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The Basic Equations of a Submanifold

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Book cover Submanifold Theory

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Abstract

In this chapter we establish several basic facts of the theory of submanifolds that will be used throughout the book. We first introduce the second fundamental form and normal connection of an isometric immersion by means of the Gauss and Weingarten formulas. Then we derive their compatibility conditions, namely, the Gauss, Codazzi and Ricci equations. The main result of the chapter is the Fundamental theorem of submanifolds, which asserts that these data are sufficient to determine uniquely a submanifold of a Riemannian manifold with constant sectional curvature, up to isometries of the ambient space. As an application, we classify totally geodesic and umbilical submanifolds of space forms. We introduce the relative nullity distribution as well as the notion of principal normal vector fields of an isometric immersion, and derive some of their elementary properties. Submanifolds with flat normal bundle are briefly discussed.

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References

  1. Alexander, S.: Locally convex hypersurfaces of negatively curved spaces. Proc. Am. Math. Soc. 64, 321–325 (1977)

    Article  MathSciNet  Google Scholar 

  2. Alexander, S., Currier, R.: Non-negatively curved ends of Euclidean hypersurfaces. Geom. Dedicata 40, 29–43 (1991)

    Article  MathSciNet  Google Scholar 

  3. Allendoerfer, C.: The imbedding of Riemann spaces in the large. Duke Math. J. 3, 317–333 (1937)

    Article  MathSciNet  Google Scholar 

  4. Aminov, J.: Isometric immersions of n-dimensional Lobachevsky space in (2n − 1)-dimensional Euclidean space. Math. URSS Sbornik 39, 359–386 (1981)

    Article  Google Scholar 

  5. Andrade, R., Lima, R.: Convexity, rigidity and reduction of codimension of isometric immersions into space forms. Bull. Braz. Math. Soc. 50, 119–136 (2019)

    Article  MathSciNet  Google Scholar 

  6. Baldin, Y., Mercuri, F.: Isometric immersions in codimension two with nonnegative curvature. Math. Z. 173, 111–117 (1980)

    Article  MathSciNet  Google Scholar 

  7. Baldin, Y., Mercuri, F.: Codimension two nonorientable submanifolds with nonnegative curvature. Proc. Am. Math. Soc. 103, 918–920 (1988)

    Article  MathSciNet  Google Scholar 

  8. Berndt, J., Console, S., Olmos, C.: Submanifolds and Holonomy, 2nd edn. Research Notes in Mathematics. Chapman and Hall/CRC, Boca Raton (2016)

    Google Scholar 

  9. Bonini, V., Ma, S., Qing, J.: On nonnegatively curved hypersurfaces in \(\mathbb {H}^{n+1}\). Math. Ann. 372, 1103–1120 (2018)

    Article  MathSciNet  Google Scholar 

  10. Cartan, E.: Leçons sur la Géométrie des Espaces de Riemann, Gauthier-Villars, Paris (1951)

    MATH  Google Scholar 

  11. Cecil, T., Ryan, P.: Focal set of submanifolds. Pac. J. Math. 78, 27–39 (1978)

    Article  MathSciNet  Google Scholar 

  12. Cheng, S., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)

    Article  MathSciNet  Google Scholar 

  13. Dajczer, M., Florit, L., Tojeiro, R.: On a class of submanifolds carrying an extrinsic totally umbilical foliation. Israel J. Math. 125, 203–220 (2001)

    Article  MathSciNet  Google Scholar 

  14. Dajczer, M., Florit, L., Tojeiro, R.: Reducibility of Dupin submanifolds. Ill. J. Math. 49, 759–791 (2005)

    Article  MathSciNet  Google Scholar 

  15. Dajczer, M., Florit, L., Tojeiro, R.: The vectorial Ribaucour transformation for submanifolds and applications. Trans. Am. Math. Soc. 359, 4977–4997 (2007)

    Article  MathSciNet  Google Scholar 

  16. do Carmo, M., Lima, E.: Isometric immersions with semi-definite second quadratic forms. Arch. Math. 20, 173–175 (1969)

