Nota: A tradução desta entrada está atualmente em revisão de qualidade, portanto parte do conteúdo é exibida temporariamente apenas em inglês.
Este verbete ainda não foi traduzido para o seu idioma, portanto o original é exibido abaixo.
submanifold
This term is used exclusively within the specialized register of differential geometry and topology. It describes a specific relationship where a smaller geometric space is nested within a larger one while maintaining the same structural properties, such as smoothness or continuity. It is a technical term of art and is virtually never encountered in general conversation or non-mathematical writing.
In a mathematical context, the term implies a rigorous inheritance of properties from the ambient manifold. While a subset might be any collection of points, a submanifold must satisfy strict local conditions to ensure it behaves like a manifold in its own right, distinguishing it from arbitrary subsets or singular spaces.
Meanings
Examples
The sphere is a two dimensional submanifold of three dimensional Euclidean space.
The sphere is a two dimensional submanifold of three dimensional Euclidean space.
Is this particular subset actually a smooth submanifold?
Is this particular subset actually a smooth submanifold?
I need to determine if the image of this map is an embedded submanifold.
The intersection of two transverse submanifolds is also a submanifold.
The intersection of two transverse submanifolds is also a submanifold.
Wait, the boundary of the manifold is itself a submanifold of one lower dimension.
He wondered if the critical points formed a closed submanifold.
We can define a Riemannian metric on the submanifold by restricting the metric of the ambient space.
We can define a Riemannian metric on the submanifold by restricting the metric of the ambient space.
Let us assume that M is a compact submanifold of R n.