# ON THE STABILITY OF IMMERSED MANIFOLDS IN $E^4$

• Abdel-all, Nassar H. ;
• Hussein, Rawya A.
• Published : 1999.07.01
• 28 4

#### Abstract

This work is concerned mainly with the variational problem on an immersion x:M $\rightarrow$} $E^4$ . A new approach is introduced depends on the normal variation in any arbitrary normal direction in the normal bundle. The results of this work are considered as a continuation and an extension to that obtained in [1], [2] and [3], [4] respectively. The methods adapted here are based on Cartan's methods of moving frames and the calculus of variations.

#### Keywords

stability;immersion;normal variation

#### References

1. Total mean curvature and submanifolds of finite type Chen, B.Y.
2. Osaka J. Math. no.4 Stability of surface with constant mean curvature in 3-dimensional Riemannian Manifolds Sakaki, M.
3. Journal v.27 On a problem of Chem Joell, Weiner;Willmore(et al.)
4. J. Univ. Kuwait Sci. v.8 Generalised conformal mappings Svec, A.
5. Some questions about theory of surfaces in elliptic spase Pogorolov, A.V.
6. J. Amer. of Math. VXCIV no.3 On the total curvature of immersed manifolds, Ⅱ: Mean curvature and Length of second fundamental form Chen, B.Y.
7. Kodal Math. SEM. REP v.25 Submanifolds umbilical with respect ot a nonparallel normal subbundle Chen, B.Y.;Kentaro Yano
8. Foundations of differential Geometry v.Ⅱ Kobayashi, S.;Nomizu, N.
9. J. London Math. Sco. v.6 no.2 On a variational problem on gypersurfaces Chen, B.Y.
10. Bulletin Faculty of Science v.24(2-C) Soliman, M.A.;Abdel-All, N.H.;Rawya A.;Hussien,
11. Quart. J. Math. v.26 no.2 On Stable Submanifolds with parallel mean curvature Chen, B.Y.;Houh, C.S.