Nonparametric Statistics on Manifolds Data on Manifolds Directional and Axial Data Similarity Shape Data and Size and Shape Data Digital Camera Images Stereo Imaging Data of the Eye Fundus CT Scan Data DTI Data Data Tables Basic Nonparametric Multivariate Inference Basic Probability Theory Integration on Euclidean Spaces Random Vectors Sampling Distributions of Estimators Consistency and Asymptotic Distributions of Estimators The Multivariate Normal Distribution Convergence in Distribution Limit Theorems Elementary Inference Comparison of Two Mean Vectors Principal Components Analysis (PCA) Multidimensional Scaling Nonparametric Bootstrap and Edgeworth Expansion Nonparametric Function Estimation Data Analysis on Hilbert Spaces Exercises Geometry and Topology of Manifolds Manifolds, Submanifolds, Embeddings, Lie Groups Riemannian Structures, Curvature, Geodesics The Laplace-Beltrami Operator Topology of Manifolds Manifolds as Spaces of Objects in Data Analysis Exercises Consistency of Fréchet Moments on Manifolds Introduction Fréchet Means and Cartan Means Exercises Nonparametric Distributions of Fréchet Means Introduction Fréchet Total Sample Variance-Nonparametrics Elementary CLT for Extrinsic Means CLT and Bootstrap for Fréchet Means CLT for Extrinsic Sample Means Exercises Inference for Two Samples on Manifolds Introduction Two-Sample Test for Total Extrinsic Variances Bhattacharya''s Two-Sample Test for Means Test for Mean Change in Matched Pairs on Lie Groups Two-Sample Test for Simply Transitive Group Actions Nonparametric Bootstrap for Two-Sample Tests Exercises Function Estimation on Manifolds Introduction Statistical Inverse Estimation Proofs of Main results Kernel Density Estimation Asymptotic Theory and Nonparametric Bootstrap on Special Manifolds Statistics on Homogeneous Hadamard Manifolds Introduction Considerations for Two-Sample Tests Intrinsic Means on Hadamard Manifolds Two-Sample Tests for Intrinsic Means Analysis on Stiefel Manifolds Stiefel Manifolds Special Orthogonal Groups Intrinsic Analysis on Spheres Asymptotic Distributions on Projective Spaces Total Variance of Projective Shape Asymptotics Asymptotic Distributions of VW-Means Asymptotic Distributions of VW-Means of k -ads Inference for Projective Shapes of k -ads Two-Sample Tests for Mean Projective Shapes Nonparametric Statistics on Hilbert Manifolds Introduction Hilbert Manifolds Extrinsic Analysis of Means on Hilbert Manifolds A One-Sample Test of the Neighborhood Hypothesis Analysis on Spaces of Congruences of k -ads Introduction Equivariant Embeddings of S S k 2 and RS S km ,0 Extrinsic Means and Their Estimators Asymptotic Distribution of Extrinsic Sample Mean Mean Size-and-Shape of Protein Binding Sites Similarity Shape Analysis Introduction Equivariant Embeddings of S k 2 and R S km ,0 Extrinsic Mean Planar Shapes and Their Estimators Asymptotic Distribution of Mean Shapes A Data-Driven Example Statistics on Grassmannians Equivariant Embeddings of Grassmann Manifolds Dimitric Mean of a Random Object on a Grassmannian Extrinsic Sample Covariance Matrix on a Grassmannian Applications in Objecton Nonparametric Function Estimation Data Analysis on Hilbert Spaces Exercises Geometry and Topology of Manifolds Manifolds, Submanifolds, Embeddings, Lie Groups Riemannian Structures, Curvature, Geodesics The Laplace-Beltrami Operator Topology of Manifolds Manifolds as Spaces of Objects in Data Analysis Exercises Consistency of Fréchet Moments on Manifolds Introduction Fréchet Means and Cartan Means Exercises Nonparametric Distributions of Fréchet Means Introduction Fréchet Total Sample Variance-Nonparametrics Elementary CLT for Extrinsic Means CLT and Bootstrap for Fréchet Means CLT for Extrinsic Sample Means Exercises Inference for Two Samples on Manifolds Introduction Two-Sample Test for Total Extrinsic Variances Bhattacharya''s Two-Sample Test for Means Test for Mean Change in Matched Pairs on Lie Groups Two-Sample Test