Differential topology

From Mathematics wiki

Contents 1 Manifolds 2 Tangent and cotangent bundles 2.1 Tangent spaces 2.2 Cotangent spaces 3 Tensors 4 Flows and Lie derivatives 5 Differential forms 6 Index theorems 7 Morse theory 8 Resources

Manifolds

• Manifold
  • Curve
  • Surface
• Coordinate chart
• Diffeomorphism

Tangent and cotangent bundles

Tangent spaces

• Tangent vector field
• Integral curve
• Tangent space
• Tangent basis
• Coordinate basis
• Tangent bundle

Cotangent spaces

• Cotangent vector field
• Gradient
• Cotangent space
• Normal one-form
• Cotangent basis
• Coordinate cobasis
• Cotangent bundle

Tensors

• Inner product
• Tensor
• Abstract index notation
• Fiber bundle
• Field
• Vector field
• Tensor field

Flows and Lie derivatives

• Flow
• Lie dragging
• Lie derivative
• Lie bracket
• Isometry

Differential forms

• Differential form
• Exterior product (wedge product)
• Interior product
• one-form
• Exterior derivative
• Exact form
• Closed form

Index theorems

• Fredholm index

Morse theory

Resources

• Books
• Lectures