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My book says that the cusp y=x^2/3 is not an embedded submanifold of R². Why is that?
mathwonk said:every line through the origin of that set has intersection number ≥ 2 with the set. for a manifold, the generic intersection number will be one.
jostpuur said:Isn't the content of the posts #4 and #5 the same?
The cusp is not a submanifold because it does not satisfy the definition of a submanifold, which states that it must be locally diffeomorphic to Euclidean space. The cusp fails to meet this criteria because it has a singularity at the tip, making it non-smooth.
In some cases, the cusp may be considered a submanifold if it is given a different smooth structure. For example, if we consider the cusp as a subset of a different space, such as complex space, it may satisfy the definition of a submanifold. However, in the standard definition of a submanifold in Euclidean space, the cusp does not qualify.
The fact that the cusp is not a submanifold has important implications in differential geometry and topology. It highlights the importance of smoothness and the limitations of the definition of a submanifold. It also demonstrates the intricacies and complexities of studying geometric objects.
Yes, there are many other geometric objects that do not qualify as submanifolds. Some examples include cones, self-intersecting curves, and fractals. These objects also fail to meet the criteria of being locally diffeomorphic to Euclidean space.
The cusp challenges our understanding of submanifolds and reminds us that there are exceptions to every rule. It also encourages us to think more deeply about the definitions and properties of submanifolds and how they can be extended or modified to accommodate more complex geometric objects.