## 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. |

### What people are saying - Write a review

User Review - Flag as inappropriate

vdevfdv

### 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 | |

### Other editions - View all

### Common terms and phrases

angle of rotation called Chapter compact oriented containing convex coordinate vector fields critical point cylinder denote diffeomorphism differential equations domain dot product Example Exercise Figure focal locus formula Gauss map Gauss-Kronecker curvature Gaussian curvature given global parametrization grad Hence Hint integral curve isometry Lemma let p e level set linear Lp(v matrix maximal geodesic maximal integral curve n-plane n-sphere n-surface in R"+1 normal curvature normal vector field Note open set orientation vector field oriented n-surface oriented plane oriented plane curve orthogonal orthonormal basis parametrized curve parametrized n-surface plane curve Poincare metric principal curvatures real number Riemannian metric rigid motion Show singular smooth function smooth map smooth tangent vector smooth unit tangent smooth vector field subset surface tangent space tangent vector field total angle unique unit normal vector unit tangent vector v e Sp variation velocity Weingarten map xu x2

### Popular passages

Page 246 - Hurewicz, W. (1958). Lectures on Ordinary Differential Equations, Cambridge, Mass., MIT Press.