tokens: We introduce the Manifold Geometry Matching Generative Adversarial Network ( MGM GAN ) , which adds two novel mechanisms to facilitate GANs sampling from the geometry of the manifold rather than the density and then aligning two manifold geometries : ( 1 ) an importance sampling technique that reweights points based on their density on the manifold , making the discriminator only able to discern geometry and ( 2 ) a penalty adapted from traditional manifold alignment literature that explicitly enforces the geometry to be preserved .