Tohoku Mathematical Journal

Total curvature of complete submanifolds of Euclidean space

Franki Dillen and Wolfgang Kühnel

Full-text: Open access


The classical Cohn-Vossen inequality states that for any complete 2-dimensional Riemannian manifold the difference between the Euler characteristic and the normalized total Gaussian curvature is always nonnegative. For complete open surfaces in Euclidean 3-space this curvature defect can be interpreted in terms of the length of the curve "at infinity''. The goal of this paper is to investigate higher dimensional analogues for open submanifolds of Euclidean space with cone-like ends. This is based on the extrinsic Gauss-Bonnet formula for compact submanifolds with boundary and its extension "to infinity''. It turns out that the curvature defect can be positive, zero, or negative, depending on the shape of the ends "at infinity''. We give an explicit example of a 4-dimensional hypersurface in Euclidean 5-space where the curvature defect is negative, so that the direct analogue of the Cohn-Vossen inequality does not hold. Furthermore we study the variational problem for the total curvature of hypersurfaces where the ends are not fixed. It turns out that for open hypersurfaces with cone-like ends the total curvature is stationary if and only if each end has vanishing Gauss-Kronecker curvature in the sphere "at infinity''. For this case of stationary total curvature we prove a result on the quantization of the total curvature.

Article information

Tohoku Math. J. (2), Volume 57, Number 2 (2005), 171-200.

First available in Project Euclid: 27 June 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space


Dillen, Franki; Kühnel, Wolfgang. Total curvature of complete submanifolds of Euclidean space. Tohoku Math. J. (2) 57 (2005), no. 2, 171--200. doi:10.2748/tmj/1119888334.

