An immersed submanifold is a manifold that is mapped into another manifold via a smooth immersion, meaning the map is smooth and its differential is injective at every point. Unlike embedded submanifolds, immersed submanifolds may have self-intersections or fail to be homeomorphic to their image in the ambient manifold.

Immersed Submanifold

Smooth immersion

An ambient manifold is a higher-dimensional space in which a lower-dimensional manifold is embedded or immersed. It provides the context or environment that allows the properties and behaviors of the embedded manifold to be studied and understood within a broader geometric framework.

Ambient Manifold

Embedded submanifolds

Self-intersections

Homeomorphic

A differential represents an infinitesimally small change in a function's value with respect to changes in its input, essentially capturing the function's rate of change at a particular point. It is a fundamental concept in calculus, underpinning the derivation of derivatives and integrals, and is crucial for understanding continuous change in various fields of science and engineering.

Differential

Injective

A manifold is a topological space that locally resembles Euclidean space, allowing for the application of calculus and other mathematical tools. Manifolds are fundamental in mathematics and physics, providing the framework for understanding complex structures like curves, surfaces, and higher-dimensional spaces.

Concepts

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