The application of Auto-Encoder (AE) to multi-view representation learning has gained traction due to advancements in deep learning. While some current AE-based multi-view representation learning algorithms incorporate the geometric structure of the input data into their feature representation learning process, their use of a shallow structured graph regularization term can be restrictive when used in conjunction with deep models. Furthermore, current multi-view representation learning algorithms do not fully utilize the diversity and consistency presented in different views, leading to a reduction in the efficacy of feature learning. This paper introduces a novel approach, reconstructed graph constrained auto-encoders (RGCAE), for multi-view representation learning. Unlike existing methods, our approach incorporates deep adaptive graph regularization based on multi-layer perceptron to ensure the preservation of the geometric similarity graph, which is constructed based on the local invariance principle. By decoupling the feature representation learning from the preservation of the geometric structure among different views, our approach can better leverage the diversity presented in multi-view data. We obtain view-specific representations that preserve the geometric structure and then combine them by averaging to obtain a common representation. To ensure the consistency of the multi-view data, we minimize the loss between the view-specific and common representations. Consequently, our RGCAE approach can maintain the geometric structure of multi-view data and is better suited for integration with deep models. Extensive experiments on six datasets demonstrate that RGCAE obtained promising performance, compared with the state-of-the-art methods.