Export citation


  • C. B. Allendoerfer, The Euler number of a Riemannian manifold, Amer. J. Math. 62 (1940), 243--248.
  • C. B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943), 101--129.
  • S. C. de Almeida and F. Brito, Minimal hypersurfaces of $S\sp 4$ with constant Gauss-Kronecker curvature, Math. Z. 195 (1987), 99--107.
  • S. C. de Almeida and F. Brito, Closed 3-dimensional hypersurfaces with constant mean curvature and constant scalar curvature, Duke Math. J. 61 (1990), 195--206.
  • S. C. de Almeida and F. Brito, Closed hypersurfaces of $S^4$ with two constant symmetric curvatures, Ann. Fac. Sci. Toulouse Math. (6) 6 (1997), 187--202.
  • W. Ballmann, M. Gromov and V. Schroeder, Manifolds of nonpositive curvature, Progr. Math. 61, Birkhäuser Boston, Inc., Boston, Mass., 1985.
  • T. F. Banchoff and W. Kühnel, Tight submanifolds, smooth and polyhedral, Tight and taut submanifolds (Berkeley, Calf., 1994), 51--118, Math. Sci. Res. Inst. Publ. 32, Cambridge Univ. Press, Cambridge, 1997.
  • V. Bangert, Total curvature and the topology of complete surfaces, Compos. Math. 41 (1980), 95--105.
  • A. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin, 1987.
  • D. Bleecker, The Gauss-Bonnet inequality and almost-geodesic loops, Adv. Math. 14 (1974), 183--193.
  • T. E. Cecil and P. J. Ryan, Tight and taut immersions of manifolds, Res. Notes Math. 107, Pitman, Boston, Mass., 1985.
  • S. Chang, A closed hypersurface with constant scalar and mean curvatures in $S^4$ is isoparametric, Comm. Anal. Geom. 1, (1993), 71--100.
  • J. Cheeger and M. Gromov, Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), 1--34.
  • B.-Y. Chen, G-total curvature of immersed submanifolds, J. Differential Geom. 7 (1972), 371--391.
  • S.-S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747--752.
  • S.-S. Chern, On the curvatura integra in a Riemannian manifold, Ann. of Math. (2) 46 (1945), 674--684.
  • S.-S. Chern and R. K. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306--318, Part II: Michigan Math. J. 5 (1958), 5--12.
  • S. Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compos. Math. 2 (1935), 69--133.
  • F. Dillen and W. Kühnel, Total curvature for open submanifolds of Euclidean spaces, Differential geometry (Sakado, 2001), 139--148, Josai Math. Monogr. 3, Josai Univ., Sakado, 2001.
  • W. Fenchel, On total curvatures of Riemannian manifolds: I, J. London Math. Soc. 15 (1940), 15--22.
  • D. Ferus, Totale Absolutkrümmung in Differentialgeometrie und- topologie, Lecture Notes in Math. 66, Springer-Verlg, Berlin-New York, 1968.
  • D. Ferus, On the type number of hypersurfaces in spaces of constant curvature, Math. Ann. 187 (1970), 310--316.
  • D. Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313--316.
  • F. Fiala, Le problème isopérimètres sur les surfaces ouvertes à courbure positive, Comment. Math. Helv. 13 (1941), 293--346.
  • A. Fialkow, Hypersurfaces of a space of constant curvature, Ann. of Math. (2) 39 (1938), 762--785.
  • R. Finn, On a class of conformal metrics, with application to differential geometry in the large, Comment. Math. Helv. 40 (1965), 1--30.
  • M. van Gemmeren, Totale Krümmung und totale Absolutkrümmung von Untermannigfaltigkeiten des $\mathbbR^m$, Dissertation, Universität Stuttgart, Stuttgart, 1995.
  • M. van Gemmeren, Total absolute curvature and tightness of noncompact manifolds, Trans. Amer. Math. Soc. 348 (1996), 2413--2426.
  • D. H. Gottlieb, All the way with Gauss-Bonnet and the sociology of mathematics, Amer. Math. Monthly 103 (1996), 457--469.
  • P. Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 706--727.
  • H. Hopf, Über die Curvatura integra geschlossener Hyperflächen, Math. Ann. 95 (1926), 340--367.
  • H. Hopf, Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten, Math. Ann. 96 (1927), 225--250.
  • H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen, Math. Nachr. 4 (1951), 232--249.
  • A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13--72.
  • A. Huber, Vollständige konforme Metriken und isolierte Singularitäten subharmonischer Funktionen, Comment. Math. Helv. 41 (1966/1967), 105--136.
  • T. Ishihara, The Euler characteristics and Weyl's curvature invariants of submanifolds in spheres, J. Math. Soc. Japan 39 (1987), 247--256.
  • G. Ishikawa, M. Kimura and R. Miyaoka, Submanifolds with degenerate Gauss mappings in spheres, Lie groups, geometric structures and differential equations--one hundred years after Sophus Lie (Kyoto/Nara, 1999), 115--149, Adv. Stud. Pure Math. 37, Math. Soc. Japan, Tokyo, 2002.
  • L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203--221.
