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Lee, J.M. (2012) Introduction to Smooth Manifolds. Springer.
https://doi.org/10.1007/978-1-4419-9982-5

has been cited by the following article:

• TITLE: A Unified Geometric and Energetic Framework for Deep Neural Networks via RKHS Embeddings

  AUTHORS: Heriony Rapelanoro-Rabenja

  KEYWORDS: RKHS Geometry, Riemannian Learning, Extrinsic Curvature, Mean Curvature, Geometric Regularization, Energy-Based Models

  JOURNAL NAME: Applied Mathematics, Vol.17 No.8, August 28, 2026

  ABSTRACT: This article develops a unified geometric and energetic framework for the analysis of deep neural networks, based on embedding the output manifold into a Reproducing Kernel Hilbert Space (RKHS). This embedding induces a natural Riemannian metric, a Levi-Civita connection, a second fundamental form, and a mean curvature vector, allowing the construction of a complete geometric energy model. We show how these tools lead to intrinsic learning dynamics, coherent geometric regularization, and physically interpretable energy flows. Experiments demonstrate improvements in stability, robustness, and generalization.