Application of Elementary Differential Geometry to Influence Analysis

With linear algebra and vector calculus as pre-requisites, the first part of this textbook presents an introduction to the geometry of graphs, encompassing the concepts of normal curvature, sectional curvature, Ricci curvature, and Gaussian curvature. The second part of the book provides background statistical concepts and basic models that form the fundamental knowledge necessary for better comprehension of the concept of local influence; while the third part focuses on the application of differential geometry to local influence analysis, and discusses when and how geometric concepts can be used in an effective manner to develop measures for assessing local influence. This textbook is intended for the use of senior undergraduate and graduate students in mathematics or statistics. For mathematics students, it illustrates how elementary differential geometry of graphs can be used effectively to tackle concrete problems outside mathematics. For statistics students, it facilitates an understanding of and direct access to differential geometric concepts currently used in statistics research. The inter-disciplinary nature of the concepts and terminologies presented herein help to bridge crucial knowledge gaps for mathematics and statistics students, and to facilitate further interaction, collaboration, and innovative research activities among mathematical scientists.|With linear algebra and vector calculus as pre-requisites, the first part of this textbook presents an introduction to the geometry of graphs, encompassing the concepts of normal curvature, sectional curvature, Ricci curvature, and Gaussian curvature. The second part of the book provides background statistical concepts and basic models that form the fundamental knowledge necessary for better comprehension of the concept of local influence; while the third part focuses on the application of differential geometry to local influence analysis, and discusses when and how geometric concepts can be used in an effective manner to develop measures for assessing local influence. This textbook is intended for the use of senior undergraduate and graduate students in mathematics or statistics. For mathematics students, it illustrates how elementary differential geometry of graphs can be used effectively to tackle concrete problems outside mathematics. For statistics students, it facilitates an understanding of and direct access to differential geometric concepts currently used in statistics research. The inter-disciplinary nature of the concepts and terminologies presented herein help to bridge crucial knowledge gaps for mathematics and statistics students, and to facilitate further interaction, collaboration, and innovative research activities among mathematical scientists.