  We prove that the automorphism group of an odd dimensional Calabi-Yau
manifold of Picard number two is always a finite group. This makes a sharp
contrast to the automorphism groups of K3 surfaces and hyperk\"ahler manifolds
and birational automorphism groups, as we shall see. We also clarify the
relation between finiteness of the automorphism group (resp. birational
automorphism group) and the rationality of the nef cone (resp. movable cone)
for a hyperk\"ahler manifold of Picard number two. We will also discuss a
similar conjectual relation together with exsistence of rational curve,
expected by the cone conjecture, for a Calabi-Yau threefold of Picard number
two,

Algebraic Geometry