1.2 Tangent spaces and cotangent spaces

Tangent spaces and cotangent spaces are crucial tools in understanding manifolds. They help us grasp how things move and change on curved surfaces, giving us a way to measure and describe motion at each point.

These concepts are key to working with smooth functions on manifolds. They let us talk about directions, rates of change, and measurements in a way that works even when our space isn't flat like regular Euclidean space.

Tangent Spaces

Tangent Vectors and Tangent Spaces

Top images from around the web for Tangent Vectors and Tangent Spaces

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

1 of 3

Top images from around the web for Tangent Vectors and Tangent Spaces

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

• 

  4.1 Displacement and Velocity Vectors | University Physics Volume 1 View original

  Is this image relevant?

1 of 3

• Tangent vector represents a direction of motion along a curve on a manifold at a specific point
• Can be thought of as a velocity vector of a curve passing through a point (position, velocity)
• Tangent space at a point $p$ on a manifold $M$, denoted as $T_pM$, is the set of all tangent vectors to $M$ at $p$
• $T_pM$ forms a vector space with the same dimension as the manifold $M$
• Tangent spaces at different points on a manifold are distinct vector spaces (different directions of motion at each point)

Derivations and Vector Fields

• Derivation at a point $p$ on a manifold $M$ is a linear map from the set of smooth functions on $M$ to $\mathbb{R}$ that satisfies the Leibniz rule (product rule for derivatives)
• Tangent vectors can be identified with derivations (every tangent vector corresponds to a unique derivation and vice versa)
• Vector field on a manifold $M$ assigns a tangent vector to each point of $M$ (smooth assignment of velocities)
• Smooth vector field $X$ on $M$ is a smooth map that assigns to each point $p \in M$ a tangent vector $X_p \in T_pM$
• Examples of vector fields include gradient fields (assign gradient vector at each point) and velocity fields in fluid dynamics (assign velocity vector at each point)

Cotangent Spaces and Duality

Cotangent Spaces and Dual Spaces

• Cotangent space at a point $p$ on a manifold $M$, denoted as $T_p^*M$, is the dual space of the tangent space $T_pM$
• Elements of the cotangent space are called cotangent vectors or 1-forms
• Dual space $V^*$ of a vector space $V$ is the set of all linear maps (functionals) from $V$ to its underlying field ($\mathbb{R}$ for real manifolds)
• Cotangent vectors act on tangent vectors to produce real numbers (measure tangent vectors in some sense)
• Basis for $T_p^*M$ can be constructed from a basis of $T_pM$ using the dual basis (basis vectors in $T_p^*M$ are dual to basis vectors in $T_pM$)

Pushforwards and Pullbacks

• Pushforward (differential) of a smooth map $F: M \to N$ between manifolds is a linear map $dF_p: T_pM \to T_{F(p)}N$ that maps tangent vectors from $T_pM$ to $T_{F(p)}N$
• Pushforward describes how tangent vectors are transformed under the map $F$ (how velocities change)
• Pullback (codifferential) of a smooth map $F: M \to N$ is a linear map $F^*: T_{F(p)}^*N \to T_p^*M$ that maps cotangent vectors from $T_{F(p)}^*N$ to $T_p^*M$
• Pullback describes how cotangent vectors (1-forms) are transformed under the map $F$ (how measurements change)
• Pushforward and pullback are adjoint (dual) to each other with respect to the natural pairing between tangent and cotangent spaces
• Examples include the pullback of a differential form (how a differential form changes under a coordinate transformation) and the pushforward of a vector field (how a vector field is transformed under a diffeomorphism)