Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus
of (G). A torus manifold with (G)-action is defined to be a smooth connected
closed oriented manifold of dimension (2\dim T) with an almost effective action
of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold
(M) with (G)-action then the action of a finite covering group of (G) factors
through (\tilde{G}=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod
SO(2l_i)\times T^{l_0}). The action of (\tilde{G}) on (M) restricts to an
action of (\tilde{G}'=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod
U(l_i)\times T^{l_0}) which has the same orbits as the (\tilde{G})-action.
We define invariants of torus manifolds with (G)-action which determine their
(\tilde{G}')-equivariant diffeomorphism type. We call these invariants
admissible 5-tuples. A simply connected torus manifold with (G)-action is
determined by its admissible 5-tuple up to (\tilde{G})-equivariant
diffeomorphism. Furthermore we prove that all admissible 5-tuples may be
realised by torus manifolds with (\tilde{G}")-action where (\tilde{G}") is a
finite covering group of (\tilde{G}').Comment: 56 pages; a mistake in section 6 corrected; accepted for publication
in Trans. Am. Math. So
