Differential Geometry: A Geometric Introduction, 1/e
David W. Henderson, Cornell University
Published July, 1997 by Prentice Hall Engineering/Science/Mathematics
Copyright 1998, 250 pp.
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This is the only text that introduces differential geometry by combining
an intuitive geometric foundation, a rigorous connection with the
standard formalisms, computer exercises with Maple, and a problems-based
approach. Text has running theme of intrinsic vs. extrinsic ways of looking at curves and surfaces. Starting with basic geometric ideas and proceeding to the
analytic and algebraic formalisms, this text provides a common and
accessible foundation on which all of the various formalisms of differential
geometry can be based and from which they can be assessed.
Uses basic intuitive geometry as a starting point
which makes the material more accessible and the formalism more meaningful.
Topics are based on and introduced through 55 core problems.
Working through these problems provides students with a deeper and
personal understanding of the material.
The ribbon test for geometrically finding geodesics is introduced
in Chapter 1. Then it is proven that it works in Chapter 3. Finally,
using ruled surfaces in Chapter 7, it is proven that almost all geodesics
can be found this way.
Introduces hyperbolic geometry in the first chapter
rather than in a closing chapter as in other books.
Supports an intuitive grasp of concepts. For example,
it introduces the analytic notions of differentiability and smoothness
through the use of the geometric notions of field of view and zooming.
Includes 19 computer projects for use with Maple.
An Instructor's Manual with complete solutions for each
problem is available.
How to Use This Book.
1. Surfaces and Straightness.
When Do You Call a Straight Line? How Do You Construct a
Straight Line? Local (and Infinitesimal) Straightness. Intrinsic Straight
Lines on Cylinders. Geodesics on Cones. Is Shortest Always
Straight? Locally Isometric Surfaces. Local Coordinates
for Cylinders and Cones. Geodesics in Local Coordinates. What Is Straight
on a Sphere? Intrinsic Curvature on a Sphere. Local Coordinates on
a Sphere. Strakes, Augers, and Helicoids. Surfaces of Revolution.
Hyperbolic Plane. Surface as Graph of a Function z=f(x,y)
2. Extrinsic Curves.
Introduction. Give Examples of F.O.V.'s. Archimedian Property. Vectors and Affine
Linear Space. Smoothness and Tangent Directions. Curvature of a Curve
in Space. Curvature of the Graph of a Function. Osculating Circle.
Strakes. When a Curve Does Not Lie in a Plane.
3. Extrinsic Descriptions of Intrinsic Curvature.
Smooth Surfaces and Tangent Planes. Extrinsic Curvature
- Geodesics on Sphere. Intrinsic Curvature - Curves on Sphere. Intrinsic
(Geodesic) Curvature. Geodesics on Surfacesthe Ribbon Test. Ruled
Surfaces and the Converse of the Ribbon Test.
4. Tangent Space, Metric, Directional Derivative.
The Tangent Space. Mean Value Theorem: Curves, Surfaces.
Natural Parametrizations of Curves. Riemannian Metric. Riemannian
Metric in Local Coordinates on a Sphere. Riemannian Metric in Local
Coordinates on a Strake. Vectors in Extrinsic Local Coordinates. Measuring
Using the Riemannian Metric. Directional Derivatives. Directional
Derivative in Local Coordinates. Differentiating a Metric. Expressing
Normal Curvature. Geodesic Local Coordinates. Differential Operator.
Metric in Geodesic Coordinates.
5. Area, Parallel Transport, Intrinsic Curvature.
The Area of a Triangle on a Sphere. Introducing Parallel
Transport. The Holonomy of a Small Geodesic Triangle. Dissection of
Polygons into Triangles. Gauss-Bonnet for Polygons on a Sphere. Parallel
Fields and Intrinsic Curvature. Holonomy on Surfaces. Holonomy Explains
Foucault's Pendulum. Intrinsic Curvature of a Surface.
6. Gaussian Curvature Extrinsically Defined.
Pep Talk to the Reader. Gaussian Curvature, Extrinsic Definition. Second Fundamental
Form. The Gauss Map. Gauss-Bonnet and Intrinsic Curvature. Matrix of the Second
Fundamental. Mean Curvature and Minimal Surfaces. Celebration of Our Hard Work.
7. Applications of Gaussian Curvature.
Gaussian Curvature in Local Coordinates. Curvature on Sphere,
Strake, Catenoid. Circles, Polar Coordinates, and Curvature. Exponential
Map and Shortest Is Straight. Ruled Surfaces and Ribbons. Surfaces
with Constant Curvature. Curvature of the Hyperbolic Plane.
8. Intrinsic Local Descriptions and Manifolds.
Covariant Derivative and Connections. ManifoldsIntrinsic
and Extrinsic. Christoffel Symbols, Intrinsic Descriptions. Intrinsic
Curvature and Geodesics. Lie Brackets, Coordinate Vector Fields.
Riemann Curvature Tensors. Calculation of Curvature Tensors in Local
Coordinates. Intrinsic Calculations in Examples.
Appendix A. Linear Algebraa Geometric Point of View.
Where Do We Start? Geometric Affine Spaces. Vector Spaces. Inner ProductLengths
and Angles. Linear Transformations and Operators. Areas, Cross Products,
and Triple Products. Volumes, Orientation, and Determinants. Eigenvalues
and Eigenvectors. Introduction to Tensors.
Appendix B. Analysis from a Geometric Point of View.
Smooth Functions. Invariance of Domain. Inverse Function
Theorem. Implicit Function Theorem.
Appendix C. Computer Scripts.
Standard Functions. Strake. Surfaces of Revolution. Surfaces as Graph of a Function. Tangent Vectors to Curves. Curvature and Tangent Vectors. Osculating Planes. Osculating Circles. Frenet Frame. Tangent Planes to Surfaces. Curves on a Surface. Extrinisic Curvature Vectors. The Three Curvature Vectors. Ruled Surfaces. Non-dissectable Polyhedron. Sign of (Gaussian) Curvature. Mulitple Principle Directions. Gauss Map. Helicoid to Catenoid.