    Article  MathSciNet  Google Scholar 

  17. do Carmo, M., Warner, F.: Rigidity and convexity of hypersurfaces in spheres. J. Differ. Geom. 4, 133–144 (1970)

    Google Scholar 

  18. Eisenhart, L.P.: Riemannian Geometry. Princeton University Press, Princeton (1964)

    MATH  Google Scholar 

  19. Fabricius-Bjerre, Fr., Sur les variétés à torsion nulle. Acta Math. 66, 49–77 (1936)

    Article  MathSciNet  Google Scholar 

  20. Ferapontov, E.: Surfaces with flat normal bundle: an explicit construction. Differ. Geom. Appl. 14, 15–37 (2001)

    Article  MathSciNet  Google Scholar 

  21. Fialkow, A.: Hypersurfaces of a space of constant curvature. Ann. Math. 39, 762–785 (1938)

    Article  MathSciNet  Google Scholar 

  22. Florit, L., Ziller, W.: Nonnegatively curved Euclidean submanifolds in codimension two. Comment. Math. Helv. 91, 629–651 (2016)

    Article  MathSciNet  Google Scholar 

  23. Gromov, M.: Partial Differential Relations. Springer, Berlin (1986)

    Book  Google Scholar 

  24. Gromov, M., Rokhlin, V.: Embeddings and immersions in Riemannian geometry. Russ. Math. Surv. 25, 1–57 (1970)

    Article  Google Scholar 

  25. Han, Q., Hong, J-X.: Isometric Embedding of Riemannian Manifolds in Euclidean Spaces. Mathematical Surveys and Monographs, vol. 130. American Mathematical Society, Providence (2006)

    Google Scholar 

  26. Ivanova-Karatopraklieva, I., Markov, P., Sabitov, I.: Bending of surfaces III. J. Math. Sci. 149, 861–895 (2008)

    Article  MathSciNet  Google Scholar 

  27. Jacobowitz, H.: The Gauss-Codazzi equations. Tensor 39, 15–22 (1982)

    MathSciNet  MATH  Google Scholar 

  28. Leung, D., Nomizu, K.: The axiom of spheres in Riemannian geometry. J. Differ. Geom. 5, 487–489 (1971

    Article  MathSciNet  Google Scholar 

  29. Lira, J., Tojeiro, R., Vitório, F.: A Bonnet theorem for isometric immersions into products of space forms. Arch. Math. 95, 469–479 (2010)

    Article  MathSciNet  Google Scholar 

  30. Mendonça, B., Tojeiro, R.: Umbilical submanifolds of \(\mathbb {S}^n\times \mathbb {R}\). Can. J. Math. 66, 400–428 (2014)

    Article  MathSciNet  Google Scholar 

  31. Menninga, N.: Immersions of positively curved manifolds into manifolds with curvature bounded above. Trans. Am. Math. Soc. 318, 809–821 (1990)

    Article  MathSciNet  Google Scholar 

  32. Milnor, J.: Morse Theory. Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton (1963)

    Google Scholar 

  33. Moore, J.D.: Conformally flat submanifolds of Euclidean space. Math. Ann. 225, 89–97 (1977)

    Article  MathSciNet  Google Scholar 

  34. Moore, J.D.: Codimension two submanifolds of positive curvature. Proc. Am. Math. Soc. 70, 72–74 (1978)

    Article  MathSciNet  Google Scholar 

  35. Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63, 20–63 (1956)

    Article  MathSciNet  Google Scholar 

  36. Nomizu, K.: Characteristic roots and vectors of a differentiable family of symmetric matrices. Linear Multilinear Algebra 1, 159–162 (1973)

    Article  MathSciNet  Google Scholar 

  37. Nomizu, K., Yano, K.: On circles and spheres in Riemannian geometry. Math. Ann. 210, 163–170 (1974)

    Article  MathSciNet  Google Scholar 

  38. O’Neill, B.: Semi-Riemannian Geometry With Applications to Relativity. Pure and Applied Mathematics, vol. 103. Academic, New York (1983)

