In differential geometry, pushforward and pullback are operations associated with smooth maps between manifolds that allow the transfer of geometric structures such as vector fields and differential forms. The pushforward acts on tangent vectors, while the pullback acts on differential forms, facilitating the study of how structures transform under mappings.

Pushforward And Pullback

Differential geometry is the field of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry, particularly those involving curves and surfaces. It plays a crucial role in understanding the geometry of differentiable manifolds and has applications in physics, particularly in the theory of general relativity and modern theoretical physics.

differential geometry

Pushforward operations

Pullback operations

smooth maps

Manifolds are mathematical spaces that locally resemble Euclidean space and are used to generalize concepts from calculus and geometry to more complex shapes. They play a crucial role in fields like differential geometry, topology, and theoretical physics, where they provide a framework for understanding complex structures and spaces.

manifolds

Geometric structures are mathematical frameworks that study the properties and relations of points, lines, surfaces, and solids in space. They are foundational in fields such as topology, algebraic geometry, and differential geometry, providing essential insights into the nature of shapes and spaces.

geometric structures

A vector field is a mathematical construct that assigns a vector to every point in a subset of space, often used to represent physical quantities like velocity fields in fluid dynamics or electromagnetic fields. They are essential in understanding and visualizing the behavior of vector quantities across different regions in space, providing insights into the direction and magnitude of forces or flows.

vector fields

Differential forms provide a unified approach to multivariable calculus and are essential in fields such as differential geometry and topology. They generalize the concepts of gradients, divergences, and curls, allowing for the integration over manifolds of any dimension and offering a coordinate-free framework for calculus on manifolds.

differential forms

Tangent vectors are mathematical constructs that represent directions of curves or surfaces in a given space, typically within differential geometry. They provide crucial local approximations of manifolds and are foundational elements in understanding how spaces curve and change around particular points.

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