 - 151 (1971).
J. P. Wilson, The total absolute curvature of immersed manifolds. J. London Math. Soc. 40, 362 - 366 (1966).
J. P. Wilson, Some minimal imbeddings of homogeneous spaces. J. London Math. Soc. 335–340 (1969).
P. Wintgen, Zur Integralkrimmung verknoteter Sphären, Thesis, Humboldt-Universität, Berlin, 1976.
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Kuiper, N.H. (1980). Tight Embeddings and Maps. Submanifolds of Geometrical Class Three in E N . In: Hsiang, WY., Kobayashi, S., Singer, I.M., Wolf, J., Wu, HH., Weinstein, A. (eds) The Chern Symposium 1979. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8109-9_6
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