Products of Embedded Submanifolds

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In summary, Products of Embedded Submanifolds are a way of constructing a new submanifold by taking the Cartesian product of two or more embedded submanifolds. They have various applications in mathematics, including in differential geometry, topology, and algebraic geometry. They can be generalized to higher dimensions and have connections to other mathematical concepts such as direct product and tensor product. Additionally, Products of Embedded Submanifolds have real-world applications in fields such as computer graphics, robotics, and physics.
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Arkuski
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I'm trying to come up with a simple proof that if [itex]M[/itex] is an embedded submanifold of [itex]N[/itex], and [itex]P[/itex] is an embedded submanifold of [itex]Q[/itex], then [itex]M×P[/itex] is an embedded submanifold of [itex]N×Q[/itex]. I'm thinking this could be easily done using showing that [itex]M×P[/itex] satisfies the local [itex]k[/itex]-slice condition, or that the product of smooth embeddings (from the respective inclusion maps) is also a smooth embedding.
 
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  • #2
Yes, both the slice-condition as the map-condition work. But what did you try? Where are you stuck?
 

Related to Products of Embedded Submanifolds

1. What are Products of Embedded Submanifolds?

Products of Embedded Submanifolds refer to the mathematical concept of constructing a new submanifold by taking the Cartesian product of two or more embedded submanifolds. It is a way of combining multiple submanifolds into a single structure.

2. How are Products of Embedded Submanifolds useful in mathematics?

Products of Embedded Submanifolds have various applications in mathematics, such as in differential geometry, topology, and algebraic geometry. They provide a way to study and understand complex structures by breaking them down into simpler components.

3. Can Products of Embedded Submanifolds be generalized to higher dimensions?

Yes, Products of Embedded Submanifolds can be generalized to higher dimensions. In fact, the concept of Cartesian product can be extended to any number of submanifolds, regardless of their dimensions.

4. How do Products of Embedded Submanifolds relate to other mathematical concepts?

Products of Embedded Submanifolds are closely related to the concept of direct product in abstract algebra. They also have connections to the tensor product in linear algebra and the direct sum in vector spaces.

5. Are there any real-world applications of Products of Embedded Submanifolds?

Yes, Products of Embedded Submanifolds have various real-world applications, such as in computer graphics, robotics, and physics. They have been used to model and analyze complex systems and structures in these fields.

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