Mathematical Foundations of Artificial Intelligence Basics of Manifold Theory (Momiao Xiong)（Z-Library）

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Mathematical Foundations of Artificial Intelligence Mathematical Foundations of Artificial Intelligence: Basics of Manifold Theory is the first volume in a two‑part series. Together, they establish a unifying mathematical framework based on smooth manifold theory and Riemannian geometry—essential tools for representing, analyzing, and integrating the growing complexity of modern artificial intelligence (AI) systems and scientific models. Differential geometry now plays a central role across AI, biology, physics, and medi‑ cine. From deep learning, generative modeling, and manifold learning to reasoning algo‑ rithms and physical AI, manifolds offer a coherent geometric language that bridges theory and practice. This volume introduces key concepts—topological and smooth manifolds, Riemannian metrics, differential forms, Lie derivatives, and statistical geometry—along‑ side illustrative applications to data science, genomics, drug discovery, and AI‑driven systems. Unlike traditional texts, this book combines rigor with intuition, integrating formal theory, computational methods, and interdisciplinary insights, and is ideal for graduate students and professionals in mathematics, statistics, computer science, AI, physics, bioin‑ formatics, and biomedical sciences. It also serves as a foundational reference for research‑ ers developing AI systems grounded in geometry, scientific modeling, and data‑driven discovery. Key Features • Unifies core manifold concepts to support integrated thinking across disciplines • Treats manifolds as natural geometric domains for data representation in AI and the sciences • Bridges abstract theory with practical algorithms and real‑world applications • Develops Lie derivative aware graphical neural networks for adaptive‑AI and molecular property prediction • Develops Lie derivative enhanced reaction‑diffusion equations for disease gene identification and treatment design • Develops probabilistic modeling and information geometry for modern learning systems • Applies geometric insight to AI fields, including generative models, graph learn‑ ing, and reasoning • Applies the Gauss map and Chen–Gauss–Bonnet theorem to physical AI incor‑ porating geometric constraints for robotics and tumor cell location and range identification

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• Features step‑by‑step examples, case studies, and visual explanations to support understanding • Serves as an advanced educational and skill‑building resource in the age of AI, leveraging the capabilities of emerging AI tools for automatic programming and self‑study Momiao Xiong is a retired Professor in the Department of Biostatistics and Data Science, University of Texas School of Public Health, and a regular member of the Genetics & Epigenetics (G&E) Graduate Program at The University of Texas MD Anderson Cancer Center, UTHealth Graduate School of Biomedical Science. He is President of the Society of Artificial Intelligence Research.

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Mathematical Foundations of Artificial Intelligence Basics of Manifold Theory Momiao Xiong

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Designed Cover Image: Momiao Xiong First edition published 2026 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2026 Momiao Xiong Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.co.uk For Product Safety Concerns and Information please contact our EU representative GPSR@taylorandfrancis.com. Taylor & Francis Verlag GmbH, Kaufingerstraße 24, 80331 München, Germany Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 9781041076254 (hbk) ISBN: 9781041076278 (pbk) ISBN: 9781003641452 (ebk) DOI: 10.1201/9781003641452 Typeset in Palatino by codeMantra

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v Contents Preface ............................................................................................................................................ xii Author Biography ........................................................................................................................ xvi 1. Smooth Manifold....................................................................................................................1 1.1 Introduction ...................................................................................................................1 1.1.1 Manifold ............................................................................................................1 1.2 Charts ..............................................................................................................................2 1.2.1 Basics of Charts ................................................................................................2 1.2.2 Method to Compute a Chart ..........................................................................4 1.2.3 Transition Map .................................................................................................5 1.3 Smooth Functions and Smooth Maps ........................................................................8 1.3.1 Smooth Functions ............................................................................................8 1.3.2 Smooth Maps .................................................................................................. 10 1.4 Manifolds with Boundary ......................................................................................... 11 1.5 Finite‑Dimensional Vector Spaces ............................................................................ 12 1.6 Open Submanifolds .................................................................................................... 14 1.7 Regular Value Theorem ............................................................................................. 17 1.8 Smooth Manifold Chart Lemma............................................................................... 20 1.9 Grassmann Manifolds: Definition and Properties .................................................22 1.9.1 Quotient Space................................................................................................22 1.9.2 Grassmann Manifold ....................................................................................23 1.9.3 Chaos Game Representation (CGR), Frequency Chaos Game Representation (FCGR) and Grassmannians ............................................. 27 1.9.4 Extension of CGR for SNP Genotype Data ................................................28 Appendix 1A: Grassmann Manifold Representation ....................................................... 37 Conclusion .............................................................................................................................. 39 Exercises .................................................................................................................................. 39 References ...............................................................................................................................40 2. Riemannian Geometry ........................................................................................................ 41 2.1 Riemannian Metrics ................................................................................................... 41 2.2 Local Expression of Riemannian Metrics ................................................................44 2.3 Frames and Dual Coframes .......................................................................................45 2.4 Dual Coframe under a Change of Frame ................................................................45 2.5 Riemannian Metric in Tensor Notation ...................................................................46 2.5.1 Tensor Product ...............................................................................................46 2.6 Riemannian Metric Under a Change of Frame ......................................................48 2.7 Riemannian Submanifolds and Isometric Immersions ........................................ 49 2.8 Product Riemannian Manifolds ...............................................................................50 2.9 Real Hyperbolic Space ................................................................................................ 51 2.9.1 The Lorentzian Inner Product ..................................................................... 51 2.9.2 Hyperboloid Model (Lorentzian Model) .................................................... 51 2.9.3 Poincaré Disk Model .....................................................................................53

