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Isometric of harmonic mappings of complete Riemannian manifolds. (English) Zbl 1075.53528

Pitiş, Gheorghe (ed.) et al., Proceedings of the 4th international workshop on differential geometry, Braşov, Romania, September 16–22, 1999. Braşov: Transilvania University Press (ISBN 973-9474-68-3). 56-61 (2000).
In the first part of the present paper the authors make some remarks on one of their previous papers [see Publ. Math. 55, No. 1–2, 211–219 (1999; Zbl 0955.53034)]. There the authors investigated isometric mappings \(\varphi\) of complete Riemannnian manifolds \(V^n=(M,g)\) into Euclidean space \(E^{n+m}\) (respectively, into the unit sphere \(S^{n+m-1}\subseteq E^{n+m})\), where \(\varphi (V^n)\)-cannot be pinched by certain geodesic balls \(B(r)\) of radius \(r:\varphi (V^n)\not\subseteq B(r)\subseteq E^{n+m}\) (respectively, \(\varphi (V^n)\not\subseteq B(r)\subseteq S^{n+m-1}\)).
In the present paper, the authors weaken a condition of the results obtained in the above mentioned paper and they point out that some of them are sharp. In the second part of the present paper, in the above described problem the authors replace the isometric mappings by harmonic ones and obtain a necessary condition in order that \(\varphi (V^n)\) cannot be pinched by any ball \(B(r)\) of \(E^{n+m}\) or, equivalently, \(\varphi (V^n)\) stretches out to infinity in \(E^{n+m}\).
For the entire collection see [Zbl 1066.53004].

MSC:

53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 0955.53034
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