Steiner's formula for the volume of a parallel hypersurface in a riemannian manifold

Abbena, E.; Gray, A.; Vanhecke, L.

Abbena, E. ; Gray, A. ; Vanhecke, L.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 8 (1981) no. 3, pp. 473-493.

MR Zbl

@article{ASNSP_1981_4_8_3_473_0,
     author = {Abbena, E. and Gray, A. and Vanhecke, L.},
     title = {Steiner's formula for the volume of a parallel hypersurface in a riemannian manifold},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {473--493},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 8},
     number = {3},
     year = {1981},
     mrnumber = {634858},
     zbl = {0472.53056},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_1981_4_8_3_473_0/}
}

TY  - JOUR
AU  - Abbena, E.
AU  - Gray, A.
AU  - Vanhecke, L.
TI  - Steiner's formula for the volume of a parallel hypersurface in a riemannian manifold
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 1981
SP  - 473
EP  - 493
VL  - 8
IS  - 3
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_1981_4_8_3_473_0/
LA  - en
ID  - ASNSP_1981_4_8_3_473_0
ER  -

%0 Journal Article
%A Abbena, E.
%A Gray, A.
%A Vanhecke, L.
%T Steiner's formula for the volume of a parallel hypersurface in a riemannian manifold
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 1981
%P 473-493
%V 8
%N 3
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_1981_4_8_3_473_0/
%G en
%F ASNSP_1981_4_8_3_473_0

Abbena, E.; Gray, A.; Vanhecke, L. Steiner's formula for the volume of a parallel hypersurface in a riemannian manifold. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 8 (1981) no. 3, pp. 473-493. http://www.numdam.org/item/ASNSP_1981_4_8_3_473_0/

Bibliographie
Cité par

[AB] J. Abram, Tensor calculus through differential geometry, Butterworths, London, 1965. | MR | Zbl

[AL1] C.B. Allendoerfer, The Euler number of a Riemannian manifold, Amer. J. Math., 62 (1940), pp. 243-248. | JFM | MR | Zbl

[AL2] C.B. Allendoerfer, Steiner's formula on a general Sn+1, Bull. Amer. Math. Soc., 54 (1948), pp. 128-135. | MR | Zbl

[AL3] C.B. Allendoerfer, Global differential geometry of imbedded manifolds, Problemi di geometria differenziale in grande, C.I.M.E., Cremonese, Roma, 1958. | Zbl

[AW] C.B. Allendoerfer - A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc., 53 (1943), pp. 101-129. | MR | Zbl

[BGM] M. Berger - P. Gauduchon - E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, 1971. | MR | Zbl

[BG] M. Berger - B. Gostiaux, Géométrie Différentielle, Armand Colin, Paris, 1972. | MR | Zbl

[BDP] J. Bertrand - C.F. Diguet - V. Puiseux, Demonstration d'un théorème de Gauss, Journal de Mathématiques, 13 (1848), pp. 80-90.

[BS] A.L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik, vol. 93, Springer-Verlag, 1978. | MR | Zbl

[BC] R. Bishop - R. Crittenden, Geometry of manifolds, Academic Press, 1964. | MR | Zbl

[BK] W. Blaschke, Vorlesungen über Integratgeometrie, Deutscher Verlag der Wissenschaften, 1955. | MR | Zbl

[BR] W. Blaschke - H. Reichardt, Einführung in die Differentialgeometrie, Springer-Verlag, 1960. | MR | Zbl

[BF] T. Bonnesen - W. Fenchel, Theorie der konvexen Körper, Ergebnisse der Mathematik, Springer-Verlag, 1934, Chelsea reprint, 1948. | MR | Zbl

[CH1] S.S. Chern, On the kinematic formula in the Euclidean space of N dimensions, Amer. J. Math., 74 (1952), pp. 227-236. | MR | Zbl

[CH2] S.S. Chern, On the kinematic formula in integral geometry, J. Math. Mech., 16 (1966), pp. 101-118. | MR | Zbl

[DC] M. Do Carmo, Differential geometry of curves and surfaces, Prentice Hall, 1978. | MR | Zbl