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vi Contents 2.9.4 Poincaré Disk Model of Hyperbolic Geometry and Its Use in Phylogenetics ................................................................................58 2.9.5 Exponential Map ............................................................................................66 2.9.6 Logarithmic (Inverse Exponential) Map .................................................... 71 2.9.7 Exponential and Logarithmic Maps for Poincaré Ball Model ................. 73 2.9.8 Hyperbolic Message Passing Networks (HMPNs) ................................... 74 2.10 Spheres .......................................................................................................................... 79 2.10.1 Definition ........................................................................................................ 79 2.10.2 Coordinates ..................................................................................................... 79 2.10.3 Curvatures (All Radii = 1) ..............................................................................80 2.10.4 The Special Case 3S ........................................................................................80 Exercises ..................................................................................................................................80 References ............................................................................................................................... 81 3. Differential Forms ................................................................................................................ 82 3.1 Introduction ................................................................................................................. 82 3.2 Differential 1‑Forms in n .........................................................................................84 3.2.1 Introduction ....................................................................................................84 3.2.2 Differential 1‑Form Associated with the Force Field ................................85 3.2.3 The Integration of a 1‑Form Along a Smooth Curve ................................86 3.3 An n‑Form on n .........................................................................................................88 3.3.1 Definition of an n‑Form on n .....................................................................88 3.3.2 Explanation ..................................................................................................... 89 3.3.3 Properties of n‑Forms .................................................................................... 89 3.3.4 Intuitive Understanding ............................................................................... 89 3.4 Volume and Determinant ..........................................................................................90 3.4.1 Mathematical Definition of Cross Product ................................................90 3.4.2 Properties of the Cross Product ................................................................... 91 3.4.3 Geometric Interpretation .............................................................................. 91 3.4.4 Applications .................................................................................................... 91 3.4.5 Relationship between the Cross Product and the Wedge Product ......... 92 3.5 Differential k‑Forms in n ......................................................................................... 95 3.5.1 Basics of Differential Forms ......................................................................... 95 3.5.2 Differential k‑Forms in n ............................................................................ 96 3.5.3 Exterior Derivative ......................................................................................... 98 3.6 Mathematical Formulas for Integrating Differential Forms ............................... 100 3.6.1 Line Integrals: Integrate 1‑Forms over Curves ........................................ 101 3.6.2 Higher‑Dimensional Integrals: Integrate k‑Forms over k‑Dimensional Manifolds ........................................................................... 106 3.7 Stokes’ Theorem ........................................................................................................ 108 3.7.1 Classical Stokes’ Theorem .......................................................................... 108 3.7.2 The General Stokes’ Theorem on Manifolds ........................................... 111 3.8 The Poincaré Lemma and Poincaré Map: Explanation and Significance ........................................................................................................ 114 3.8.1 Introduction .................................................................................................. 114 3.8.2 The Poincaré Lemma ................................................................................... 114 3.8.3 The Poincaré Lemma ................................................................................... 120 3.9 The Divergence Theorem on Manifolds ................................................................ 127

