Image classification is widely used in many fields. Traditional metric learning based classification methods always maximize between-class distances and minimize within-class distances based on features calculated from each individual. Different from traditional methods, this paper takes each class as a distribution and try to maximize the distances among different distributions using information geometry. In order to minimize the distance among individuals within a class, the paper assumes that each class follows a joint Gaussian distribution and takes an exploratory study on the relation between a within-class distance and the determinant of the covariance matrix of the distribution. It is found that under some assumptions, the average within-class distance among the same class is proportional to the standard deviation (for a random variable) or the product of standard deviations of each feature (for a random vector). As a result, the standard deviation (for a random variable) or the determinant of the covariance matrix (for a random vector) is used to substitute the within-class distance in the metric learning. The proposed method thereinafter saves a lot of computational cost. The method is then applied to person re-identification, which is a very important application in a 5G time, such as smart city. To our surprise, the proposed method is very competitive compared with many state-of-the-art methods while saving the computational cost in the learning stage. Experimental results demonstrate the effectiveness of the proposed method.