[ER] J. Erbacher, Riemannian manifolds of constant curvature and growth functions of submanifolds, Michigan Math. J., 19 (1972), pp. 215-223. | MR | Zbl

[FD1] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), pp. 418-491. | MR | Zbl

[FD2] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. | MR | Zbl

[FE] E. Ffrmi, Sopra i fenomeni che avvengono in vicinanza di una linea oraria, Atti R. Accad. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 31 (1922), pp. 21-23, 51-52, 101-103. Also The collected works of E. Fermi, vol. 1, pp. 17-19, The University of Chicago Press, 1962. | JFM

[FI] F. Fiala, Le problème des isopérimètres sur les surfaces ouvertes à coubure positive, Comment. Math. Helv., 13 (1941), pp. 293-341. | JFM | MR | Zbl

[FL1] F.J. Flaherty, The volume of a tube in complex projective space, Illinois J. Math., 16 (1972), pp. 627-638. | MR | Zbl

[FL2] F.J. Flaherty, Curvature measures for piecewise linear manifolds, Bull. Amer. Math. Soc., 79 (1973), pp. 100-102. | MR | Zbl

[GR1] A. Gray, Minimal varieties and almost Hermitian submanifolds, Mich. Math. J., 12 (1965), pp. 273-287. | MR | Zbl

[GR2] A. Gray, A generalization of F. Schur's theorem, J. Math. Soc. Japan, 21 (1969), pp. 454-457. | MR | Zbl

[GR3] A. Gray, Some relations between curvature and characteristic classes, Math. Ann., 184 (1970), pp. 257-267. | MR | Zbl

[GR4] A. Gray, Weak holonomy groups, Math. Z., 123 (1971), pp. 290-300. | MR | Zbl

[GR5] A. Gray, The volume of a small geodesic ball in a Riemannian manifold, Michigan Math. J., 20 (1973), pp. 329-344. | MR | Zbl

[GR6] A. Gray, Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Topology (to appear). | MR | Zbl

[GV1] A. Gray - L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math., 142 (1979), pp. 157-198. | MR | Zbl

[GV2] A. Gray - L. Vanhecke, The volume of tubes about curves in a Riemannian manifold, Proc. London Math. Soc. (to appear). | MR | Zbl

[GV3] A. Gray - L. Vanhecke, The volume of tubes in a Riemannian manifold, (to appear). | MR | Zbl

[GV4] A. Gray - L. Vanhecke, Oppervlakten van geodetische cirkels op oppervlakken, Med. Konink. Acad. Wetensch. Lett. Schone Kunst. België Kl. Wetensch., 42, No. 1 (1980), pp. 1-17. | MR | Zbl

[GS] P.A. Griffiths, Complex differential geometry and curvature integrals associated to singularities of complex analytic varieties, Duke Math. J., 45 (1978), pp. 427-512. | MR | Zbl

[GN1] N. Grossman, Polynomial estimates for Betti numbers of loop spaces of symmetric and close to symmetric Riemannian manifolds, Indiana Math. J., 20 (1971), pp. 717-731. | MR | Zbl

[GN2] N. Grossman, On characterization of Riemannian manifolds by growth of tubular neigborhoods, Proc. Amer. Math. Soc., 32 (1972), pp. 556-560. | MR | Zbl

[GU] H.W. Guggenheimer, Differential geometry, McGraw Hill, 1963. | MR | Zbl

[HA1] H. Hadwiger, Die erweiterten Steinerschen Formeln für ebene und sphitrische Bereiche, Comment. Math. Helv., 18 (1945), pp. 59-72. | MR | Zbl

[HA2] H. Hadwiger, Inhaltsungleichungen für innere und äusere Parallelmengen, Experientia (Basel), 2 (1946), p. 490. | MR | Zbl

[HA3] H. Hadwiger, Über das volumen der Parallelmengen, Mitt. Naturforsch. Ges. Bern., N.F., 3 (1946), pp. 121-125. | MR | Zbl

[HA4] H. Hadwiger, Über die erweiterten Steinerschen Formeln für Parallelmengen, Rev. Mat. Hispano-Americana, 4a serie, 6 (1946), pp. 160-163. | MR | Zbl

[HA5] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, 1957. | MR | Zbl

[HS] S. Haas - H.L. Schmid, Mathematisches Wörterbuch, Akademie-Verlag, B. G. Teubner, Berlin, 1967.