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viiContents 3.9.1 Introduction .................................................................................................. 127 3.9.2 Mathematical Background ......................................................................... 127 3.9.3 Gauss–Green (Divergence) Theorem on a Riemannian Manifold ....... 131 3.10 The Laplace–Beltrami Operator on Riemannian Manifold ................................ 132 3.10.1 From Euclidean ∆ to the Laplace–Beltrami Operator ............................ 132 3.10.2 Laplace Equation on M ............................................................................... 132 3.10.3 Heat Equation on M .................................................................................... 133 3.11 The Laplace–Beltrami Operator on Hyperbolic Space ........................................ 133 3.11.1 Upper Half–Space Model UHPHn .................................................................. 133 3.11.2 Poincaré Disk Model diskHn .......................................................................... 134 3.11.3 Geodesic‑Polar Form of the Hyperbolic Laplacian ................................. 135 3.12 Laplace–Beltrami Operator on Hyperbolic Space to Reasoning Tree and Graphs ........................................................................................................ 135 3.12.1 Why Hyperbolic Hk + Laplace–Beltrami Is Natural for Trees ............... 135 3.12.2 Laplace–Beltrami Operator in the Poincaré Ball ..................................... 135 3.12.3 From Continuous ∆Η to a Discrete Hyperbolic Graph Laplacian ......... 135 3.13 Hyperbolic‑Laplacian Transformer (HL‑Former) for Reasons over Hierarchies, Trees and Other Negatively‑Curved Data ...................................... 136 3.13.1 Where to Inject LH Inside a Transformer .................................................. 137 3.13.2 Putting It Together—HL‑Former block .................................................... 137 3.13.3 Toy Reasoning Example—Ancestor Queries on a Binary Tree ............. 137 3.14 The Hyperbolic‑Laplacian Transformer in Reinforcement‑Learning (RL) ....... 138 3.14.1 Why Marry Laplacian + Transformer with RL? ....................................... 138 3.14.2 State Representation Pipeline ..................................................................... 138 3.14.3 HL‑Former Policy/Value Network ............................................................ 138 3.14.4 Toy Reasoning Task (Ancestry Environment) ......................................... 139 Appendix 3A: Product ωiX (Contraction of a Differential Form with a Vector Field) ...139 3A1 Definition ...................................................................................................... 139 3A2 Mathematical Formula for Calculation .................................................................. 140 Appendix 3B: Additional Information for Proving Poincare Lemma .......................... 142 Appendix 3C: Induced Surface Measure dSg on a Hypersurface ................................. 144 3C1 General Formula .......................................................................................... 144 3C2 Coordinate Restriction Formula .............................................................................. 144 Exercises ................................................................................................................................ 145 References ............................................................................................................................. 146 4. Lie Derivatives .................................................................................................................... 147 4.1 Introduction ............................................................................................................... 147 4.2 The Pullback .............................................................................................................. 147 4.2.1 Definition of Pullback ................................................................................. 148 4.2.2 Pullback of Differential 1‑Forms ............................................................... 150 4.2.3 General Algorithm for Computing the Pullback *ωf ............................ 151 4.2.4 General Definition of the Pullback *∅ Tt of a Tensor Field T by the Flow ∅t .......................................................................................................... 153 4.3 Lie Derivative ............................................................................................................. 156 4.3.1 Definition ...................................................................................................... 157 4.3.2 Lie Derivatives for Different Types of Tensor Fields .............................. 160 4.3.3 Lie Derivative of a General Tensor Field .................................................. 168 4.4 Applications ............................................................................................................... 177

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viii Contents 4.4.1 Practical Uses of Lie Derivatives................................................................ 177 4.4.2 Graph Neural Network (Message Passing Neural Networks) .............. 180 4.4.3 Temporal and Space Varying Graphs ....................................................... 193 Exercises ................................................................................................................................ 204 References ............................................................................................................................. 205 5. Advanced Topics in Riemannian Geometry ................................................................. 206 5.1 Riemannian Submanifolds and Fundamental Theorem .................................... 206 5.1.1 Riemannian Submanifolds ......................................................................... 206 5.1.2 Fundamental Theorem of Riemannian Geometry ................................. 207 5.1.3 Fundamental Forms and Principal Curvatures ...................................... 209 5.1.4 The Gauss Map in Differential Geometry ................................................ 214 5.1.5 The Principal Curvatures ........................................................................... 217 5.1.6 Gaussian Curvature ..................................................................................... 220 5.2 The Gauss–Bonnet Theorem (Compact Orientable Surface, No Boundary) ....222 5.2.1 The Gauss–Bonnet Theorem ......................................................................223 5.2.2 Extended Gauss–Bonnet Formula ............................................................. 226 5.2.3 Chern‑Gauss–Bonnet Theorem .................................................................228 5.3 Jacobi Fields ................................................................................................................229 5.3.1 Variations of Geodesics and the Definition of a Jacobi Field .................229 5.3.2 The Jacobi Equation .....................................................................................230 5.3.3 Applications for Jacobi Equation ...............................................................238 Appendix 5A: The Gauss Map and Spherical Geometry for a Geodesic Triangle ..... 239 Appendix 5B: The Commutation Formula for Covariant Derivatives ........................ 242 5B1 Preliminaries ............................................................................................................. 243 5B1.1 Covariant Derivative ................................................................................... 243 5B1.2 Lie Bracket ..................................................................................................... 243 5B1.3 Riemann Curvature Tensor ........................................................................ 243 Exercises ................................................................................................................................253 References .............................................................................................................................254 6. Statistical Theory on Manifolds ......................................................................................255 6.1 Introduction ...............................................................................................................255 6.1.1 Statistical Theories on Manifolds ..............................................................255 6.1.2 Asymptotic Theory for Statistics on Manifolds ......................................255 6.1.3 Sampling Methods on Manifolds ..............................................................256 6.1.4 Limitations and Future Research Challenges .........................................256 6.2 Sample Statistics on Manifold .................................................................................256 6.2.1 The Fréchet (Intrinsic) Mean and Karcher Mean .................................... 257 6.2.2 Empirical Distribution Functions on Manifolds ..................................... 257 6.2.3 Variance of the Empirical Distribution on a Manifold ...........................258 6.2.4 Central Limit Theorem for the Fréchet Mean ..........................................258 6.3 Two‐Sample Tests Designed for Data on a Riemannian Manifold .................... 259 6.3.1 Tangent Space Two‑Sample Test (Hotelling’s T2 Test on a Manifold) .. 259 6.3.2 Energy Distance‑Based Two‑Sample Test on a Manifold ...................... 261 6.4 Probability Distributions on Manifold .................................................................. 262 6.4.1 Derivation of Probability Distribution Functions on Manifolds .......... 262 6.4.2 Binomial Distribution on Manifold .......................................................... 263 6.4.3 Negative Binomial Distribution on Manifold .......................................... 265

