  A torus manifold is a closed smooth manifold of dimension $2n$ having an
effective smooth $T^n = (S^1)^n$-action with non-empty fixed points. Petrie
\cite{petrie:1973} has shown that any homotopy equivalence between a complex
projective space $\CP^n$ and a torus manifold homotopy equivalent to $\CP^n$
preserves their Pontrjagin classes. A \emph{generalized Bott manifold} is a
closed smooth manifold obtained as the total space of an iterated complex
projective space bundles over a point, where each fibration is a
projectivization of the Whitney sum of a finite many complex line bundles. For
instance, we obtain a product of complex projective spaces if all fibrations
are trivial. If each fiber is $\CP^1$, then we call it an (ordinary) \emph{Bott
manifold}. In this paper, we investigate the invariance of Pontrjagin classes
for torus manifolds whose cohomology ring is isomorphic to that of generalized
Bott manifolds. We show that any cohomology ring isomorphism between two torus
manifolds whose cohomology ring is isomorphic to that of a product of
projective spaces preserves their Pontrjagin classes, which generalizes the
Petrie's theorem. In addition, we show that any cohomology ring isomorphism
between two torus cohomology Bott manifolds preserves their Pontrjagin classes.
As a corollary, there are at most a finite number of torus manifolds homotopy
equivalent to either a given product of complex projective space or a given
Bott manifold.
