| Title:
|
A pointwise inequality in submanifold theory (English) |
| Author:
|
De Smet, P. J. |
| Author:
|
Dillen, F. |
| Author:
|
Verstraelen, L. |
| Author:
|
Vrancken, L. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
35 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
115-128 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $N^m(c)$. (English) |
| Keyword:
|
submanofolds of real space froms |
| Keyword:
|
scalar curvature |
| Keyword:
|
normal curvature |
| Keyword:
|
mean curvature |
| Keyword:
|
inequality |
| MSC:
|
53C40 |
| idZBL:
|
Zbl 1054.53075 |
| idMR:
|
MR1711669 |
| . |
| Date available:
|
2008-06-06T22:22:46Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107688 |
| . |
| Reference:
|
[A] Abe, K.: The complex version of Hartman-Nirenberg cylinder theorem.J. Differ. Geom 7 (1972), 453–460. MR 0383307 |
| Reference:
|
[B] Blair, D. E.: Contact Manifolds in Riemannian Geometry.Lecture Notes in Mathematics, vol. 509, Springer, Berlin, 1976. Zbl 0319.53026, MR 0467588 |
| Reference:
|
[C1] Chen, B. Y.: Geometry of submanifolds.Marcel Dekker, New York, 1973. Zbl 0262.53036, MR 0353212 |
| Reference:
|
[C2] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds.Archiv Math. 60 (1993), 568–578. Zbl 0811.53060, MR 1216703 |
| Reference:
|
[C3] Chen, B. Y.: Mean curvature and shape operator of isometric immersions in real-space-forms.Glasgow Math. J. 38 (1996), 87–97. Zbl 0866.53038, MR 1373963 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[Ch] Chern, S. S.: Minimal submanifolds in a Riemannian manifold.Univ. of Kansas, Lawrence, Kansas, 1968. MR 0248648 |
| Reference:
|
[Da1] Dajczer, M., Florit, L. A.: A class of austere submanifolds.preprint. |
| Reference:
|
[Da2] Dajczer, M., Florit, L. A.: On Chen’s basic equality.Illinois Jour. Math 42 (1998), 97–106. MR 1492041 |
| Reference:
|
[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 |
| Reference:
|
[DN] Dillen, F., Nölker, S.: Semi-parallelity, multi-rotation surfaces and the helix-property.J. Reine. Angew. Math. 435 (1993), 33–63. MR 1203910 |
| Reference:
|
[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 |
| Reference:
|
[GR] Guadalupe, I. V., Rodriguez, L.: Normal curvature of surfaces in space forms.Pacific J. Math. 106 (1983), 95–103. MR 0694674 |
| Reference:
|
[N] Nölker, S.: Isometric immersions of warped products.Diff. Geom. and Appl. (to appear). MR 1384876 |
| Reference:
|
[O] O’Neill, B.: The fundamental equations of a submersion.Michigan Math. J. 13 (1966), 459–469. MR 0200865 |
| Reference:
|
[W] Wintgen, P.: Sur l’inégalité de Chen-Willmore.C. R. Acad. Sc. Paris 288 (1979), 993– 995. Zbl 0421.53003, MR 0540375 |
| Reference:
|
[YI] Yano, K., Ishihara, S.: Invariant submanifolds of an almost contact manifold.Kōdai Math. Sem. Rep. 21 (1969), 350– 364. MR 0248695 |
| . |
