The Principal Extrinsic and Intrinsic Tangent Directions of GeneralisedWintgen Ideal Legendrian Submanifolds
Authors: A. Sebekovic
Keywords: Generalised Wintgen ideal Legendrian submanifolds, Ricci principal directions, Casorati principal directions.
Abstract:
Abstract: For Legendrian submanifolds Mn in Sasakian space forms ∼M 2n+1 (c), I. Mihai obtained an inequality relating (intrinsic) the normalised scalar curvature and (extrinsic) squared mean curvature and normalised normal curvature of M in ∼M , characterising also the corresponding equality case. In this paper, it’s shown that (intrinsic) Ricci principal directions and (extrinsic) Casorati principal directions, for generalised Wintgen ideal Legendrian submanifolds Mn in Sasakian space forms ∼M 2n+1 (c), do coincide.
References:
[1] D. BLAIR, Riemannian Geometry of Contact and Sympletic Manifolds, Birkkhauser, Boston, 2002.
[2] F. CASORATI, Mesure de la courbure des surfaces suivant l’idee commune, Acta Math. 14, No1 (1890), 95-110.
[3] B. Y. CHEN, Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gaus and normal curvature, Ann. Glob. Anal. Geom. 38(2), (2010), 145-160.
[4] B. Y. CHEN and L. VERSTRAELEN, A characterization of quasi-umbi-Lical submanifolds and its applications, Boll. Un. Mat. Ital 14(1977), 49-57.
[5] T. CHOI and Z. LU, On the DDVV consecture and comass in calibrated geometry (I) , Math.Z. 260 (2008), 409-429.
[6] S. DECU, M. PETROVIC-TORGASEV, A. SEBEKOVIC and L. VERSTRAELEN, Ricci and Casorati principal directions of Wintgen ideal submanifolds , Filomat 28:4 (2014), 657-661.
[7] P. J. DE SMET, F. DILLEN, L. VERSTRAELEN and L. VRANCKEN, A pointwise inquality in submanifolds theory, Arch. Math. (Brno), 35(1999), 115-128.
[8] J. GE and Z. TANG, A proof of the DDVV conjecture and its equality case, Pacific J. Math., 237(2008), 87-95.
[9] I. V. GUADALUPE and L. RODRIGUEZ, Normal curvature of surfaces in space forms, Pacific J. Math., 106(1983), 95-103.
[10] S. HAESEN, D. KOWALCZYK and L. VERSTRAELEN, On extrinsic principal directions of Riemannian submanifolds, Note Mat. 29 No. 2(2009), 41-53.
[11] C. JORDAN , Generalisation du theorema d’ Euler sur la courbure des surfaces, C.R.Acad. Sc. Paris, 79 (1874), 909-912.
[12] Z. LU , Normal scalar curvature conjecture and its applications, J. Funct. Anal. 261(2011), 1284-1308.
[13] I. MIHAI , On the generalised Wintgen inquality for Legendrian submanifolds in Sasakian space forms, Tohoku Math. J. 69(1) (2017), 45-53.
[14] M. PETROVIC-TORGASEV and L. VERSTRAELEN, On Deszcz symmetries of Wintgen ideal submanifolds, Arch. Math. (Brno) 44(2008), 57-67.
[15] B. ROUXEL , Sur quelques proprietes anallagmatiques de l’espace euclidien E4,Memoire couronne par l’Academie Royale de Belguque, 1982 (128 pp).
[16] A. SEBEKOVIC , M. PETROVIC-TORGASEV and A. PANTIC, Pseudosymmetry properties of generalised Wintgen ideal Legendrain submanifolds, Filomat 33:4 (2019), 1209-1215.
[17] P. WINTGEN , Sur l’inegalite de Chen-Wilmore, C.R. Acad. Sci. Paris 288(1979), 993-995.