## Differential Geometry and Its ApplicationsDifferential geometry has a long, wonderful history it has found relevance in areas ranging from machinery design of the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole, it mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, it is also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only “see” geodesics on surfaces, but they will also see the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. Students will also see how particles move under constraints. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract. |

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### Contents

Surfaces | 67 |

Curvatures | 107 |

Constant Mean Curvature Surfaces | 161 |

Geodesies Metrics and Isometries | 209 |

Holonomy and the GaussBonnet Theorem | 275 |

The Calculus of Variations and Geometry | 311 |

A Glimpse at Higher Dimensions | 397 |

A List of Examples | 437 |

Examples in Chapter 3 | 438 |

Examples in Chapter 7 | 439 |

B Hints and Solutions to Selected Problems | 441 |

Suggested Projects for Differential Geometry | 453 |

455 | |

461 | |

About the Author | 469 |

### Common terms and phrases

algebra angle arclength calculate catenoid Chapter circle Clairaut relation closed curve coefficients compute condition cone consider constraint coordinates cos2 cosh course covariant derivative cylinder defined definition denote determined differential equation differential geometry dot product elliptic Enneper's surface Euler-Lagrange equation Example Exercise Figure fixed endpoint problem ft(s function Gauss curvature Gauss map geodesic equations given gives graph helix Hence Hint holonomy hyperboloid integral inverse isometry Lemma linear Maple Mathematies matrix mean curvature metric minimal surface Mylar balloon normal curvature Note obtain parallel parametrization patch pendulum perpendicular plane curve plot procedure Proof radius Recall Section shape operator Show simply sinh solution solve sphere Suppose surface area surface of revolution tangent plane tangent vector Theorem torus TP(M triangle unduloid unit normal unit speed curve variable vector field w-parameter curve write zero