## Elementary Topics in Differential GeometryIn the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under standing of spaces of many dimensions. It should be apparent that courses on the junior level should draw upon and reinforce the concepts and skills learned during the previous year. Unfortunately, in differential geometry at least, this is usually not the case. Textbooks directed to students at this level generally restrict attention to 2-dimensional surfaces in 3-space rather than to surfaces of arbitrary dimension. Although most of the recent books do use linear algebra, it is only the algebra of ~3. The student's preliminary understanding of higher dimensions is not cultivated. |

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### Contents

Graphs and Level Sets | 1 |

Vector Fields | 6 |

The Tangent Space | 13 |

Surfaces | 16 |

Vector Fields on Surfaces Orientation | 23 |

The Gauss Map | 31 |

Geodesics | 38 |

Parallel Transport | 45 |

Local Equivalence of Surfaces and Parametrized Surfaces | 121 |

Focal Points | 132 |

Surface Area and Volume | 139 |

Minimal Surfaces | 156 |

The Exponential Map | 163 |

Surfaces with Boundary | 177 |

The GaussBonnet Theorem | 190 |

Rigid Motions and Congruence | 210 |

The Weingarten Map | 53 |

Curvature of Plane Curves | 62 |

Arc Length and Line Integrals | 68 |

Curvature of Surfaces | 82 |

Convex Surfaces | 95 |

Parametrized Surfaces | 108 |

Isometries | 220 |

Riemannian Metrics | 231 |

Bibliography | 245 |

247 | |

249 | |

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attains boundary called Chapter choice circle compact compute connected consistent constant containing continuous coordinate vector fields critical point curvature defined definite denned denote derivative determinant differential direction domain dot product equal equation Example Exercise exists fact Figure focal follows formula geodesic geometry given global Hence Image integral curve interval isometry Lemma length level set linear matrix maximal measures metric normal Note obtained open set oriented n-surface orthogonal orthonormal basis parallel parametrized curve parametrized n-surface particular plane curve positive principal curvatures Proof properties regular relative Remark respect restriction Show singular smooth function smooth map smooth vector field subset Suppose surface tangent space tangent vector field Theorem unique unit unit speed unit vector usual v e Sp variation volume zero