  Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal
H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the
$A$-inner product. This paper studies the geometry of the set $$ {\cal
I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It
is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of
adjointable operators, and a homogeneous space of the group of invertible
operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth
curves in ${\cal I}_A^a$ with given initial conditions, which are minimal for
the metric induced by $\langle \ , \ \rangle_A$, are presented. This result
depends on an adaptation of M.G. Krein's extension method of symmetric
contractions, in order that it works also for symmetrizable transformations
(i.e., operators which are selfadjoint for the $A$-inner product).