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ixContents 6.4.4 Poisson Distribution on Manifold ............................................................. 267 6.4.5 Exponential Families on Manifold ............................................................ 268 6.4.6 Riemannian Normal Distribution ............................................................. 270 6.4.7 Distribution Theory for Transformation of Random Vectors on Manifold .................................................................................... 272 6.4.8 Chi‑Square Distribution on Manifold ......................................................277 6.4.9 Two‐Way Contingency Tables on Manifold ............................................. 278 6.5 Frequency Chaos Game Representation (FCGR) Matrix and Multi‑locus Genetic Association Studies ....................................................................................284 6.5.1 From Genotypes to FCGR and Contingency Tables ...............................285 6.5.2 Viewing the Data in a Grassmannian Framework .................................285 6.5.3 Testing Association Using × n2 Contingency Table Statistics ...............285 6.5.4 Type 1 Error Rates ....................................................................................... 287 6.6 Manifold Diffusion Variational Autoencoder (MD‑VAE) ...................................288 6.6.1 Model Architecture ..................................................................................... 289 6.6.2 Training Objective ....................................................................................... 290 6.6.3 Latent Density Modeling and Sampling .................................................. 290 6A1 Concept and Definition of Normal Coordinates .................................................. 299 6A1.1 Examples to Illustrate Normal Coordinates ............................................ 299 6A2 Affine Connection .....................................................................................................300 6A2.1 Introduction ..................................................................................................300 6A2.2 Directional Derivative .................................................................................300 6A2.3 In Coordinates .............................................................................................. 301 6A2.4 Definition of Affine Connection ................................................................303 6A2.5 Compare Vectors at Different Points .........................................................304 6A2.6 Connection Defines Covariant Derivative ...............................................304 6B1 The Riemannian Log Map ....................................................................................... 311 6B1.1 Definition ...................................................................................................... 311 6B1.2 Intuitive Explanation ................................................................................... 312 6C1 Defining Σ in the Tangent Space ............................................................................ 313 6C1.1 Sample Version ............................................................................................. 314 6C1.2 Connections to the Central Limit Theorem ............................................. 314 6D1 Vector Field ................................................................................................................ 315 6E1 The Riemann Curvature Tensor ............................................................................. 317 6E1.1 Lower Index (Covariant Index) and Upper Index (Contravariant Index) ............................................................................................................. 317 6E1.2 Contravariant Components ........................................................................ 317 6E1.3 Covariant Components , ,1( )…x xn ............................................................. 318 6E2 Metric Relation .......................................................................................................... 318 6E2.1 Lowering an Index and Raising an Index ................................................ 318 6E2.2 Rules for Lowering an Index ...................................................................... 319 6E2.3 The Riemann Curvature Tensor ................................................................ 320 6E2.4 First Derivative of the Christoffel Symbols .............................................. 321 6E3 Exponential Map and Jacobi Fields ........................................................................ 324 6E3.1 The Exponential Map and Riemann Normal Coordinates ................... 324 6E3.2 The Pullback Metric via the Exponential Map ........................................ 324 6E3.3 Expressing the Second Derivative in Terms of Jacobi Fields ................. 325 6E3.4 Sectional Curvature for Solving the Jacobi Equation ............................. 325 6E3.5 Taylor Expansion for Solving the Jacobi Equation .................................. 326

