Manifolds and Tangent Spaces

Overview

This lecture introduces the concept of tangent spaces on manifolds, explains how they generalize the idea of tangents from curves to higher dimensions, and outlines how calculus is performed on manifolds using local coordinates and charts.

Tangent Lines and Tangent Spaces

• A tangent line lightly touches a curve at a single specific point in two-dimensional space.
• For a manifold in three dimensions, the analogous object is the tangent space at a point, which appears as a flat plane.
• The tangent space at point ( p ) on manifold ( M ) is denoted as ( T_pM ).

Manifolds and Local Coordinates

• Manifolds generalize surfaces; they need not be embedded in higher-dimensional space and exist independently.
• Local coordinates (( x_1, x_2, \ldots )) are used to describe positions and directions on a manifold.
• These coordinates change smoothly and provide a basis for defining tangent vectors.

Tangent Vectors and Vector Spaces

• Tangent vectors represent all possible directions one can move from point ( p ) on a manifold.
• In 1D, directions are along the curve; in 2D, infinitely many directions form a circle in the tangent plane.
• The tangent space at a point is a vector space of all possible tangent vectors there.

Charts, Curves, and Calculus on Manifolds

• To perform calculus, we map (via a chart ( \varphi )) a neighborhood ( U ) of ( p ) to Euclidean space ( \mathbb{R}^n ).
• A curve ( \gamma(t) ) passing through ( p ) is mapped to ( \mathbb{R}^n ) for differentiation.
• The velocity vector at ( p ) is computed as the derivative ( d(\varphi \circ \gamma)/dt ) at the relevant time.

Basis and Differential Operators

• To compare tangent spaces across points or charts, we use a basis for the tangent space at each point.
• An arbitrary function ( f ) (a scalar field) assigns a real number to each point on the manifold.
• The rate of change of ( f ) along a curve is used to define basis vectors as differential operators.
• The basis of the tangent space consists of differential operators, not just geometric arrows.

Key Terms & Definitions

• Manifold — A space that locally resembles Euclidean space but can have a different global structure.
• Tangent space (( T_pM )) — The vector space of all possible directions at a point ( p ) on a manifold ( M ).
• Chart — A map assigning local coordinates to points on a manifold, allowing calculus to be performed.
• Tangent vector — A direction of movement at a point, represented as a velocity along a curve.
• Scalar field — A function assigning a single real number to each point on the manifold.
• Differential operator — An operator acting on functions, forming the basis of tangent spaces.

Action Items / Next Steps

• Review the provided PDF summary for further study.
• Practice mapping curves from a manifold to Euclidean space and computing derivatives.
• Attempt to define tangent spaces and basis vectors for different manifolds independently.