  • N. Kleinjohann and R. Walter, Nonnegativity of the curvature operator and isotropy for isometric immersions, Math. Z. 181 (1982), 129--142.
  • S. Kobayashi and K. Nomizu, Foundations of Differential Geometry II, Interscience Tracts in Pure and Applied Mathematics 15 II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969.
  • P. Kohlmann, Uniqueness results for hypersurfaces in space forms, Habilitationsschrift Univ. Dortmund, 1995.
  • Y. Kubo, A note on the scaling limit of a complete open surface, Tokyo J. Math. 18 (1995), 179--183.
  • N. H. Kuiper, Der Satz von Gauss-Bonnet für Abbildungen in $E^N$ und damit verwandte Probleme, Jahresber. Deutsch. Math.-Verein. 69 (1967), 77--88.
  • E. Leuzinger, On the Gauss-Bonnet formula for locally symmetric spaces of noncompact type, Enseign. Math. (2) 42 (1996), 201--214.
  • R. Osserman, Global properties of minimal surfaces in $E^3$ and $E^n$, Ann. of Math. (2) 80 (1964), 340--364.
  • M. Pinl and H. Trapp, Stationäre Krümmungsdichten auf Hyperflächen des euklidischen $R_n+1$, Math. Ann. 176 (1968), 257--272.
  • E. Portnoy, Toward a generalized Gauss-Bonnet formula for complete, open manifolds, Comment. Math. Helv. 46 (1971), 324--344.
  • R. C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom. 8 (1973), 465--477.
  • S. Rosenberg, Gauss-Bonnet theorems for noncompact surfaces, Proc. Amer. Math. Soc. 86 (1982), 184--185.
  • S. Rosenberg, On the Gauss-Bonnet theorem for complete manifolds, Trans. Amer. Math. Soc. 287 (1985), 745--753.
  • R. Sacksteder, On hypersurfaces with nonnegative sectional curvature, Amer. J. Math. 82 (1960), 609--630.
  • L. A. Santaló, Total curvatures of compact manifolds immersed in Euclidean space, Symposia Mathematica XIV (Convegni di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome 1973), 363--390, Academic Press, London, 1974.
  • V. A. Sharafutdinov, Relative Euler class and the Gauss-Bonnet theorem, Siberian Math. J. 14 (1974), 930--940.
  • K. Shiohama, Busemann functions and total curvature, Invent. Math. 53 (1979), 281--297.
  • K. Shiohama, The role of total curvature on complete noncompact Riemannian 2-manifolds, Illinois J. Math. 28 (1984), 597--620.
  • K. Shiohama, Total curvature and minimal areas of complete open surfaces, Proc. Amer. Math. Soc. 94 (1985), 310--316.
  • K. Shiohama, T. Shioya and M. Tanaka, Mass of rays on complete open surfaces, Pacific J. Math. 143 (1990), 349--358.
  • T. Shioya, The ideal boundaries and global geometric properties of complete open surfaces, Nagoya Math. J. 120 (1990), 181--204.
  • T. Shioya, The ideal boundaries of complete open surfaces, Tohoku Math. J. (2) 43 (1991), 37--59.
  • T. Shioya, Geometry of total curvature, Acts de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 561--600, Semin. Congr. 1, Soc. Math. France, Paris, (1996).
  • J. J. Stoker, Über die Gestalt der positiv gekrümmten offenen Flächen im dreidimensionalen Raume, Compos. Math. 3 (1936), 55--88.
  • E. Teufel, Anwendungen der differentialtopologischen Berechnung der totalen Krümmung und totalen Absolutkrümmung in der sphärischen Differentialgeometrie, Manuscripta Math. 32 (1980), 239--262.
  • K. Voss, Einige differentialgeometrische Kongruenzsätze für geschlossene Flächen und Hyperflächen, Math. Ann. 131 (1956), 180--218.
  • K. Voss, Variations of curvature integrals, Affine differential geometry (Oberwolfach, 1991), Results Math. 20 (1991), 789--796.
  • R. Walter, A generalized Allendoerffer-Weil formula and an inequality of the Cohn-Vossen type, J. Differential Geom. 10 (1975), 167--180.
  • R. Walter, Compact hypersurfaces with a constant higher mean curvature function, Math. Ann. 270 (1985), 125--145.
  • H. Weyl, On the volume of tubes, Amer. J. Math. 61 (1939), 461--472.
  • B. White, Complete surfaces of finite total curvature, J. Differential Geom. 26 (1987), 315--326, Correction 28 (1988), 359--360.
  • T. J. Willmore, Total curvature in Riemannian geometry, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chicester, Halstead Press, New York, 1982.
  • P. Wintgen, On total absolute curvature of nonclosed submanifolds, Ann. Global Anal. Geom. 2 (1984), 55--87.
  • H. Wu, A structure theorem for complete noncompact hypersurfaces of nonnegative curvature, Bull. Amer. Math. Soc. 77 (1971), 1070--1071.
  • J.-W. Yim, Convexity of the ideal boundary for complete open surfaces, Trans. Amer. Math. Soc. 347 (1995), 687--700.
  • K. Shiohama, T. Shioya and M. Tanaka, The geometry of total curvature on complete open surfaces, Cambridge Tracts in Math. 159, Cambridge Univ. Press, Cambridge, 2003.
  • J. Milnor, On the immersion of $n$-manifolds in $(n+1)$-space, Comment. Math. Helv. 30 (1956), 275--284.