2007 # Biharmonic maps between warped product manifolds

**Abstract:** Biharmonic maps between warped products are studied. The main results are:\ud
(i) the condition for the biharmonicity of the inclusion of a Riemannian manifold N into the warped product M ×f 2 theprojectionπ : M×f2 N → M;\ud
N and of\ud
￼(ii) the construction of two new classes of non-harmonic biharmonic maps using products of harmonic maps φ = 1M × ψ : M × N → M × N and warping the metric on their domain or codomain;\ud
(iii) the study of three classes of axially symmetric biharmonic maps, using the warped pr…

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“…The last part of this paper is devoted to the study of biharmonic maps between warped product manifolds, these maps were also studied in [17]. We will give some results of the biharmonicity in other particular cases.…”

confidence: 99%

“…The last part of this paper is devoted to the study of biharmonic maps between warped product manifolds, these maps were also studied in [17]. We will give some results of the biharmonicity in other particular cases.…”

confidence: 99%

“…(A) One can check (see also [BMO2]) that for the cases (i) p = 0, and (ii) p = m, the maps φ = x |x| p are actually harmonic maps. We know that in these cases these maps are f -biharmonic for any f .…”

confidence: 99%

“…In this case, (iii) and (iv) imply that φ = x |x| p is a proper biharmonic map if and only if p = −2, or p = m − 2. Note that the case p = −2 was missed in the list of Remark 5.8 in [BMO2]. (B) For p = 0, m, and k = 0, the maps in cases (iii) and (iv) provide infinitely many example of proper f -biharmonic maps (i.e., which are neither harmonic nor biharmonic maps).…”

confidence: 99%

“…New examples of proper-biharmonic maps were also obtained by using warped product manifolds (see [7]). …”

confidence: 99%