Skip to main content
SpringerLink
Log in

Gauge Theory, Isotropy, and Surfaces in Affine 4-Space

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The surface theory in the equiaffine space R4 is developed on the basis of H. Weyl’s gauge theory. Rescaling of the Weyl geometry leads to a 1-parameter family of invariant transversal plane bundles containig former special constructions. A transversal bundle metric is gained via the notion of isotropy. The paper then proceeds with a general tensorial theory, including theorema egregium and Radon-type results and a discussion of cubic fundamental forms. Finally there is given an application to homogeneous surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Besse, A. [1987]: Einstein manifolds. I–XII a. 1–510. Springer Verlag 1987.

  • Bourbaki, N. [1971]: Variétés différentielles et analytiques. Fasc. d. rés. / Par. 1 à 7, deuxième éd. Éléments de mathématique, Fasc. XXXIII. Hermann 1971.

  • Burstin, C. und Mayer, W. [1927]: Die Geometrie zweifach ausgedehnter Mannigfaltigkeiten F 2 [m affinen Raum R 4. Math. Z. 27 (1927), 373–407

    Article  MathSciNet  Google Scholar 

  • Chern, S.S. [1944]: Laplace transforms of a class of higher dimensional varietes in a projective space of n dimensions. Proc. Nat. Acad. Sci. USA [vn30 (1944), 95–97; reprint in S.S. Chern: Selected Papers, Vol. II, p. 113–115.

    Google Scholar 

  • Chern, S.S. [1947]: Sur une classe remarquable de variétés dans l’espace projectif à n dimensions. Science Reports Tsing Hua Univ. 4 (1947), 328–336; reprint in S.S. Chern: Selected Papers, Vol. I, p. 138–146.

    MathSciNet  Google Scholar 

  • Eells, J. and Lemaire, L. [1978]: A report on harmonic maps. Bull. London Math. Soc. 10 (1978), 1–68.

    Article  MathSciNet  MATH  Google Scholar 

  • Eisenhart, L.P. [1968]: Non-Riemannian Geometry. i–viii a, 1–184 AMS Coll. Publ. VIII, sixth printing.

  • Finikow, S.P. [1959]: Theorie der Kongruenzen. I–XV u. 1–491. Akademie Verlag Berlin 1959 (German translation of the Russian original 1950).

  • Grunwald, M. [2001]: Zentroaffine Flächentheorie im vierdimensionalen Raum. Thesis preprint Dortmund 2001, 1–85.

  • Hilgert, J. und Neeb, K. [1991]: Lie-Gruppen und Lie-Algebren. 1–361. Vieweg Verlag 1991.

  • Kleinjohann, N. and Walter, R. [1982]: Nonnegativity of the Curvature Operator and Isotropy for Isometric Immersions. Math. Z. 181(1982), 129–142.

    Article  MathSciNet  MATH  Google Scholar 

  • Klingenberg, W. [1951]: Zur affinen Differentialgeometrie. Teil II: Über zweidimensionale Flächen im vierdimensionalen Raum. Math. Z. 54(1951), 184–216.

    Article  MathSciNet  Google Scholar 

  • Kobayashi, S. and Nomizu, K. [1969]: Foundations of differential geometry, Vol. II. i–xv a. 1–470. Interscience publishers 1969.

  • Laugwitz, D. [1968]: Differentialgeometrie. 2. Aufl. 1–183. Teubner Stuttgart 1968.

  • Li, J.: Harmonic Surfaces in Affine 4-Space. Preprint.

  • Nomizu, K. and Pinkall, U. [1988]: Cubic form theorem for affine immersions. Res. in Math. 13 (1988). 338–362.

    Article  MathSciNet  MATH  Google Scholar 

  • Nomizu, K. and Vrancken, L. [1993]: A new equiaffine theory for surfaces in R4. Intern. J. Math. 4 (1993), 127–165.

    Article  MathSciNet  MATH  Google Scholar 

  • O’Neill, B. [1965]: Isotropic and Kähler immersions. Canad. J. Math. 17 (1965), 907–915.

