Araştırma Makalesi
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Yıl 2018, Cilt: 11 Sayı: 2, 28 - 33, 30.11.2018

Nicholas D. Brubaker Jasmine Camero Oscar Rocha Rocha Bogdan D. Suceava

Öz

Kaynakça

  • [1] Casorati, Felice, Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne, Acta Math. 14 (1) (1890), 95–110.
  • [2] Chen, Bang-Yen, Total mean curvature and submanifolds of finite type, World Scientific, 1984.
  • [3] Chen, Bang-Yen, Pseudo-Riemannian geometry, δ−invariants and applications, World Scientific, 2011.
  • [4] Brzycki, B.; Giesler, M.D.; Gomez, K.; Odom L.H.; and Suceava, B.D., A ladder of curvatures for hypersurfaces in the Euclidean ambient space, Houston Journal of Mathematics, 40 (2014). pp. 1347–1356.
  • [5] Conley, C. T. R.; Etnyre, R.; Gardener, B.; Odom L.H.; and Suceava, B.D., New Curvature Inequalities for Hypersurfaces in the Euclidean Ambient Space, Taiwanese Journal of Mathematics, 17 (2013), 885–895.
  • [6] Decu, S.; Haesen, S.; and Verstraelen, L., Optimal inequalities involving Casorati curvatures, Bull. Transylv. Univ. Bra¸sov Ser. B 14 (2007), 85–93.
  • [7] Decu, S.; Haesen, S.; Verstraelen, L.; Vîlcu, G.-E., Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant -Sectional Curvature, Entropy 20 (2018) 20, 529.
  • [8] do Carmo, M.P., Riemannian Geometry, Birkhäuser, 1992.
  • [9] Euler, Leonhard, Recherches sur la courbure des surfaces, Memoires de l’academie des sciences de Berlin (written in 1760, published in 1767), 16: 119 –143.
  • [10] Gauss, C.F. , Disquisitiones circa superficies curvas, Typis Dieterichianis, Goettingen, 1828.
  • [11] Germain, Sophie, Mémoire sur la courbure des surfaces, Journal für die reine und andewandte Mathematik, Herausgegeben von A. L. Crelle, Siebenter Band, pp. 1–29, Berlin, 1831.
  • [12] Kenmotsu, Katsuei, Surfaces with Constant Mean Curvature, translated by Katsuhiro Moriya, Translations of Mathematical Monographs, Vol. 221, American Mathematical Society, 2003.
  • [13] Knoebel, Art; Laubenbacher, Reinhard; Lodder, Jerry; Pengelley, David, Mathematical Masterpieces: Further Chronicles by the Explorers, Undergraduate Texts in Mathematics, Springer-Verlag, 2007.
  • [14] Spivak, Michael, A Comprehensive Introduction to Differential Geometry, third edition, Publish or Perish, 1999.
  • [15] Suceava, B. D., The spread of the shape operator as conformal invariant, J. Inequal. Pure Appl. Math, 4 (2003), issue 4, article 74.
  • [16] Suceava, B. D., The amalgamatic curvature and the orthocurvatures of three dimensional hypersurfaces in E4, Publicationes Mathematicae, 87(2015), no. 1-2, 35–46.
  • [17] Suceava, B.D., A Geometric Interpretation of Curvature Inequalities on Hypersurfaces via Ravi Substitutions in the Euclidean Plane, Math Intelligencer 40 (2018), 50–54.
  • [18] Verstraelen, L., Geometry of submanifolds I. The first Casorati curvature indicatrices, Kragujevac J. Math. 37(2013), 5–23.
  • [19] Weingarten, J., Über eine Klasse auf einander abwickelbarer Flächen, Journal für die Reine und Angewandte Mathematik 59 (1861), 382–393.
  • [20] Weingarten, J., Über die Flächen, derer Normalen eine gegebene Fläche berühren, Journal für die Reine und Angewandte Mathematik 62 (1863), 61–63.

A Ladder of Curvatures in the Geometry of Surfaces

Yıl 2018, Cilt: 11 Sayı: 2, 28 - 33, 30.11.2018

Nicholas D. Brubaker Jasmine Camero Oscar Rocha Rocha Bogdan D. Suceava

Öz

Many investigations in the local differential geometry of surfaces focused on Gaussian curvature
and mean curvature. Besides these classical curvature invariants, are there any other geometric
quantities that deserve to be investigated? In the recent decades, there have been important
developments in the area of new curvature invariants for submanifolds, mostly included in Bang-
Yen Chen’s important monograph Pseudo-Riemannian geometry, δ−invariants and applications, World
Scientific, 2011. These developments are inviting us to look at the classical content from a different
perspective, exploring other quantities that might be of interest.

