Pushforward

Definition

The pushforward is a mathematical operation that takes a function defined on one manifold and produces a function on another manifold via a smooth map between them. This operation is crucial in understanding how geometrical and topological properties transform when mapping from one space to another, linking concepts of differential geometry and symplectic geometry.

5 Must Know Facts For Your Next Test

1. The pushforward is denoted as $f_*: T_pM \to T_{f(p)}N$, where $f: M \to N$ is the smooth map from manifold $M$ to manifold $N$, and $T_pM$ and $T_{f(p)}N$ are the tangent spaces at points $p$ and $f(p)$ respectively.
2. The pushforward takes a vector in the tangent space of the source manifold and maps it to the tangent space of the target manifold, reflecting how vectors transform under the mapping.
3. One application of the pushforward is in calculating how differential forms transform under a smooth map, which is essential for integration on manifolds.
4. In symplectic geometry, pushforwards can be used to study how symplectic structures behave under smooth mappings, particularly in terms of preserving or altering their properties.
5. The pushforward operation is closely related to the concept of Lie derivatives, which measure how tensor fields change along vector fields.

Review Questions

• How does the pushforward operation relate to the concept of smooth maps between manifolds?
  • The pushforward operation directly relies on smooth maps between manifolds, as it transforms vectors from the tangent space of one manifold to another through these mappings. Specifically, given a smooth map $f: M \to N$, the pushforward $f_*$ allows us to take a vector at a point in $M$ and find its corresponding image in the tangent space at the mapped point in $N$. This relationship emphasizes how geometric structures are preserved or altered under smooth transformations.
• Discuss the importance of pushforwards in relation to differential forms and integration on manifolds.
  • Pushforwards are crucial for understanding how differential forms change when passed through a smooth map between manifolds. They allow us to compute integrals over transformed manifolds by translating forms from one space into another. This operation ensures that properties such as orientation and volume are correctly adjusted according to the mapping, making it possible to apply integral calculus in more complex geometrical contexts, particularly in symplectic geometry.
• Evaluate how understanding pushforwards enhances our ability to study symplectic structures and their properties under mappings.
  • Understanding pushforwards is fundamental for studying symplectic structures as they provide insight into how these structures behave under various smooth mappings. By analyzing the transformation of symplectic forms through pushforwards, we can determine whether certain properties are preserved or altered during these mappings. This knowledge plays a key role in applications such as Hamiltonian dynamics, where recognizing how phase spaces change under transformations leads to deeper insights into the underlying physical phenomena.

Related terms

smooth map: A smooth map is a function between manifolds that has continuous derivatives of all orders, ensuring the smoothness of the mapping process.

tangent bundle: The tangent bundle is a manifold that encapsulates all the tangent vectors at each point of another manifold, essential for analyzing the pushforward of vector fields.

differential: The differential of a function describes how the function changes in response to small changes in its inputs, playing a vital role in calculating pushforwards.