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x Contents 6E4 Volume Expansion .................................................................................................... 328 6E4.1 Definition of Volume on a Riemannian Manifold .................................. 328 6E4.2 The Metric Tensor Expansion .................................................................... 329 6E4.3 Deriving the Volume Element Expansion ................................................ 329 6E4.4 Examples .......................................................................................................330 6F1 Setup in Normal Coordinates .................................................................................330 6F2 Expansion of the Metric ........................................................................................... 331 6F3 Expansion of the Determinant ................................................................................ 331 6F4 Expansion of the Square‐Root ................................................................................. 331 6F5 Integrate in Polar Coordinates ................................................................................ 331 6G1 Cell‐Proportion Probabilities with Curvature Correction ..................................333 6G2 Multinomial Covariance without Curvature ........................................................334 6G3 Expansion with 0δ ≠ji ..............................................................................................334 6G4 Expressing ∆ ij in Terms of Curvature ....................................................................335 6H1 Asymptotic Normality of the Cell‐Proportion Vector .........................................335 6H2 Rewriting Pearson’s 2X as a Quadratic Form ........................................................336 6H3 Asymptotic Distribution via the Delta Method ...................................................336 6H4 Weighted Chi‐Square via Eigen‐Decomposition ..................................................336 6I1 Principal Angles between Two Subspaces ............................................................ 337 6I2 Geodesic Distance on Gr(k, n) .................................................................................. 337 6I3 Test Statistic ................................................................................................................338 6I4 Asymptotic Distribution via the Delta Method ...................................................338 6J1 Setting and Notation ................................................................................................ 339 6J2 Reverse–Time Coordinates ...................................................................................... 339 6J3 Generators and the Conditional Drift ....................................................................340 6J3.1 Forward Generator ......................................................................................340 6J3.2 Backward Conditional Expectation ..........................................................340 6J3.3 Insert a Conditional Expectation (Law of Total Expectation) ............... 341 6K1 Background and Intuition .......................................................................................343 6K2 Definition....................................................................................................................343 6K3 Fundamental Properties ..........................................................................................344 6K4 How to Compute Itô Integrals in Practice .............................................................345 6K4.1 Direct Isometry ............................................................................................345 6K4.2 Itô’s Formula/Integration by Parts ............................................................345 6K4.3 Product Rule (Itô’s Multiplication Table) ..................................................346 6L1 Introduction ...............................................................................................................346 6L2 An Itô Diffusion ........................................................................................................347 6L3 Definition of the Generator ......................................................................................347 6L3.1 Core Idea .......................................................................................................347 6L3.2 Explicit Formula ...........................................................................................347 6L4 Examples ....................................................................................................................349 6M1 Motivation ..................................................................................................................350 6M2 Computing t * by integration by parts ...................................................................350 6M3 Interpretation and Why the Adjoint Matters ........................................................ 351 6M3.1 Kolmogorov Forward/Fokker–Planck Equation .................................... 351 6M3.2 Coordinate Form .......................................................................................... 352 6M4 Conservation form ....................................................................................................353 6M5 Backwards Kolmogorov PDE ..................................................................................353 6M5.1 Define the Backward Value Function .......................................................353

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xiContents 6M5.2 Markov Property and Dynamic Programming Identity .......................353 6M5.3 Take an Infinitesimal Time Step ................................................................353 6N1 The Integrating‑Factor Trick for Linear SDEs .......................................................354 Exercises ................................................................................................................................ 356 References ............................................................................................................................. 357 Index ..................................................................................................................................... 359

