Elementary differential geometry /
O'Neill, Barrett.
Elementary differential geometry / Barrett O'Neill. - Rev. 2nd ed. - Amsterdam : Boston : Elsevier Academic Press, c2006. - xi, 503 p. ill. ; 24 cm.
Includes bibliographical references (p. 467) and index.
Introduction
Chapter 1: calculus on euclidean space:
Euclidean space. Tangent vectors. Directional derivatives. Curves in r3. 1-forms. Differential forms. Mappings;
Chapter 2: frame fields:
Dot product. Curves. The frenet formulas. Arbitraryspeed curves. Covariant derivatives. Frame fields. Connection forms. The structural equations. ;
Chapter 3: euclidean geometry:
Isometries of r3. The tangent map of an isometry. Orientation. Euclidean geometry. Congruence of curves. ;
Chapter 4: calculus on a surface:
Surfaces in r3. Patch computations. Differentiable functions and tangent vectors. Differential forms on a surface. Mappings of surfaces. Integration of forms. Topological properties. Manifolds.;
Chapter 5: shape operators:
The shape operator of m r3. Normal curvature. Gaussian curvature. Computational techniques. The implicit case. Special curves in a surface. Surfaces of revolution. ;
Chapter 6: geometry of surfaces in r3:
The fundamental equations. Form computations. Some global theorems. Isometries and local isometries. Intrinsic geometry of surfaces in r3. Orthogonal coordinates. Integration and orientation. Total curvature. Congruence of surfaces. ;
Chapter 7: riemannian geometry: geometric surfaces. Gaussian curvature. Covariant derivative. Geodesics. Clairaut parametrizations. The gauss-bonnet theorem. Applications of gauss-bonnet. ;
Chapter 8: global structures of surfaces: length-minimizing properties of geodesics. Complete surfaces. Curvature and conjugate points. Covering surfaces. Mappings that preserve inner products. Surfaces of constant curvature. Theorems of bonnet and hadamard. ;
Appendix
Bibliography
Answers to odd-numbered exercises
Subject index
0120887355 (acidfree paper) 9780120887354 (hbk)
Geometry, Differential.
516.36 / N411
Elementary differential geometry / Barrett O'Neill. - Rev. 2nd ed. - Amsterdam : Boston : Elsevier Academic Press, c2006. - xi, 503 p. ill. ; 24 cm.
Includes bibliographical references (p. 467) and index.
Introduction
Chapter 1: calculus on euclidean space:
Euclidean space. Tangent vectors. Directional derivatives. Curves in r3. 1-forms. Differential forms. Mappings;
Chapter 2: frame fields:
Dot product. Curves. The frenet formulas. Arbitraryspeed curves. Covariant derivatives. Frame fields. Connection forms. The structural equations. ;
Chapter 3: euclidean geometry:
Isometries of r3. The tangent map of an isometry. Orientation. Euclidean geometry. Congruence of curves. ;
Chapter 4: calculus on a surface:
Surfaces in r3. Patch computations. Differentiable functions and tangent vectors. Differential forms on a surface. Mappings of surfaces. Integration of forms. Topological properties. Manifolds.;
Chapter 5: shape operators:
The shape operator of m r3. Normal curvature. Gaussian curvature. Computational techniques. The implicit case. Special curves in a surface. Surfaces of revolution. ;
Chapter 6: geometry of surfaces in r3:
The fundamental equations. Form computations. Some global theorems. Isometries and local isometries. Intrinsic geometry of surfaces in r3. Orthogonal coordinates. Integration and orientation. Total curvature. Congruence of surfaces. ;
Chapter 7: riemannian geometry: geometric surfaces. Gaussian curvature. Covariant derivative. Geodesics. Clairaut parametrizations. The gauss-bonnet theorem. Applications of gauss-bonnet. ;
Chapter 8: global structures of surfaces: length-minimizing properties of geodesics. Complete surfaces. Curvature and conjugate points. Covering surfaces. Mappings that preserve inner products. Surfaces of constant curvature. Theorems of bonnet and hadamard. ;
Appendix
Bibliography
Answers to odd-numbered exercises
Subject index
0120887355 (acidfree paper) 9780120887354 (hbk)
Geometry, Differential.
516.36 / N411
