I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$ (where $Isom_{h}(X)$ means the isometries of $h$ that preserves orientation)? What are the obstructions? Any ideas on how to measure this? Is this true at least for Riemann surfaces?
Thanks in advance.
EDIT: As a friend of mine has just suggested, what about the complexification $Isom_{h}(X)^{\mathbb{C}}$? Will $Isom_{h}(X)^{\mathbb{C}} \cong Aut(X)$ for some special class of complex manifolds?