Anahtar Kelimeler

principal curvatures Gaussian curvature mean curvature Casorati curvature smooth surfaces conformal Gauss map

Kaynakça

  • [1] Casorati, Felice, Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne, Acta Math. 14 (1) (1890), 95–110.
  • [2] Chen, Bang-Yen, Total mean curvature and submanifolds of finite type, World Scientific, 1984.
  • [3] Chen, Bang-Yen, Pseudo-Riemannian geometry, δ−invariants and applications, World Scientific, 2011.
  • [4] Brzycki, B.; Giesler, M.D.; Gomez, K.; Odom L.H.; and Suceava, B.D., A ladder of curvatures for hypersurfaces in the Euclidean ambient space, Houston Journal of Mathematics, 40 (2014). pp. 1347–1356.
  • [5] Conley, C. T. R.; Etnyre, R.; Gardener, B.; Odom L.H.; and Suceava, B.D., New Curvature Inequalities for Hypersurfaces in the Euclidean Ambient Space, Taiwanese Journal of Mathematics, 17 (2013), 885–895.
  • [6] Decu, S.; Haesen, S.; and Verstraelen, L., Optimal inequalities involving Casorati curvatures, Bull. Transylv. Univ. Bra¸sov Ser. B 14 (2007), 85–93.
  • [7] Decu, S.; Haesen, S.; Verstraelen, L.; Vîlcu, G.-E., Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant -Sectional Curvature, Entropy 20 (2018) 20, 529.
  • [8] do Carmo, M.P., Riemannian Geometry, Birkhäuser, 1992.
  • [9] Euler, Leonhard, Recherches sur la courbure des surfaces, Memoires de l’academie des sciences de Berlin (written in 1760, published in 1767), 16: 119 –143.
  • [10] Gauss, C.F. , Disquisitiones circa superficies curvas, Typis Dieterichianis, Goettingen, 1828.
  • [11] Germain, Sophie, Mémoire sur la courbure des surfaces, Journal für die reine und andewandte Mathematik, Herausgegeben von A. L. Crelle, Siebenter Band, pp. 1–29, Berlin, 1831.
  • [12] Kenmotsu, Katsuei, Surfaces with Constant Mean Curvature, translated by Katsuhiro Moriya, Translations of Mathematical Monographs, Vol. 221, American Mathematical Society, 2003.
  • [13] Knoebel, Art; Laubenbacher, Reinhard; Lodder, Jerry; Pengelley, David, Mathematical Masterpieces: Further Chronicles by the Explorers, Undergraduate Texts in Mathematics, Springer-Verlag, 2007.
  • [14] Spivak, Michael, A Comprehensive Introduction to Differential Geometry, third edition, Publish or Perish, 1999.
  • [15] Suceava, B. D., The spread of the shape operator as conformal invariant, J. Inequal. Pure Appl. Math, 4 (2003), issue 4, article 74.
  • [16] Suceava, B. D., The amalgamatic curvature and the orthocurvatures of three dimensional hypersurfaces in E4, Publicationes Mathematicae, 87(2015), no. 1-2, 35–46.
  • [17] Suceava, B.D., A Geometric Interpretation of Curvature Inequalities on Hypersurfaces via Ravi Substitutions in the Euclidean Plane, Math Intelligencer 40 (2018), 50–54.
  • [18] Verstraelen, L., Geometry of submanifolds I. The first Casorati curvature indicatrices, Kragujevac J. Math. 37(2013), 5–23.
  • [19] Weingarten, J., Über eine Klasse auf einander abwickelbarer Flächen, Journal für die Reine und Angewandte Mathematik 59 (1861), 382–393.
  • [20] Weingarten, J., Über die Flächen, derer Normalen eine gegebene Fläche berühren, Journal für die Reine und Angewandte Mathematik 62 (1863), 61–63.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Nicholas D. Brubaker Bu kişi benim

Jasmine Camero Bu kişi benim

Oscar Rocha Rocha Bu kişi benim

Bogdan D. Suceava

Yayımlanma Tarihi 30 Kasım 2018
Yayımlandığı Sayı Yıl 2018 Cilt: 11 Sayı: 2

Kaynak Göster

APA Brubaker, N. D., Camero, J., Rocha, O. R., Suceava, B. D. (2018). A Ladder of Curvatures in the Geometry of Surfaces. International Electronic Journal of Geometry, 11(2), 28-33.
AMA Brubaker ND, Camero J, Rocha OR, Suceava BD. A Ladder of Curvatures in the Geometry of Surfaces. Int. Electron. J. Geom. Kasım 2018;11(2):28-33.
Chicago Brubaker, Nicholas D., Jasmine Camero, Oscar Rocha Rocha, ve Bogdan D. Suceava. “A Ladder of Curvatures in the Geometry of Surfaces”. International Electronic Journal of Geometry 11, sy. 2 (Kasım 2018): 28-33.
EndNote Brubaker ND, Camero J, Rocha OR, Suceava BD (01 Kasım 2018) A Ladder of Curvatures in the Geometry of Surfaces. International Electronic Journal of Geometry 11 2 28–33.
IEEE N. D. Brubaker, J. Camero, O. R. Rocha, ve B. D. Suceava, “A Ladder of Curvatures in the Geometry of Surfaces”, Int. Electron. J. Geom., c. 11, sy. 2, ss. 28–33, 2018.
ISNAD Brubaker, Nicholas D. vd. “A Ladder of Curvatures in the Geometry of Surfaces”. International Electronic Journal of Geometry 11/2 (Kasım 2018), 28-33.
JAMA Brubaker ND, Camero J, Rocha OR, Suceava BD. A Ladder of Curvatures in the Geometry of Surfaces. Int. Electron. J. Geom. 2018;11:28–33.
MLA Brubaker, Nicholas D. vd. “A Ladder of Curvatures in the Geometry of Surfaces”. International Electronic Journal of Geometry, c. 11, sy. 2, 2018, ss. 28-33.
Vancouver Brubaker ND, Camero J, Rocha OR, Suceava BD. A Ladder of Curvatures in the Geometry of Surfaces. Int. Electron. J. Geom. 2018;11(2):28-33.
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