for Simply Transitive Group Actions Nonparametric Bootstrap for Two-Sample Tests Exercises Function Estimation on Manifolds Introduction Statistical Inverse Estimation Proofs of Main results Kernel Density Estimation Asymptotic Theory and Nonparametric Bootstrap on Special Manifolds Statistics on Homogeneous Hadamard Manifolds Introduction Considerations for Two-Sample Tests Intrinsic Means on Hadamard Manifolds Two-Sample Tests for Intrinsic Means Analysis on Stiefel Manifolds Stiefel Manifolds Special Orthogonal Groups Intrinsic Analysis on Spheres Asymptotic Distributions on Projective Spaces Total Variance of Projective Shape Asymptotics Asymptotic Distributions of VW-Means Asymptotic Distributions of VW-Means of k -ads Inference for Projective Shapes of k -ads Two-Sample Tests for Mean Projective Shapes Nonparametric Statistics on Hilbert Manifolds Introduction Hilbert Manifolds Extrinsic Analysis of Means on Hilbert Manifolds A One-Sample Test of the Neighborhood Hypothesis Analysis on Spaces of Congruences of k -ads Introduction Equivariant Embeddings of S S k 2 and RS S km ,0 Extrinsic Means and Their Estimators Asymptotic Distribution of Extrinsic Sample Mean Mean Size-and-Shape of Protein Binding Sites Similarity Shape Analysis Introduction Equivariant Embeddings of S k 2 and R S km ,0 Extrinsic Mean Planar Shapes and Their Estimators Asymptotic Distribution of Mean Shapes A Data-Driven Example Statistics on Grassmannians Equivariant Embeddings of Grassmann Manifolds Dimitric Mean of a Random Object on a Grassmannian Extrinsic Sample Covariance Matrix on a Grassmannian Applications in Object;/P> Inference for Two Samples on Manifolds Introduction Two-Sample Test for Total Extrinsic Variances Bhattacharya''s Two-Sample Test for Means Test for Mean Change in Matched Pairs on Lie Groups Two-Sample Test for Simply Transitive Group Actions Nonparametric Bootstrap for Two-Sample Tests Exercises Function Estimation on Manifolds Introduction Statistical Inverse Estimation Proofs of Main results Kernel Density Estimation Asymptotic Theory and Nonparametric Bootstrap on Special Manifolds Statistics on Homogeneous Hadamard Manifolds Introduction Considerations for Two-Sample Tests Intrinsic Means on Hadamard Manifolds Two-Sample Tests for Intrinsic Means Analysis on Stiefel Manifolds Stiefel Manifolds Special Orthogonal Groups Intrinsic Analysis on Spheres Asymptotic Distributions on Projective Spaces Total Variance of Projective Shape Asymptotics Asymptotic Distributions of VW-Means Asymptotic Distributions of VW-Means of k -ads Inference for Projective Shapes of k -ads Two-Sample Tests for Mean Projective Shapes Nonparametric Statistics on Hilbert Manifolds Introduction Hilbert Manifolds Extrinsic Analysis of Means on Hilbert Manifolds A One-Sample Test of the Neighborhood Hypothesis Analysis on Spaces of Congruences of k -ads Introduction Equivariant Embeddings of S S k 2 and RS S km ,0 Extrinsic Means and Their Estimators Asymptotic Distribution of Extrinsic Sample Mean Mean Size-and-Shape of Protein Binding Sites Similarity Shape Analysis Introduction Equivariant Embeddings of S k 2 and R S km ,0 Extrinsic Mean Planar Shapes and Their Estimators Asymptotic Distribution of Mean Shapes A Data-Driven Example Statistics on Grassmannians Equivariant Embeddings of Grassmann Manifolds Dimitric Mean of a Random Object on a Grassmannian Extrinsic Sample Covariance Matrix on a Grassmannian Applications in Objectsic Means Analysis on Stiefel Manifolds Stiefel Manifolds Special Orthogonal Groups Intrinsic Analysis on Spheres Asymptotic Distributions on Projective Spaces Total Variance of Projective Shape Asymptotics Asymptotic Distributions of VW-Means Asymptotic Distributions of VW-Means of k-ads Inference for Projective Shapes of k-ads Two-Sample Tests for Mean Projective Shapes Nonparametric Statistics on Hilbert Manifolds Introduction Hilbert Manifolds Extrinsic Analysis of Means on Hilbert Manifolds