This study applies information geometry of normal distribution to model Japanese vowels on the basis of the first and second formants. The distribution of Kullback-Leibler (KL) divergence and its decomposed components were investigated to reveal the statistical invariance in the vowel system. The results suggest that although significant variability exists in individual KL divergence distributions, the population distribution tends to converge into a specific log-normal distribution. This distribution can be considered as an invariant distribution for the standard-Japanese speaking population. Furthermore, it was revealed that the mean and variance components of KL divergence are linearly related in the population distribution. The significance of these invariant features is discussed.