HelpA problem about embedded surface.

Thread starter aquarius
Start date Apr 30, 2009
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Surface

In summary, an embedded surface is a curved two-dimensional surface that is placed in a higher dimensional space. The topology of an embedded surface is defined by its genus, which represents the number of holes or handles it contains. Unlike regular surfaces, embedded surfaces have a specific placement in a higher dimensional space. They have various applications in fields like computer graphics, physics, and engineering. However, working with embedded surfaces can be challenging due to their complex mathematical descriptions and visual representations, as well as topological changes that can occur.

Apr 30, 2009

aquarius

I have a parametrization of a torus T^2:t(u,v)=((2+cosu)cosv,(2+cosu)sinv,sinu). How to show that T^2 is an embedded surface in R^3? How to show that the tangent bundleT(T^2) is trivial? Thanks.

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Apr 30, 2009

zhentil

There's only one way to show that's it's an embedded surface. Show it's a 1-1 immersion, and note that the torus is compact.

1. What is an embedded surface?

An embedded surface is a mathematical concept that describes a two-dimensional surface that is embedded in a higher dimensional space. In other words, it is a surface that is curved and occupies a three-dimensional space.

2. How do you define the topology of an embedded surface?

The topology of an embedded surface is defined by its genus, which is the number of holes or handles it contains. For example, a sphere has a genus of 0, while a torus has a genus of 1.

3. What is the difference between an embedded surface and a regular surface?

An embedded surface is a type of regular surface that has a specific embedding or placement in a higher dimensional space. Regular surfaces can also include non-embedded surfaces, such as self-intersecting surfaces or surfaces with singularities.

4. What are some real-world applications of embedded surfaces?

Embedded surfaces have many applications in fields such as computer graphics, physics, and engineering. They are used to model and simulate surfaces in 3D environments, such as in video games and computer-aided design. They are also used in physics to study the properties of curved space and in engineering to design and optimize surfaces for various purposes.

5. What are some challenges in studying and working with embedded surfaces?

One of the main challenges in working with embedded surfaces is the complexity of their mathematical descriptions and calculations. Another challenge is visualizing and representing these surfaces accurately, as they exist in a higher dimensional space. Additionally, topological changes in the surface, such as self-intersections, can make it difficult to analyze and work with embedded surfaces.