More on tight isometric immersions of codimension two

Chen, Chang Shing

doi:10.1090/S0002-9939-1973-0375169-1

More on tight isometric immersions of codimension two
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by Chang Shing Chen

Proc. Amer. Math. Soc. 40 (1973), 545-553

DOI: https://doi.org/10.1090/S0002-9939-1973-0375169-1

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Abstract:

We continue our investigation on tight isometric immersion of a nonnegatively curved compact manifold ${M^n}$ into ${R^{n + 2}}$. Under some minor restrictions, we prove that the immersion is a product embedding of convex hypersurfaces. For surfaces in ${R^4}$, the restrictions are unnecessary.

References

C. S. Chen, On tight isometric immersion of codimension two, Amer. J. Math. 94 (1972), 974–990. MR 375168, DOI 10.2307/2373561
Shiing-shen Chern and Nicolaas H. Kuiper, Some theorems on the isometric imbedding of compact Riemann manifolds in euclidean space, Ann. of Math. (2) 56 (1952), 422–430. MR 50962, DOI 10.2307/1969650
Shiing-shen Chern and Richard K. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79 (1957), 306–318. MR 84811, DOI 10.2307/2372684

Tight immersions in higher dimensions

Nicolaas H. Kuiper, Minimal total absolute curvature for immersions, Invent. Math. 10 (1970), 209–238. MR 267597, DOI 10.1007/BF01403250
Nicolaas H. Kuiper, Tight topological embeddings of the Moebius band, J. Differential Geometry 6 (1971/72), 271–283. MR 314057
John A. Little and William F. Pohl, On tight immersions of maximal codimension, Invent. Math. 13 (1971), 179–204. MR 293645, DOI 10.1007/BF01404629
John Douglas Moore, Isometric immersions of riemannian products, J. Differential Geometry 5 (1971), 159–168. MR 307128
Barrett O’Neill, Isometric immersion of flat Riemannian manifolds in Euclidean space, Michigan Math. J. 9 (1962), 199–205. MR 152970