In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian \Delta _N, written {\text {VMO}}_{\Delta _N}({\mathbb {R}^n}). We first describe it with the classical {\text {VMO}}({\mathbb {R}^n}) and certain {\text {VMO}} on the half-spaces. Then we demonstrate that {\text {VMO}}_{\Delta _N}({\mathbb {R}^n}) is actually {\text {BMO}}_{\Delta _N}({\mathbb {R}^n})-closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. By means of the functional analysis, we also obtain the duality between certain {\text {VMO}} and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for {\text {BMO}} functions on the space of homogeneous type, which can be applied to our argument and otherwhere.

The first author acknowledges financial support from the Spanish Ministry of Science and Innovation, through the \Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554) and from the Spanish National Research Council, through the \Ayuda extraordinaria a Centros de Excelencia Severo Ochoa" (20205CEX001). He is also supported by China Postdoctoral Science Foundation (2018M643280).