The fundamental building block in many learning models is the distance measure that is used. Usually, the linear distance is used for simplicity. Replacing this stiff distance measure with a flexible one could potentially give a better representation of the actual distance between two points. I will present how the normal distribution changes if the distance measure respects the underlying structure of the data. In particular, a Riemannian manifold will be learned based on observations. The geodesic curve can then be computed—a length-minimizing curve under the Riemannian measure. With this flexible distance measure we get a normal distribution that locally adapts to the data. A maximum likelihood estimation scheme is provided for inference of the parameters mean and covariance, and also, a systematic way to choose the parameter defining the Riemannian manifold. Results on synthetic and real world data demonstrate the efficiency of the proposed model to fit non-trivial probability distributions.