  A well-known theorem of W. Fischer and H. Grauert states that analytic fiber
spaces with all fibers isomorphic to a fixed compact connected complex manifold
are locally trivial. Motivated by this result, we show that if $k$ is an
algebraically closed field of infinite transcendence degree over its prime
field, then every smooth projective family over a reduced $k$-scheme of finite
type with isomorphic fibers having reduced automorphism group schemes is
locally trivial in the \'etale topology. We do so by reducing the problem to
the case when the base is a smooth integral curve, and then, using the
vanishing of the Kodaira-Spencer map, we prove formal triviality of such
families at every geometric point of the base. We also provide examples of
smooth projective fibrewise trivial families in positive characteristic whose
Kodaira-Spencer map are nowhere vanishing.
