# Differential Geometry Notes Spring 1991

### Based on undergraduate linear algebra and multivariable calculus

Copyright 2007: conversion to LaTeX/PDF format and complete edit and expansion in progress: dgn1991-2008.pdf

### Part 1 Algebra

 dg0.pdf Table of contents and summary of still missing motivating introduction (unrecorded for posterity: 2006 insert plus Maple visualization) 0 CHAPTER 1: dg1.pdf Index conventions 1 A vector space and its dual space 2 Linear transformations of a vector space into itself (and tensors) 10 Tensor product and matrix multiplication 15c Worked Problems 16 Remark 19 dg2.pdf Linear transformations of V into itself and a change of basis 20 Linear transformations between V and V* 27 Invertible maps between V and V* 28 Problems 29 Index shifting (and inner products) 31 Geometric interpretation of index shifting 33 Problem 34 Cute fact 35 Index shifting conventions 37 Problem 40 Partial evaluation of a tensor and index shifting 41 Contraction of tensors 41 Rn with usual inner product 42 Problem 43

 CHAPTER 2 dg3.pdf Measure Motivation 48b Symmetry properties and volume (measure) 49 The algebra of antisymmetric tensors 59 CHAPTER 3 Whoa! Review of what we've done so far 62 Problem set (p.43) discussed 71 Remark on generalized Kronecker deltas  [MOVED TO CHAPTER 2] 74 CHAPTER 4 The wedge product 75 Subspace orientation and antisymmetric tensors (duality operation) 81 Exercises 85 Inner product for antisymmetric tensors 88 The unit n-form on an oriented vector space with inner product 89 Linear maps 93 Exercises 97

### Part 2 Calculus

 CHAPTER 5 with extra  section of flow lines of vector fields dg4.pdf The tangent space in multivariable calculus 1 Some problems in 3d calculus 9 More motivation for the re-interpretation of the tangent space 10 Frames and dual frames 15 More on Lie brackets 17b Exercises 17b More on tangent covectors, the differential, and vector fields 17c dg5.pdf Non-Cartesian coordinates on Rn 18 Cylindrical and spherical coordinates on R3 28 Exercise 34 Exercise answer 34b Exercises p.17b,c answered 34c Summarizing remark, exercise 35 Exercise p.35 answer 36 spherical coordinates in detail 37 Exercise p.40 worked 40b Exercises 40c

 CHAPTER 6 dg6.pdf Covariant derivatives on Rn with Euclidean metric 44 Notation for covariant derivatives 47 Exercise 50 Exercises pp.30,40,42,43 worked 52 Duality and extension to tensors of a covariant derivative 54 Note on class of Cartesian coordinate systems 56 Exercise 57 Exercise pp.49,50 worked 58 Exercise 59 The clever way of evaluating the components of the covariant derivative 60 Exercises 62 Noncoordinate frames 63 Exercises 64

 CHAPTER 7 dg7.pdf Laplacian and divergence and gradient 65 Exercises 66 Aside on determinant of differentials 68 Exercises 69 Second covariant derivatives 71 Exercises 72 More practice evaluating components of the covariant derivative 74 Aside on orthogonal matrices 78 Exercise p.40c worked 84 Exercise p.57 worked 85 Exercise p.59 worked 86 Exercise p.62 worked 87 Exercise p.64 worked 89 Exercise 90 CHAPTER 8 dg8.pdf Covariant differentiation along a curve and parallel translation 91 Exercise 92 Exercise 93 Exercise 99 dg9.pdf Geodesics 100 Exercise 101 The 2-sphere of radius ro 103 Exercise 106 CHAPTER 9 Describing intrinsic curvature 107 Exercises 110 Interpretation of curvature 111 CHAPTER 10 TO DO extrinsic curvature

 CHAPTER 11 dg10.pdf Integration of differential forms 114 Parametrized p-surface 115 Pulling back differential forms 117 Exercises 121 Change of variables 123 dg11.pdf The exterior derivative d 126 Exercise 131 The exterior derivative and a metric 136 Exercise 142 dg12.pdf Induced orientation 145 Stokes' theorem 148 CHAPTER 12 Final remarks 151 dg13.pdf Final exam 156 Final exam worked 162 Last page 171 APPENDIX A. From trigonometry to hyperbolic functions and hyperbolic geometry APPENDIX B. Special relativity and hyperbolic geometry APPENDIX C. Worked problems

Index of PDF files, page totals and filesizes

## Preface

These handwritten notes from a course given at Villanova University in the spring semester of 1991 were scanned and posted on the web in 2006 at http://www34.homepage.villanova.edu/robert.jantzen/notes/dg1991
and were converted to a LaTeX compuscript and revised in 2007 with the help of Hans Kuo of Taiwan through a serendipitous internet collaboration. Most undergraduate courses on differential geometry are leftovers from the early part of the last century, focusing on curves and surfaces in space, which is not very useful for the most important application of the twentieth century: general relativity and field theory in theoretical physics. Most mathematicians who teach such courses are not well versed in physics, so perhaps this is a natural consequence of the distancing of mathematics from physics, two fields which developed together in creating these ideas from Newton to Einstein and beyond.

The idea of these notes is to develop the essential tools of modern differential geometry while bypassing more abstract notions like manifolds, which although important for global questions, are not essential for local differential geometry and therefore need not steal precious time from a first course aimed at undergraduates.

Part 1 (Algebra) develops the vector space structure of \$\mathbb{R}^n\$ and its dual space of real-valued linear functions, and builds the tools of tensor algebra on that structure, getting the index manipulation part of tensor analysis out of the way first. Part 2 (Calculus) then develops \$\mathbb{R}^n\$ as a manifold first analyzed in Cartesian coordinates, beginning by redefining the tangent space of multivariable calculus to be the space of directional derivatives at a point, so that all of the tools of Part 1 then can be applied pointwise in the space. Non-Cartesian coordinates and the Euclidean metric are then used as a shortcut to what would be the consideration of more general manifolds with Riemannian metrics in a more ambitious course, followed by the covariant derivative and parallel transport, leading naturally into curvature. The exterior derivative and integration of differential forms is the final topic, showing how conventional vector analysis fits into a more elegant unified framework.

The theme of Part 1 is that one needs to distinguish the linearity properties from the inner product (``metric") properties of linear algebra. The inner product geometry governs lengths and angles, and the determinant then enables one to extend the linear measure of length to area and volume in the plane or 3-dimensional space, and to \$p\$-dimensional objects in \$\mathbb{R}^n\$. The determinant also tests linear independence of a set of vectors and hence is key to characterizing subspaces independent of the particular set of vectors we use to describe them while assigning an actual measure to the \$p\$-parallelepipeds formed by a particular set, once an inner product sets the length scale for orthogonal directions. By appreciating the details of these basic notions in the setting of \$\mathbb{R}^n\$, one is ready for the tools needed point by point in the tangent spaces to \$\mathbb{R}^n\$, once one understands the relationship between each tangent space and the simpler enveloping space.

The LaTeX version of these notes is in progress.