  We study the tt*-geometry with vanishing endormorphism $\mathcal{U}$. Given
an integrable harmonic Higgs bundle $(E, h, \Phi, \mathcal{U},\mathcal{Q})$ on
a complex manifold $M$, Firstly we prove that, under the \emph{IS} condition,
vanishing $\mathcal{U}$ implies vanishing Higgs field $\Phi$ and the Chern
connection of the Hermitian Einstein metric $h$ is a holomorphic connection, so
the metric $h$ and $\mathcal{Q}$ are invariant. Secondly, without the \emph{IS}
condition, we show that vanishing $\mathcal{U}$ will imply vanishing Higgs
field $\Phi$ if we assume that the Chern connection of $h$ is a holomorphic
connection. Finally, we add real structure $\kappa$. Given any
\emph{CV}-structure, we prove that super-symmetric operator $\mathcal{Q}$ must
have $0$ as an eigenvalue when the underlying bundle has odd rank.
