Abstract
For an oriented space-like surface M in a four-dimensional indefinite space form\({R^4_2(c)}\), there is a Wintgen type inequality; namely, the Gauss curvature K, the normal curvature K D and mean curvature vector H of M in \({R^4_2(c)}\) satisfy the general inequality: \({K+K^D \geq \langle H,H \rangle+c}\). An oriented space-like surface in \({R^4_2(c)}\) is called Wintgen ideal if it satisfies the equality case of the inequality identically. In this paper, we study Wintgen ideal surfaces in \({R^4_2(c)}\) . In particular, we classify Wintgen ideal surfaces in \({R^4_2(c)}\) with constant Gauss and normal curvatures. We also completely classify Wintgen ideal surfaces in \({\mathbb E^4_2}\) satisfying |K| = |K D| identically.
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References
Byrd P.F., Friedman M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer, New York, Heidelberg (1971)
Chen B.Y.: Geometry of Submanifolds. Mercer Dekker, New York (1973)
Chen B.Y.: Total Mean Curvature and Submanifolds of Finite Type. World Scientific, New Jersey (1984)
Chen B.Y.: Exact solutions of a class of differential equations of Lamé’s type and its applications to contact geometry. Rocky Mount. J. Math. 30, 497–506 (2000)
Chen B.Y.: A minimal immersion of hyperbolic plane in neutral pseudo-hyperbolic 4-space and its characterization. Arch. Math. 94, 291–299 (2010)
Chen B.Y.: Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann. Global Anal. Geom. 38, 145–160 (2010)
Chen B.Y.: Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension. J. Geom. Phys. 60, 260–280 (2010)
Chen B.Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications. World Scientific Publ., Hackensack, New Jersey (2011)
Chen, B.Y., Suceava, B.D.: Classification theorems for space-like surfaces in 4-dimensional indefinite space forms with index 2. Taiwanese J. Math. 15 (2011) (in press)
Decu S., Petrović–Torgašev M., Verstraelen L.: On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds. Tamkang J. Math. 41, 109–116 (2010)
Guadalupe I.V., Rodriguez L.: Normal curvature of surfaces in space forms. Pacific J. Math. 106, 95–103 (1983)
Kenmotsu K.: Minimal surfaces with constant curvature in 4-dimensional space forms. Proc. Am. Math. Soc. 89, 133–138 (1983)
Lawden D.F.: Elliptic Functions and Applications. springer, Berlin (1989)
O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1982)
Petrović-Torgašev M., Verstraelen L.: On Deszcz symmetries of Wintgen ideal submanifolds. Arch. Math. (Brno). 44, 57–67 (2008)
Sasaki M.: Spacelike maximal surfaces in 4-dimensional space forms of index 2. Tokyo J. Math. 25, 295–306 (2002)
Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288 993–995 (1979)
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Chen, BY. Wintgen Ideal Surfaces in Four-dimensional Neutral Indefinite Space Form \({R^4_2(c)}\) . Results. Math. 61, 329–345 (2012). https://doi.org/10.1007/s00025-011-0119-8
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DOI: https://doi.org/10.1007/s00025-011-0119-8