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xii Preface In recent years, the remarkable advances in artificial intelligence (AI), especially deep learning and reasoning, have outpaced the development of a unified mathematical lan‑ guage to describe and analyze its diverse models and applications. As AI increasingly interacts with complex scientific domains—such as biology, physics, and medicine—there is a growing need for theoretical frameworks that can transcend individual algorithms and offer a common mathematical foundation. This book, Basics of Manifolds, is the first in a two‑part series designed to fill that need. Our central thesis is that smooth manifold theory and Riemannian geometry provide a unifying mathematical language for representing, analyzing, and integrating a wide variety of AI models and applications. By treating data, functions, models, and learning processes as geometric objects on manifolds, we gain the tools to understand AI not just as computation, but as geometry in action. This geometric viewpoint enables deeper insight into model structure, generalization, robustness, interpretability, and domain adaptation— properties of critical importance in the era of interdisciplinary AI. Like traditional calculus, which is the mathematical foundation of the traditional natu‑ ral sciences, differential manifold is not only the foundation of AI and machine creativity but also the mathematical foundation of modern all‑natural sciences, including physics, engineering, chemistry, molecular biology, medicine, and health care. Furthermore, recent developments in AI have demonstrated the potential for AI to contribute back to math‑ ematics and statistics by offering flexible algorithmic frameworks, powerful abstract rea‑ soning capabilities, and self‑adapting AI, which support various aspects of AI research and discovery in natural sciences. By embracing the smooth manifold framework and tools from Riemannian geometry, we gain a powerful, unifying viewpoint that can bring together various strands of AI, all‑natural science research, and their interactions. This book serves as the foundational volume. It introduces the key mathematical struc‑ tures underpinning modern geometric learning: topological and smooth manifolds, Riemannian metrics, differential forms, Lie derivatives, and the statistical geometry of curved spaces. Rather than developing these topics in abstract isolation, we emphasize intuitive motivations, computational tools, and real‑world relevance, including applica‑ tions in biological data modeling, hyperbolic representation learning, graph neural net‑ works, and manifold‑based statistical inference. The chapters are organized to balance mathematical rigor with accessibility: • Chapter 1 provides a foundational introduction to smooth manifolds, a central concept in modern geometry and its applications across mathematics, physics, and data science, with examples ranging from circles and spheres to projective and Grassmann spaces. We describe how local Euclidean structure enables cal‑ culus and coordinate transformations, forming the basis for smooth charts and atlases. Applications to geometry, matrix spaces, and even biological data repre‑ sentations, such as chaos game representations (CGR) of genomic sequences, are presented, demonstrating the manifold framework’s versatility. By the end, the reader gains both an intuitive and rigorous understanding of smooth manifolds, laying the groundwork for further study in differential geometry and its diverse applications.

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xiiiPreface • Chapter 2 provides a comprehensive introduction to Riemannian geometry—the study of metrics on manifolds, bridging classical differential geometry with mod‑ ern applications in data science and deep learning. Core concepts such as coordi‑ nate charts, tensor notation, pullbacks, and isometries are developed alongside visual illustrations and local expressions. Emphasis is placed on model spaces— Euclidean, spherical, and especially hyperbolic geometry—where the hyperboloid and Poincaré disk models are explored in depth. We explain tools such as the exponential map and logarithm map. Specialized topics include isometric immer‑ sions, exponential and logarithmic maps, and the construction of product mani‑ folds. The chapter culminates with practical applications to bioinformatics and geometric deep learning, including hyperbolic phylogenetic tree inference and Hyperbolic Message Passing Networks (HMPNs). These models exploit negative curvature to embed tree‑like and relational data with high fidelity, offering pow‑ erful tools for representation learning on complex biological and graph‑structured datasets. By connecting theoretical foundations to machine learning architectures, this chapter equips readers across disciplines to leverage Riemannian geometry in data‑driven discovery. • Chapter 3 introduces differential forms as a unifying mathematical framework for integration across diverse geometric objects—curves, surfaces, and manifolds— extending classical calculus into higher‑dimensional, coordinate‑independent set‑ tings and modern manifold calculus. Beginning with motivation from line and surface integrals, the chapter defines differential 1‑forms and general k‑forms, demonstrating how they encode quantities such as work, flux, and volume in a geometrically meaningful way. The wedge product and exterior derivative are introduced to build higher‑degree forms and perform generalized differentiation. Through concrete examples, including integration on spheres and tori, the chapter connects differential forms to physical and biological interpretations. Key theorems such as Stokes’ Theorem and the Poincaré Lemma are presented both in classical and manifold contexts, showing how topological structure influences exactness and conservation laws. The chapter also explores the diver‑ gence theorem and the Laplace–Beltrami operator, highlighting their relevance in Riemannian geometry and manifold‑based modeling. Applications include hyperbolic graph Laplacians, spectral graph theory, and the development of the Hyperbolic‑Laplacian Transformer (HL‑Former) for hierarchical reasoning, rein‑ forcement learning, and single‑cell space transcriptomics. • Chapter 4 introduces the Lie derivative as a geometric tool for quantifying how scalar, vector, or tensor fields change along flows generated by vector fields, extending classical scalar derivatives in classical calculus to modern general sca‑ lar, vector, and tensor derivatives. Rooted in differential geometry, the Lie deriva‑ tive captures dynamic evolution, symmetry, and directional change on manifolds, providing a foundation for analyzing continuous transformations in complex systems. Emphasis is placed on applications to dynamic systems, biology, and geometric deep learning. Use cases include modeling thought flows in cognitive graphs, assessing RNA‑velocity fields in single‑cell genomics, and understanding gene regulatory networks through reachability and control theory. The chapter culminates in the integration of Lie derivatives into message‑passing neural net‑ works, enhancing their geometric expressiveness and interpretability in drug dis‑ covery, molecular graph prediction, and tissue dynamics.

