12.20.5 Tensor Pullback Pushforward Notation

Tensor Pullback and Pushforward Notation explores how tensors transform under mappings, essential in differential geometry and algebraic structures.

Tensor Pullback Pushforward Notation is the system of symbols used to describe how tensors are transported between manifolds, or between tangent and cotangent spaces at different points, through a smooth map connecting those spaces. The pullback operates on covariant objects such as covector fields and differential forms, carrying them backward along the map, while the pushforward operates on contravariant objects such as tangent vectors, carrying them forward along the map.

The Underlying Map

Smooth Map Between Manifolds

The notation begins with a smooth map between two manifolds, typically denoted with a lowercase letter such as $φ$ , sending points of one manifold to points of another.

φ : M \to N

Differential of the Map

The differential of $φ$ at a point, often written with a star subscript or an asterisk, is the linear map between tangent spaces that underlies the pushforward operation.

φ_{*} : T_{p} M \to T_{φ (p)} N

Pushforward Notation

Notation for Vectors

Given a tangent vector $v$ at a point of $M$ , its pushforward under $φ$ is written with the star notation applied to the vector.

φ_{*} v \in T_{φ (p)} N

Notation for Contravariant Tensor Fields

The pushforward extends to general contravariant tensors when $φ$ is a diffeomorphism, since a two-sided inverse is required to move tensor fields consistently across the entire manifold rather than at a single point.

(φ_{*} T)^{i} = \frac{\partial y^{i}}{\partial x^{j}} T^{j}

Pullback Notation

Notation for Covectors

Given a covector $ω$ at a point of $N$ , its pullback under $φ$ is written with a superscript star applied to the covector.

φ^{*} ω \in T_{p}^{*} M

Pullback Defined Through Pushforward

The pullback of a covector acting on a tangent vector is defined by first pushing the vector forward and then evaluating the original covector on the result, which is expressed notationally as a composition.

(φ^{*} ω) (v) = ω (φ_{*} v)

Notation for Covariant Tensor Fields and Forms

The pullback extends naturally to arbitrary covariant tensor fields and differential forms without requiring $φ$ to be invertible, since covariant components transform using the Jacobian in a way that only requires a map into, rather than a diffeomorphism between, the two manifolds.

(φ^{*} T)_{i} = \frac{\partial y^{j}}{\partial x^{i}} T_{j}

Star and Asterisk Conventions

Subscript Star for Pushforward

The convention of placing the star as a subscript on the map symbol, as in $φ_{*}$ , signals an operation that moves objects forward, matching the direction of $φ$ itself.

Superscript Star for Pullback

The convention of placing the star as a superscript, as in $φ^{*}$ , signals an operation that moves objects backward, opposite to the direction of $φ$ .

Composition Behavior in Notation

Pushforward Composition Order

For two composable smooth maps, the pushforward of the composition equals the composition of the individual pushforwards taken in the same order as the maps.

(ψ \circ φ)_{*} = ψ_{*} \circ φ_{*}

Pullback Composition Order Reversal

For the same two composable maps, the pullback of the composition equals the composition of the individual pullbacks taken in reversed order, which is a defining structural feature distinguishing pullback notation from pushforward notation.

(ψ \circ φ)^{*} = φ^{*} \circ ψ^{*}

Distinction from General Tensor Transformation Notation

Pullback and pushforward notation differs from ordinary coordinate transformation notation in that it explicitly references a map between two possibly distinct manifolds, rather than describing a change of coordinates on a single fixed manifold. When the map is taken to be the identity on a single manifold, pullback and pushforward notation reduces to the ordinary notation for covariant and contravariant index transformation under a change of basis.