  In this paper, we discuss the concepts of bifunction and geodesic convexity
for vector valued functions on Hadamard manifold. The Hadamard manifold is a
particular type of Riemannian manifold with non-positive sectional curvature.
Using bifunction, we introduce a definition of generalized geodesic convexity
in the context of the Hadamard manifold. To support the definition, we
construct a non-trivial example that demonstrates the property of geodesic
convexity on Hadamard manifold. Additionally, we define the geodesic
$h$-convexity, geodesic $h$-pseudoconvexity and geodesic $h$-quasiconvexity for
vector valued function using bifunction and study their several properties.
Furthermore, we demonstrate the uniqueness of the solution for nonsmooth vector
variational inequality problem (NVVIP) and prove the characterization property
for the solution of NVVIP and the Minty type NVVIP (MNVVIP) on Hadamard
manifold in terms of bifunction. Afterward, we consider a nonsmooth vector
optimization problem (NVOP) and investigate the relationships among the
solutions of NVOP, NVVIP, and MNVVIP.