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xiv Preface • Chapter 5 surveys advanced tools that link local curvature to global structure in Riemannian geometry. It opens with the notion of Riemannian submanifolds, showing how an immersion inherits the ambient metric and motivating the Levi‑Civita connection via the fundamental theorem. Two quadratic forms—the first and second fundamental forms—lead to the shape operator, whose eigenval‑ ues are the principal curvatures; mean and Gaussian curvature follow immedi‑ ately. The Gauss map is introduced as a bridge between surface normals and the unit sphere, paving the way to the Gauss–Bonnet theorem and its extended and higher‑dimensional (Chern) versions, which tie the integral of curvature to the Euler characteristic and can be applied to physical AI. The second half develops Jacobi fields: variation vectors of geodesic families that satisfy the Jacobi (geode‑ sic‑deviation) equation, a cornerstone for understanding conjugate points, stabil‑ ity, and curvature comparison. Detailed derivations, coordinate formulas, and didactic examples—saddle, sphere, paraboloid, implicit surfaces—are backed by visual figures and step‑by‑step algorithms, while appendices supply full proofs of the Gauss–Bonnet triangle formula, curvature commutation, and Ricci‑tensor computations. Together, these topics furnish a coherent toolkit for analyzing cur‑ vature, topology, and geodesic behavior on smooth manifolds. • Chapter 6 explores statistical inference on manifolds, where classical tools must be redefined to respect the underlying non‑Euclidean geometry. The chapter begins with the Fréchet mean, a natural generalization of the Euclidean average, and examines its uniqueness, existence, and computation on curved spaces. It extends hypothesis testing frameworks to Riemannian manifolds, introducing Hotelling’s T2‑type statistics, tangent space approximations, and permutation methods for dis‑ tribution‑free inference. We also extend variable transformation and contingency table tests from Euclidean space to manifold space and develop manifold‑based contingency table tests for genome‑wide association studies, exploring genomic space features. Manifold‑based statistical analysis is presented through applications to shape spaces, directional data, symmetric positive definite matrices, and biological structures. Special focus is given to diffusion tensor imaging, phylogenetic tree spaces, and the geometry of covariance matrices. The chapter integrates tools from differential geometry, such as exponential and logarithmic maps, to enable computation and model comparison across curved spaces. It concludes with modern applications in machine learning and biomedi‑ cal domains, including the analysis of tumor morphologies and manifold‑valued neural representations. This chapter provides a rigorous yet practical foundation for statisticians, data scien‑ tists, and biologists working with data that naturally reside on nonlinear spaces, bridging geometry and inference for scientific discovery and data‑driven modeling. The second book in this series, Mathematical Foundations of Artificial Intelligence, builds upon this geometric base to address manifold‑based learning algorithms, geometric deep learning architectures, generative models, reinforcement learning in non‑Euclidean spaces, and AI reasoning systems grounded in differential geometry, in general, math‑ ematical foundation of AI. Our intended audience includes graduate students, researchers, and professionals in mathematics, statistics, computer science, AI, biology, medicine, and healthcare. Readers are assumed to have some background in multivariable calculus and linear

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xvPreface algebra, but no prior exposure to differential geometry is required. Numerous visual‑ izations, examples, and computational strategies are provided throughout to support intuition and real‑world modeling. In an era where scientific inquiry, intelligent computation, and data complexity are becoming deeply intertwined, a geometric perspective offers more than elegance—it offers unity. We hope this book not only illuminates the mathematical underpinnings of modern AI but also inspires its readers to think geometrically, reason topologically, and model intelligently. I am deeply grateful to my editor, David Grubbs, for his encouragement and patience during the process of creating this book. Momiao Xiong Houston, Texas June 2025

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xvi Author Biography Momiao Xiong is a retired Professor in the Department of Biostatistics and Data Science, University of Texas School of Public Health, and a regular member of the Genetics & Epigenetics (G&E) Graduate Program at The University of Texas MD Anderson Cancer Center, UTHealth Graduate School of Biomedical Science. He is President of the Society of Artificial Intelligence Research.

