Geodesic
A pointwise inequality in submanifold theory
Archivum mathematicum, Tome 35 (1999) no. 2, pp. 115-128 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We obtain a pointwise inequality valid for all submanifolds $M^n$ of all real space forms $N^{n+2}(c)$ with $n\ge 2$ and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of $M^n$, and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of $M^n$ in $N^m(c)$.
We obtain a pointwise inequality valid for all submanifolds $M^n$ of all real space forms $N^{n+2}(c)$ with $n\ge 2$ and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of $M^n$, and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of $M^n$ in $N^m(c)$.
Classification : 53C40
Keywords: submanofolds of real space froms; scalar curvature; normal curvature; mean curvature; inequality 
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De Smet, P. J.; Dillen, F.; Verstraelen, L.; Vrancken, L. A pointwise inequality in submanifold theory. Archivum mathematicum, Tome 35 (1999) no. 2, pp. 115-128. http://geodesic.mathdoc.fr/item/ARM_1999_35_2_a2/

[A] Abe, K.: The complex version of Hartman-Nirenberg cylinder theorem. J. Differ. Geom 7 (1972), 453–460. | MR

[B] Blair, D. E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509, Springer, Berlin, 1976. | MR | Zbl

[C1] Chen, B. Y.: Geometry of submanifolds. Marcel Dekker, New York, 1973. | MR | Zbl

[C2] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds. Archiv Math. 60 (1993), 568–578. | MR | Zbl

[C3] Chen, B. Y.: Mean curvature and shape operator of isometric immersions in real-space-forms. Glasgow Math. J. 38 (1996), 87–97. | MR | Zbl

[CDVV1] Chen, B. Y., Dillen, F., Verstraelen, L. and Vrancken, L.: Two equivariant totally real immersions into the nearly Kähler 6-sphere and their characterization. Japanese J. Math. 21 (1995), 207–222. | MR

[CDVV2] Chen, B.Y., Dillen, F., Verstraelen, L. and Vrancken, L.: Characterizing a class of totally real submanifolds of $S^6(1)$ by their sectional curvatures. Tôhoku Math. J. 47 (1995), 185–198. | MR

[CDVV3] Chen, B.Y., Dillen, F., Verstraelen, L. and Vrancken, L.: An exotic totally real minimal immersion of $S^3$ into $\mathbb{C}P^3$ and its characterization. Proc. Roy. Soc. Edinburgh 126A (1996), 153–165. | MR

[CDVV4] Chen, B.Y., Dillen, F., Verstraelen, L. and Vrancken, L.: Totally real minimal immersion of $\mathbb{C}P^3$ satisfying a basic equality. Arch. Math. 63 (1994), 553–564. | MR

[CY] Chen, B. Y., Yang, J.: Elliptic functions, Theta function and hypersurfaces satisfying a basic equality. Math. Proc. Camb. Phil. Soc. 125 (1999), 463–509. | MR

[Ch] Chern, S. S.: Minimal submanifolds in a Riemannian manifold. Univ. of Kansas, Lawrence, Kansas, 1968. | MR

[Da1] Dajczer, M., Florit, L. A.: A class of austere submanifolds. preprint.

[Da2] Dajczer, M., Florit, L. A.: On Chen’s basic equality. Illinois Jour. Math 42 (1998), 97–106. | MR

[DDVV] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L.: The normal curvature of totally real submanifolds of $S^6(1)$. Glasgow Mathematical Journal 40 (1998), 199–204. | MR

[DN] Dillen, F., Nölker, S.: Semi-parallelity, multi-rotation surfaces and the helix-property. J. Reine. Angew. Math. 435 (1993), 33–63. | MR

[DV] Dillen, F., Vrancken, L.: Totally real submanifolds in $S^6$ satisfying Chen’s equality. Trans. Amer. Math. Soc. 348 (1996), 1633–1646. | MR

[GR] Guadalupe, I. V., Rodriguez, L.: Normal curvature of surfaces in space forms. Pacific J. Math. 106 (1983), 95–103. | MR

[N] Nölker, S.: Isometric immersions of warped products. Diff. Geom. and Appl. (to appear). | MR

[O] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. | MR

[W] Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sc. Paris 288 (1979), 993– 995. | MR | Zbl

[YI] Yano, K., Ishihara, S.: Invariant submanifolds of an almost contact manifold. Kōdai Math. Sem. Rep. 21 (1969), 350– 364. | MR