    Article  MathSciNet  MATH  Google Scholar 

  • Pedersen, K. and Swann, T. [1993]: Riemannian submersions, four-manifolds and Einstein-Weyl geometry. Proc. London Math. Soc. 66(1993), 381–399.

    Article  MathSciNet  MATH  Google Scholar 

  • Scharlach, C. and Vrancken, L. [1992]: Affine transversal planes for surfaces in R4. In: Geometry and Topology of Submanifolds V, World Scientific Singapore 1992, 249–253.

    Google Scholar 

  • [Schonten, J.A. [1924]: Der Ricci-Kalkül. I-X u. 1–311. Springer Verlag Berlin 1924.

  • Walter, R. [1967.a]: Über zweidimensionale parabolische Flächen im vierdimensionalen affinen Raum, Teil I. Allgemeine Flächentheorie. J. reine angew. Math. 227 (1967), 178–208.

    MathSciNet  MATH  Google Scholar 

  • Walter, R. [1961.b]: Über zweidimensionale parabolische Flächen im vierdimensionalen affinen Raum, Teil II. Spezielle Flächen. J. reine angew. Math. 228 (1967), 71–92.

    Google Scholar 

  • Walter, R. [1989]: Differentialgeometrie. 2. Aufl., i-v u. 1–336. Vieweg 1989.

  • Walter, R. [1995]: On the role of the Burstin-Mayer metric for surfaces in R4. Res. in Math. 27 (1995), 194–197.

    Article  MATH  Google Scholar 

  • Walter, R. [1998]: Homogeneity for Surfaces in Four-Dimensional Vector Space Geometry. Geom. Dedicata [vn71 (1998), 129–178

    Google Scholar 

  • Walter, R. [2000.a]: Homogeneous parabolic surfaces in R4. Contr. Alg. Geom. 41 (2000), 159–180

    MATH  Google Scholar 

  • Walter, R. [2000.b]: Homogeneous Surfaces in the Equiaffine Space R4. Banach Center Publications, Proc. Conf. TDEs, submanifolds and affine differential geometry’, Banach Center, Warzaw, Sept. 2000. To appear.

  • Wang, C. P. [1994]: A Unified Equiaffine Theory for Surfaces in R4. Part I: The Definite Surfaces. In: Geometry and Topology of Submanifolds VI, World Scientific Singapore [dy1994, 210–223.

    Google Scholar 

  • Wang, C. P. [1995.a]: A Unified Equiaffine Theory for Surfaces in R4. Part II: Indefinite surfaces. Result. Math. 27 (1995), 198–205.

    Article  MATH  Google Scholar 

  • Wang, C. P. [1995.b]: Equiaffine Theory of Surfaces in R4. Dissertation TU Berlin 1995, i–vii u. 1–77.

  • Wang, C.P. [1997]: The classification of equiaffine indefinite flat homogeneous surfaces in R4. Geom. Dedicata 65 (1997), 323–353.

    Article  MathSciNet  MATH  Google Scholar 

  • Wermann, M. [2001]: Homogene Hyperflächen im vierdimensionalen äqui-affinen Raum. Dissertation Universität Dortmund 2001. Shaker Verlag Aachen. To appear.

  • Weyl, H. [1918]: Reine Infinitesimalgeometrie. Math. Z. 2 (1918), 384–411.

    Article  MathSciNet  MATH  Google Scholar 

  • Weyl, H. [1922]: Zur Infinitesimalgeometrie: p dimensionale Fläche im n dimensionalen Raum. Math. Z. 12 (1922), 154–160.

    Article  MathSciNet  MATH  Google Scholar 

  • Wolf, J.A. [1967]: Spaces of constant curvature. i–xv a. 1–408. McGraw-Hill 1967.

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität, D-44221, Dortmund, Germany

    Rolf Walter

Authors
  1. Rolf Walter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rolf Walter.

Additional information

Dedicated to Professor Shiing-shen Chern on the occasion of his 90th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Walter, R. Gauge Theory, Isotropy, and Surfaces in Affine 4-Space. Results. Math. 42, 139–176 (2002). https://doi.org/10.1007/BF03323561

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03323561

Keywords

MSC 2000

Search

Navigation