Cartesian product of orientable manifolds

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Start date Apr 28, 2009
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Cartesian Manifolds Product

In summary, the Cartesian product of two orientable manifolds is a new manifold formed by taking the product of their individual points, with a dimension equal to the sum of the dimensions of the individual manifolds. The orientation of the Cartesian product is determined by the orientations of the individual manifolds, and it is always orientable. The dimension of the Cartesian product is equal to the sum of the individual dimensions. Some applications of this concept include its use in differential geometry and topology to study the properties and structure of spaces, and in physics to model spacetime.

Apr 28, 2009

EgUaLuEsRs07

The problem is to prove that if M and N are orientable manifolds, then MxN is an orientable manifold

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Apr 28, 2009

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EgUaLuEsRs07 said:

The problem is to prove that if M and N are orientable manifolds, then MxN is an orientable manifold

Hi EgUaLuEsRs07! Welcome to PF!

Show us how far you get, and where you're stuck, and then we'll know how to help!

1. What is the definition of the Cartesian product of orientable manifolds?

The Cartesian product of two orientable manifolds M and N is a new manifold M x N formed by taking the product of their individual points, resulting in a new manifold with the dimension of the sum of the dimensions of M and N.

2. How is the orientation of the Cartesian product determined?

The orientation of the Cartesian product is determined by the orientations of the individual manifolds. If M and N have orientations defined by orientation forms ω and η respectively, then the orientation of M x N is given by the orientation form ω ∧ η, where ∧ denotes the exterior product.

3. Can the Cartesian product of orientable manifolds be non-orientable?

No, the Cartesian product of orientable manifolds is always orientable. This is because the product of two orientable manifolds results in a new manifold with an orientation form that can be defined by the product of the orientation forms of the individual manifolds.

4. How does the Cartesian product of orientable manifolds relate to the concept of dimension?

The dimension of the Cartesian product of two orientable manifolds is equal to the sum of their individual dimensions. For example, if M is a m-dimensional manifold and N is a n-dimensional manifold, then the Cartesian product M x N will be a (m+n)-dimensional manifold.

5. What are some applications of the Cartesian product of orientable manifolds?

The Cartesian product of orientable manifolds is a useful concept in mathematics and physics. It is used in differential geometry to study the properties of manifolds and in topology to study the structure of spaces. In physics, it is used in the study of spacetime, where spacetime is often modeled as a product of 3-dimensional space and 1-dimensional time.