- #1
hadi amiri 4
- 98
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show that there is no immersion of [/n] into [R][/n]
An immersion of S^n into R^n is a smooth map that preserves the local structure of the n-dimensional sphere S^n in the n-dimensional Euclidean space R^n. In other words, it is a way of embedding the sphere into the space without any self-intersections or overlapping points.
This is an important question in differential topology, as it helps us understand the limitations of embedding objects into higher-dimensional spaces. It also has implications in other fields, such as physics and computer graphics, where the ability to embed objects into higher-dimensional spaces is crucial.
No, it is not possible to give an example of an immersion of S^2 into R^2. This is because the famous Whitney embedding theorem states that there is no immersion of an n-sphere into R^n for n > 1.
The proof of this statement involves using algebraic topology and differential topology techniques. One approach is to use the Brouwer degree, which is a topological invariant that measures the number of times a map wraps around a given point. By showing that the Brouwer degree of any potential immersion is zero, we can conclude that there is no immersion of S^n into R^n.
Yes, there are other spaces into which S^n cannot be immersed. For example, there is no immersion of S^n into any space with non-trivial fundamental group, such as a torus. This is known as the Hatcher-Smale theorem. Additionally, there are spaces where the notion of immersion does not even make sense, such as non-orientable manifolds or spaces with non-integer dimensions.