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DOI: 10.1201/9781003641452‑1 1 1 Smooth Manifold 1.1 Introduction In the realm of mathematics and theoretical physics, few concepts are as powerful and versatile as that of a manifold. A manifold is a topological space that, in many ways, behaves like a Euclidean space at a small scale but can exhibit vastly different character‑ istics when viewed as a whole. This dichotomy of local simplicity and global complexity makes manifolds a cornerstone of modern geometry and its applications. 1.1.1 Manifold The simplest manifolds are the topological manifolds, which are topological spaces with the following three properties. Definition 1.1: Topological Manifold An n‑dimensional topological manifold is a topological space M that satisfies: 1. Hausdorff: For any two distinct points ∈,p q M , there exist disjoint open neigh‑ borhoods U and V with ∈p U and ∈q V. 2. Second‑Countable: There exists a countable basis for the topology on M. 3. Locally Euclidean: For every point ∈p M , there exists an open neighborhood U of p and a homeomorphism (called a chart) ϕ → ⊂: ,U V n where V is an open subset of n. Example 1.1: A Topological Manifold The circle { }( )= ∈ + =RS x y x y, | 11 2 2 2 is a classic example. It is a 1‑dimensional space that is: • Hausdorff: Any two distinct points on the circle can be separated by disjoint open arcs. • Second‑Countable: It has a countable basis (for instance, the collection of open arcs with rational endpoints). • Locally Euclidean: Every point on S1 has a neighborhood that is homeomor‑ phic to an open interval in R.

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2 Mathematical Foundations of Artificial Intelligence Example 1.2: A Non–Topological Manifold The line with two origins is a standard example. It is constructed by taking two copies of R (say, R0 and R1) and identifying every point ≠x 0 with the corresponding point in R1. The resulting space has: • Local Euclidean Property: Away from the origin, each point has a neighbor‑ hood homeomorphic to R; even around the two distinct “origin” points, one can find neighborhoods that are locally R. • Failure of the Hausdorff Property: The two distinct origins cannot be sepa‑ rated by disjoint open sets, so the space is not Hausdorff. Thus, while the line with two origins is locally Euclidean, it fails to be a topo‑ logical manifold because it does not satisfy the Hausdorff condition. 1.2 Charts 1.2.1 Basics of Charts Definition 1.2: Chart A chart on an n‑dimensional manifold M is a pair ϕ( ), U consisting of: 1. An open set ⊆U M within the manifold. 2. A homeomorphism ϕ →: .U n In other words, a chart gives a way to describe a portion of the manifold using coordinates in an n‑dimensional Euclidean space. These coordinates, often writ‑ ten as ( )…,  ,  ,  1 2x x xn , serve as a local parameterization of that region of the mani‑ fold. By covering the entire manifold with a collection of such charts—known as an atlas—and ensuring smooth transitions where these charts overlap, one can extend familiar geometric and analytic notions from n to the manifold itself ϕ −( )1 . We need charts because they allow us to translate the abstract notion of a manifold—an object that may be curved and globally complicated—into familiar Euclidean coordinates locally. Manifolds by themselves are general topological spaces that do not inherently come with a notion of “straight” lines, angles, or standard coordinate systems. By introducing charts, we effectively create a local dictionary between the manifold and a piece of n. This serves several key purposes: 1. Local Euclidean Structure: Charts enable us to treat small regions of the manifold as if they were subsets of n. This allows us to apply the tools of calculus, such as differentiation and integration, which are well defined in Euclidean space. 2. Defining Smooth Structures: With charts, we can define what it means for functions on the manifold to be smooth by checking their compositions with the coordinate maps into n. Smoothness, derivatives, and similar concepts all rely on this framework.

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3Smooth Manifold 3. Constructing Atlases and Transition Maps: A single chart generally cov‑ ers only a portion of the manifold. By using multiple overlapping charts— collectively known as an atlas—we can cover the entire manifold. The “glue” between these charts, called transition maps, must be smooth. This global compatibility is what allows us to consistently extend geometric and analytic ideas throughout the manifold. In short, charts act as the fundamental tools that bridge the abstract world of manifolds with the familiar territory of Euclidean spaces, making it possible to apply classical mathematical techniques in these more general geometric settings. Example 1.3: Graphs of Continuous Functions The graph of a function →R Rf n: can form a manifold. • Example: Let →R Rf : be the function ( ) =f x x2. • Its graph is { }( ) ∈ =Rx y y x, |2 2 , which is a 1‑dimensional submanifold embedded in R2 (Figure 1.1) • The graph is smooth and differentiable. • General Case: For any smooth function →R Rf n m: , the graph is { }( )( ) ∈ ⊆ +R Rx f x x n n m, | . This is a manifold of dimension n. FIGURE 1.1 The graph of the function ( ) = 2f x x , which forms a smooth curve (a parabola) in the xy‑plane.