# Article

Full entry | PDF   (0.3 MB)
Keywords:
submanofolds of real space froms; scalar curvature; normal curvature; mean curvature; inequality
Summary:
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)\$.
References:
[A] Abe, K.: The complex version of Hartman-Nirenberg cylinder theorem. J. Differ. Geom 7 (1972), 453–460. MR 0383307
[B] Blair, D. E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509, Springer, Berlin, 1976. MR 0467588 | Zbl 0319.53026
[C1] Chen, B. Y.: Geometry of submanifolds. Marcel Dekker, New York, 1973. MR 0353212 | Zbl 0262.53036
[C2] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds. Archiv Math. 60 (1993), 568–578. MR 1216703 | Zbl 0811.53060
[C3] Chen, B. Y.: Mean curvature and shape operator of isometric immersions in real-space-forms. Glasgow Math. J. 38 (1996), 87–97. MR 1373963 | Zbl 0866.53038
[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 1338360
[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 1329520
[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 1378838
[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 1300757
[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 1656829
[Ch] Chern, S. S.: Minimal submanifolds in a Riemannian manifold. Univ. of Kansas, Lawrence, Kansas, 1968. MR 0248648
[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 1492041
[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 1630238
[DN] Dillen, F., Nölker, S.: Semi-parallelity, multi-rotation surfaces and the helix-property. J. Reine. Angew. Math. 435 (1993), 33–63. MR 1203910
[DV] Dillen, F., Vrancken, L.: Totally real submanifolds in \$S^6\$ satisfying Chen’s equality. Trans. Amer. Math. Soc. 348 (1996), 1633–1646. MR 1355070
[GR] Guadalupe, I. V., Rodriguez, L.: Normal curvature of surfaces in space forms. Pacific J. Math. 106 (1983), 95–103. MR 0694674
[N] Nölker, S.: Isometric immersions of warped products. Diff. Geom. and Appl. (to appear). MR 1384876
[O] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. MR 0200865
[W] Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sc. Paris 288 (1979), 993– 995. MR 0540375 | Zbl 0421.53003
[YI] Yano, K., Ishihara, S.: Invariant submanifolds of an almost contact manifold. Kōdai Math. Sem. Rep. 21 (1969), 350– 364. MR 0248695

Partner of