    Google Scholar 

  39. Orjuela, J., Tojeiro, R.: Umbilical surfaces of products of space forms. Tôhoku Math. J. 68, 471–486 (2016)

    Article  MathSciNet  Google Scholar 

  40. Otsuki, T.: On principal normal vector fields of submanifolds in a Riemannian manifold of constant curvature. J. Math. Jpn. 22, 35–46 (1970)

    Article  MathSciNet  Google Scholar 

  41. Pawel, K., Reckziegel, H.: Affine submanifolds and the theorem of Cartan-Ambrose-Hicks. Kodai Math. J. 25, 341–356 (2002)

    Article  MathSciNet  Google Scholar 

  42. Pawel, K., Reckziegel, H.: On the existence of spherically bent submanifolds, an analogue of a theorem of E. Cartan. Kodai Math. J. 26, 199–220 (2003)

    Article  MathSciNet  Google Scholar 

  43. Perepelkin, D.: Sur les variétés parallèles dans un espace euclidean (ou Riemannien). C. R. (Dokl.) Acad. Sci. URSS 1935, 593–598 (1935)

    MATH  Google Scholar 

  44. Piccione, P., Tausk, D.: An existence theorem for G-structure preserving affine immersions. Indiana Univ. Math. J. 57, 1431–1465 (2008)

    Article  MathSciNet  Google Scholar 

  45. Reckziegel, H.: Krümmungsflächen von isometrischen Immersionen in Räume konstante Krümmung. Math. Ann. 223, 169–181 (1976)

    Article  MathSciNet  Google Scholar 

  46. Ros, A.: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J. Differ. Geom. 27, 215–220 (1988)

    Article  MathSciNet  Google Scholar 

  47. Ryan, P.: Homogeneity and some curvature conditions for hypersurfaces. Tôhoku Math. J. 21, 363–388 (1969)

    Article  MathSciNet  Google Scholar 

  48. Sacksteder, R.: On hypersurfaces with no negative sectional curvature. Am. J. Math. 82, 609–630 (1960)

    Article  MathSciNet  Google Scholar 

  49. Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish Inc., Houston (1979)

    MATH  Google Scholar 

  50. Szczarba, R.: On isometric immersions of Riemannian manifolds in Euclidean space. Bol. Soc. Brasil. Mat. 1, 31–45 (1970)

    Article  MathSciNet  Google Scholar 

  51. Tenenblat, K.: On isometric immersions of Riemannian manifolds. Bol. Soc. Bras. Mat. 2, 23–36 (1971)

    Article  MathSciNet  Google Scholar 

  52. Thomas, T.: On closed spaces of constant mean curvature. Am. J. Math. 58, 702–704 (1936)

    Article  MathSciNet  Google Scholar 

  53. Tompkins, C.: A flat Klein bottle isometrically embedded in Euclidean 4-space. Bull. Am. Math. Soc. 47, 208 (1941)

    Article  MathSciNet  Google Scholar 

  54. Van Lindt, D., Verstraelen, L.: A survey on axioms of submanifolds in Riemannian and Kaehlerian geometry. Colloq. Math. 54, 193–213 (1987)

    Article  MathSciNet  Google Scholar 

  55. Weinstein, A.: Positively curved n-manifolds in \(\mathbb {R}^{n+2}\). J. Differ. Geom. 192, 1–4 (1970)

    Google Scholar 

  56. Wilking, B.: Nonnegatively and positively curved manifolds. In: Grove, K., Cheeger, J. (eds.) Metric and Comparison Geometry. Surveys in Differential Geometry, vol. 11, pp. 25–62. International Press, Boston (2007)

    MATH  Google Scholar 

  57. Wu, H.: The spherical images of convex hypersurfaces. J. Differ. Geom. 9, 279–290 (1974)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. IMPA, Rio de Janeiro, Brazil

    Marcos Dajczer

  2. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, São Paulo, Brazil

    Ruy Tojeiro

Authors
  1. Marcos Dajczer
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  2. Ruy Tojeiro
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Dajczer, M., Tojeiro, R. (2019). The Basic Equations of a Submanifold. In: Submanifold Theory . Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9644-5_1

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