Fangyang Tian said:Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries??
Many Thanks!
The Regular Point Theorem of Manifolds with Boundaries is a fundamental result in topology that states that every point on a manifold with boundaries has a neighborhood that is homeomorphic to an open subset of a Euclidean space. This means that manifolds with boundaries are locally similar to flat, Euclidean spaces.
The Regular Point Theorem of Manifolds with Boundaries is used in many areas of mathematics, including differential geometry, topology, and mathematical physics. It is a key tool for studying the local properties of manifolds with boundaries and is often used to prove other theorems in these fields.
Yes, the Regular Point Theorem of Manifolds with Boundaries can be extended to higher dimensions. In fact, the theorem is a special case of a more general result known as the Whitney Embedding Theorem, which states that any smooth manifold can be embedded in a Euclidean space of sufficiently high dimension.
Yes, the Regular Point Theorem of Manifolds with Boundaries has many applications in real-world problems, particularly in physics and engineering. It is used to model and analyze various physical systems, such as fluid dynamics, electromagnetism, and general relativity.
The Regular Point Theorem of Manifolds with Boundaries is an important result for understanding the topology of space. It tells us that, locally, space can be approximated by a flat, Euclidean space. This has implications for our understanding of the global structure of space and how it can be deformed and distorted without changing its local properties.