[HZ] G. Herglotz, Über die Steinersche Formel für Parallelflächen, Abh. Math. Sem. Hansischen Univ., 15 (1943), pp. 59-72. | MR | Zbl

[HR1] R.A. Holzsager, Riemannian manifolds of finite order, Bull. Amer. Math. Soc., 78 (1972), pp. 200-201. | MR | Zbl

[HR2] R.A. Holzsager, A characterization of Riemannian manifolds of constant curvature, J. Differential Geometry, 8 (1973), pp. 103-106. | MR | Zbl

[HW] R.A. Holzsager - H. Wu, A characterization of two dimensional Riemannian manifolds of constant curvature, Michigan Math. J., 17 (1970), pp. 297-299. | MR | Zbl

[HO] H. Hotelling, Tubes and spheres in n-space and a class of statistical problems, Amer. J. Math., 61 (1939), pp. 440-460. | JFM | MR | Zbl

[OH] D. Ohmann, Eine verallgemeinerung der Steinerschen Formel, Math. Ann., 129 (1955), pp. 204-212. | MR | Zbl

[ON] B. O'Neill, Elementary differential geometry, Academic Press, 1966. | MR | Zbl

[OS1] R. Osserman, The isoperimetric inequatity, Bull. Amer. Math. Soc., 84 (1978), pp. 1182-1238. | MR | Zbl

[OS2] R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly, 86 (1979), pp. 1-29. | MR | Zbl

[SA] L.A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Addison Wesley, 1976. | MR | Zbl

[SP] M. Spivak, A comprehensive introduction to differential geometry, vol. 5, Publish or Perish, Boston, 1975. | Zbl

[STE] J. Steiner, Über parallele Flächen, Monatsbericht der Akademie der Wissenchaften zu Berlin (1840), pp. 114-118,Also Werke, vol. 2 (1882), pp. 171-176.

[SW] R. Sulanke - P. Wintgen, Differentialgeometrie und Faserbündel, Birkhäuser Verlag, 1972. | MR | Zbl

[VA1] E. Vidal Abascal, Sobre un teorema de Liouville y generalización de fórmulas de Steiner, Rev. Mat. Hispano-Americana, 4a serie, 6 (1946), pp. 3-8. | MR

[VA2] E. Vidal Abascal, Sobre fórmulas de Steiner para el área de una elipse, Rev. Geodésica, 6 (1947), pp. 425-431.

[VA3] E. Vidal Abascal, A generalization of Steiner's formula, Bull. Amer. Math. Soc., 53 (1947), pp. 841-844. | MR | Zbl

[VA4] E. Vidal Abascal, Extensión del concepto de curvas paralelas sobre una superficie. Longitud y área correspondientes a la curvatura así deducida de otra dada, Rev. Math. Hispano-Americana, 4a serie, 7 (1947), pp. 3-12.

[VA5] E. Vidal Abascal, Area engendrada sobre una superficie por un arco de geodésica cuando uno de sus extremos recorre una curva fija y longitud de la curva descrita por el otro extremo, Rev. Mat. Hispano-Americana, 4a serie, 7 (1947), pp. 132-142. | MR

[VA6] E. Vidal Abascal, Curvas paralelas sobre superficies de curvatura constante, Rev. Union Math. Argentina, 8 (1948), pp. 135-138. | MR | Zbl

[VA7] E. Vidal Abascal, Introducción a la geometría diferencial, Dossat, Madrid, 1956. | MR | Zbl

[VA8] E. Vidal Abascal, Nueva deducción de las fórmulas de Steiner en un general Sn+1, Collectanea Mathematica, 14 (1962), pp. 47-50. | MR | Zbl

[WY] H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), pp. 461-472. | JFM | MR | Zbl

[WO] R.A. Wolf, The volume of tubes in complex projective space, Trans. Amer. Math. Soc., 157 (1971), pp. 347-371. | MR | Zbl

[WU] H. Wu, A characteristic property of the Euclidean plane, Michigan Math. J., 16 (1969), pp. 141-148. | MR | Zbl