1.3 Tangent vectors, tangent spaces, and the tangent bundle

Tangent vectors are key to understanding how curves and functions behave on smooth manifolds. They represent directions and speeds at points, forming tangent spaces that mirror the manifold's dimension.

The tangent bundle unites all tangent spaces into one object, crucial for studying global properties. Smooth sections of this bundle, equivalent to vector fields, allow us to assign tangent vectors consistently across the manifold.

Tangent Vectors and Derivatives

Geometric Interpretation of Tangent Vectors

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• Tangent vector represents direction and magnitude of instantaneous velocity along a curve on a manifold
• Defines direction of motion at a specific point on the manifold
• Visualized as an arrow touching the manifold at a single point
• Mathematically expressed as equivalence class of curves passing through a point with same velocity
• Forms basis for studying local behavior of functions and curves on manifolds

Directional Derivatives and Vector Fields

• Directional derivative measures rate of change of a function in direction of tangent vector
• Generalizes concept of partial derivatives to manifolds
• Calculated by taking limit of difference quotient along curve in direction of tangent vector
• Vector field assigns tangent vector to each point on manifold
• Smooth vector field has components varying smoothly across manifold
• Applications include modeling physical phenomena (fluid flow, electromagnetic fields)

Tangent and Cotangent Spaces

Tangent Space Structure and Properties

• Tangent space consists of all tangent vectors at a specific point on manifold
• Forms vector space with operations of addition and scalar multiplication
• Dimension of tangent space equals dimension of manifold
• Basis for tangent space often chosen as partial derivatives with respect to local coordinates
• Allows representation of tangent vectors as linear combinations of basis vectors

Cotangent Space and Duality

• Cotangent space defined as dual vector space to tangent space
• Elements of cotangent space called covectors or one-forms
• Covectors map tangent vectors to real numbers
• Natural pairing between tangent and cotangent spaces denoted by angle brackets $\langle \omega, v \rangle$
• Basis for cotangent space consists of differentials of coordinate functions
• Plays crucial role in defining differential forms and integration on manifolds

Dimension and Coordinate Representations

• Dimension of tangent and cotangent spaces matches dimension of underlying manifold
• Local coordinates $(x^1, ..., x^n)$ induce basis for tangent space $\{\partial/\partial x^1, ..., \partial/\partial x^n\}$
• Corresponding basis for cotangent space given by differentials $\{dx^1, ..., dx^n\}$
• Tangent vectors and covectors expressed as linear combinations of respective basis elements
• Coordinate transformations induce change of basis in tangent and cotangent spaces

Tangent Bundle and Smooth Sections

Tangent Bundle Construction and Properties

• Tangent bundle unifies tangent spaces at all points of manifold into single geometric object
• Constructed by taking disjoint union of all tangent spaces
• Points in tangent bundle represented as pairs $(p, v)$ where $p$ belongs to manifold and $v$ belongs to tangent space at $p$
• Projection map $\pi$ sends each point in tangent bundle to its base point on manifold
• Tangent bundle inherits smooth structure from base manifold

Smooth Sections and Vector Fields

• Smooth section of tangent bundle assigns tangent vector to each point on manifold smoothly
• Equivalent to smooth vector field on manifold
• Defined mathematically as smooth map $s: M \rightarrow [TM](https://www.fiveableKeyTerm:tm)$ satisfying $\pi \circ s = id_M$
• Local expression of smooth section involves smooth functions as components in coordinate basis
• Space of smooth sections forms module over ring of smooth functions on manifold

Fiber Structure and Local Trivialization

• Fiber of tangent bundle over point $p$ consists of tangent space at $p$
• Each fiber isomorphic to $\mathbb{R}^n$ where $n$ equals dimension of manifold
• Local trivialization provides diffeomorphism between neighborhood in tangent bundle and product of open set in manifold with $\mathbb{R}^n$
• Allows representation of tangent bundle locally as product manifold
• Transition functions between different local trivializations encode global structure of tangent bundle
• Non-trivial tangent bundles (Möbius strip) demonstrate topological complexity arising from global twisting of fibers