  In this paper, we prove that there exists a $C^{\infty}$-Calabi-Yau structure
on the whole of the complexification $G^{\mathbb C}/K^{\mathbb C}$ of a rank
two symmetric space $G/K$ of compact type. The proof is performed by deriving a
relation (which differs from the known relation somewhat) between the complex
Hessian of a real-valued $C^{\infty}$-function $\psi$ on $G^{\mathbb
C}/K^{\mathbb C}$ and the Hessian of the function $\rho:=\psi|_{{\rm
Exp}_o(\mathfrak a)}\circ{\rm Exp}_o|_{\mathfrak a}$ on a maximal abelian
subspace $\mathfrak a$ of the normal space $T_o^{\perp}(G\cdot o)(=T_o(G^d\cdot
o)\subset\mathfrak g^d)$ of the orbit $G\cdot o(=G/K)$ in $G^{\mathbb
C}/K^{\mathbb C}$ at the base point $o$, where ${\rm Exp}_o$ is the exponential
map of $G^{\mathbb C}/K^{\mathbb C}$ at $o$ and $\mathfrak g^d$ is the Lie
algebra of the dual $G^d$ of $G$. This relation is derived in a new calculation
method by using the explicit descriptions of the shape operators of the orbits
of the isotropy action $K\curvearrowright G^d/K$ and the Hermann type action
$G\curvearrowright G^{\mathbb C}/K^{\mathbb C}$.
