10 Differential Geometry

Differential geometry is the study of shape, distance, and curvature on curves, surfaces, and smooth spaces.

A smooth space is a space: a collection of points that locally looks flat. Curvature is a property of a smooth space: how sharply it bends away from being flat.

Differential geometry is related to calculus and topology because the shape of a smooth space is independent of which coordinates are used to describe it. Now, here are the ideas that are fundamental to the study of differential geometry.

Coordinate independence. A smooth manifold is a space that locally looks flat. Its shape does not depend on the coordinate systems used to label its points. A coordinate system is a labeling map.

The tangent space. At each point of a smooth space there is a tangent space. A tangent space is a space: the flat local approximation to the curved space at that point.

Curvature. Curvature is a property of a smooth space: how much it departs from being flat. Intrinsic curvature is a measurement made entirely inside the space.

10.1 Topics

1. Smooth
   1. Smooth Mapping
   2. Smooth Assignment
   3. Functions
2. Tangent Space
3. Bundle
4. Linear Functional
5. Multilinear Function
   1. Alternating Function
      1. Alternating k-linear function
6. Tensors
   1. Tensor Field
   2. Type-(q,r) Tensor
   3. Type-(0,2) Tensor Field
   4. k-Tensor
   5. Symmetric Array
   6. Covariant Rank
   7. Contravariant Rank
   8. Covariant, Contravariant, and Coordinate Systems
7. Metric Tensor
   1. Inverse Metric
   2. Line Element
   3. Constant Metric Tensor
   4. Nondegenerate
   5. Nondegenerate Bilinear Form
   6. Lorentzian Manifold
8. Forms
   1. Differential Forms
   2. One-Form
   3. Two-Form
   4. k-form
   5. Differential k-Form
   6. k-Covector
   7. k-Covector Field
   8. k-Fold Product
   9. Exterior Algebra and Exterior Derivatives
9. Stoke’s Theorem and the Fundamental Theorem of Calculus
   1. Boundary Operation
   2. Edge of a Domain
10. Cartesian Product
11. Lie
    1. Lie Algebra

10.2 References

1. Lee, J. M. Introduction to Smooth Manifolds. Springer, 2013. — a manifold as a space whose geometry does not depend on coordinates.
2. Tu, L. W. An Introduction to Manifolds. Springer, 2011. — the tangent space as the flat local approximation at a point.
3. do Carmo, M. P. Differential Geometry of Curves and Surfaces. — curvature as an intrinsic measurement of departure from flatness.
4. Lovett, S. T. Differential Geometry of Manifolds. CRC Press, 2020. — shape, distance, and curvature of smooth spaces, independent of coordinates.

• Alternating Function
• Alternating k-linear function
• Boundary Operation
• Bundle
• Cartesian Product
• Constant Metric Tensor
• Contravariant Rank
• Covariant Rank
• Covariant, Contravariant, and Coordinate Systems
• Differential Forms
• Differential k-Form
• Edge of a Domain
• Exterior Algebra and Exterior Derivatives
• Forms
• Functions
• Inverse Metric
• k-Covector
• k-Covector Field
• k-Fold Product
• k-form
• k-Tensor
• Lie
• Lie Algebra
• Line Element
• Linear Functional
• Lorentzian Manifold
• Metric Tensor
• Multilinear Function
• Nondegenerate
• Nondegenerate Bilinear Form
• One-Form
• Smooth
• Smooth Assignment
• Smooth Mapping
• Stoke’s Theorem and the Fundamental Theorem of Calculus
• Symmetric Array
• Tangent Space
• Tensor Field
• Tensors
• Two-Form
• Type-(0,2) Tensor Field
• Type-(q,r) Tensor