An introduction to differential geometry with use of the tensor calculus

Title page
CHAPTER 1: CURVES IN SPACE
	1. Curves and surfaces. The summation convention
	2. Length of a curve. Linear element
	3. Tangent to a curve. Order of contact. Osculating plane
	4. Curvature. Principal normal. Circle of curvature
	5. Binormal. Torsion
	6. The Frenet formulas. The form of a curve in the neighborhood of a point
	7. Intrinsic equations of a curve
	8. Involutes and evolutes of a curve
	!J. The tangent surface of a curve. The polar surface. Osculating sphere
	10. Parametric equations of a surface. Coordinates and coordinate curves in a surface
	11. Tangent plane to a surface
	12. Developable surfaces. Envelope of a one-parameter family of surfaces
CHAPTER II: TRANSFORMATION OF COORDINATES. TENSOR CALCULUS
	13. Transformation of coordinates. Curvilinear coordinates
	14. The fundamental quadratic form of space
	15. Contravariant vectors. Scalars
	16. Length of a contravariant vector. Angle between two vectors
	17. Covariant vectors. Contravariant and covariant components of a vector
	18. Tensors. Symmetric and skew symmetric tensors
	19. Addition, subtraction and multiplication of tensors. Contraction
	20. The Christoffel symbols. The Riemann tensor
	21. The Frenet formulas in general coordinates
	22. Covariant differentiation
	23. Systems of partial differential equations of the first order. Mixed systems
CHAPTER III: INTRINSIC GEOMETRY OF A SURFACE
	24. Linear element of a surface. First fundamental quadratic form of a surface. Vectors in a surface
	25. Angle of two intersecting curves in a surface. Element of area
	26. Families of curves in a surface. Principal directions
	27. The intrinsic geometry of a surface. Isometric surfaces
	28. The Christoffel symbols for a surface. The Riemannian curvature tensor. The Gaussian curvature of a surface
	29. Differentiai parameters
	30. Isometric orthogonal nets. Isometric coordinates
	31. Isometric surfaces
	32. Geodesies
	33. Geodesic polar coordinates. Geodesic triangles
	34. Geodesic curvature
	35. The vector asociate to a given vector with respect to a curve. Paralle!ism of vectors
	36. Conformai correspondence of two surfaces
	37. Geodesic correspondence of two surfaces
CHAPTER IV: SURFACES IN SPACE
	38. The second fundamental form of a surface
	39. The equation of Gauss and the equations of Codazzi
	40. Normal curvature of a surface. Principal radii of normal curvature
	41. Lines of curvature of a surface
	42. Conjugate directions and conjugate nets. Isometric-conjugate nets
	43. Asymptotic directions and asymptotic lines. Mean conjugate directions. The Dupin indicatrix
	44. Geodesic curvaturc and geodesic torsion of a curve
	45. Parallel vectors in a surface
	46. Spherical representation of a surface. The Gaussian curvature of a surface
	47. Tangential coordinates of a surface
	48. Surfaces of center of a surface. Parallel surfaces
	49. Spherical and pseudospherical surfaces
	50. Minimal surfaces
Bib!iography
Index