Asymptotically harmonic manifolds with minimal horospheres.


SHAH, Hemangi


Let (M,g) be an asymptotically harmonic manifold with minimal horospheres. Let {ei} be an orthonormal basis of TpM and let bei be the corresponding Busemann functions on M. Then we show that :

(1) The vector space V ={bv | v ∈ TpM } is finite dimensional and dim V = dim M = n.

(2) F : M Rn defined by F(x) = (be1(x); be2(x); ; ben(x)) is an isometry and therefore, M is flat.

Thus, the flatness of
M is shown by using the strongest criterion.