A fundamental task in data exploration is to extract low dimensional representations that capture intrinsic geometry in data, especially for faithfully visualizing data in two or three dimensions. Common approaches use kernel methods for manifold learning. However, these methods typically only provide an embedding of the input data and cannot extend naturally to new data points. Autoencoders have also become popular for representation learning. While they naturally compute feature extractors that are extendable to new data and invertible (i.e., reconstructing original features from latent representation), they often fail at representing the intrinsic data geometry compared to kernel-based manifold learning. We present a new method for integrating both approaches by incorporating a geometric regularization term in the bottleneck of the autoencoder. This regularization encourages the learned latent representation to follow the intrinsic data geometry, similar to manifold learning algorithms, while still enabling faithful extension to new data and preserving invertibility. We compare our approach to autoencoder models for manifold learning to provide qualitative and quantitative evidence of our advantages in preserving intrinsic structure, out of sample extension, and reconstruction. Our method is easily implemented for big-data applications, whereas other methods are limited in this regard.