Remark on a conjecture of conformal transformations of Riemannian manifolds

Details Remark on a conjecture of conformal transformations of Riemannian manifolds Remark on a conjecture of conformal transformations of Riemannian manifolds

2004-12-17

arxiv.org/abs/math/0412341v1

Mathematics

Differential Geometry

7 pages. to appear in Rendiconti del Circolo Mat. di Palermo

Scientific paper

Ejiri gave a negative answer to a conjecture of Lichnerowicz concerning Riemannian manifolds with constant scalar curvature admitting an infinitesimal non isometric conformal transformation. With this aim he constructed a warped product of a circle of lenght $T$ and a compact manifold. But he omitted in his analysis the condition that $T$ must to be big enough. Here we give an explicit sharp bound $T_0 < T$ that will make the proof complete. Our presentation is self-contained and mainly uses bifurcation techniques. Moreover, we show that there are other such examples and contribute some results to the classification of these manifolds.

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