Orientation Forms in Different Codimension.

In summary, to define an orientation form for a codimension-1 submanifold embedded in a higher-dimensional space, one can use the nowhere-zero normal vector and an orthogonal basis for the tangent space to define a form. This method can also be extended to higher codimension submanifolds by contracting the orientation form of the ambient manifold with a set of lineally independent normal vector fields. In the case of a trivial normal bundle, the standard volume element of the ambient manifold can be contracted to obtain an orientation form. Additionally, if the normal bundle is orientable, then the submanifold must also be orientable. This can be extended to vector bundles that split as a direct sum of two subbundles.
  • #1
WWGD
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Hi, Everyone:

A way of defining an orientation form when given a codimension-1 ,

orientable n-manifold N embedded in R^{n+1} , in which

the gradient ( of the parametrized image ) is non-zero (I think n(x)

being nonzero is equivalent to N being orientable), is to

consider the nowhere-zero normal vector n(x), and to define the

form w(v)_x : = | n(x) v1 , v2 ,...,v_n-1| (**)

Where {vi}_i=1,..,n-1 is an orthogonal basis for T_x N , written

as column vectors, and n(x) is the vector normal to N at x , so that we write:

| n_1(x) v_11 v_21... v_n1|
| n_2(x) v_12 v_22...v_n2|
......

......
|n_n(x) v_1n v_2n...v_nn|

For vi= (vi1, vi2,...,vin )

Then the vectors in (##) are pairwise orthogonal, and so are

Linearly-independent.

*QUESTION* : How do we define a form for a curve of codimension-1,

and, in general, for orientable manifolds of codimension larger-

than 1 ? I have seen the expression t(x).v , meaning <t(x),v> ,for the curve. But the

tangent space of a curve is 1-dimensional, so, how is this a dot product?

Also, for codimension larger than one: do we use some sort of tensor contraction?

Thanks.

Thanks.
 
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  • #2
If the normal bundle to the submanifold is trivial then you can get an orientation form by contracting the orientation form of the ambient manifold by a smooth set of lineally independent normal vector fields.

In Euclidean space you contract the standard volume element.

If the normal bundle is not trivial i am not sure off of the top of my head but let's see if we can figure it out. It shouldn't be hard.

The normal bundle of a hypersurface (closed without boundary, codimension 1) of Euclidean space is always trivial but I do not know a direct proof and again would like to work on it with you. I do know a nasty indirect proof if you would like to see it.
 
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  • #3
lavinia said:
If the normal bundle to the submanifold is trivial then you can get an orientation form by contracting the orientation form of the ambient manifold by a smooth set of lineally independent normal vector fields.

In Euclidean space you contract the standard volume element.

If the normal bundle is not trivial i am not sure off of the top of my head but let's see if we can figure it out. It shouldn't be hard.

The normal bundle of a hypersurface (closed without boundary, codimension 1) of Euclidean space is always trivial but I do not know a direct proof and again would like to work on it with you. I do know a nasty indirect proof if you would like to see it.

Sure, I'll work on it with you. Do you have any suggestions/format in mind?
 
  • #4
WWGD said:
Sure, I'll work on it with you. Do you have any suggestions/format in mind?

Just looking at ideas plus some reading. My first thought is to see if there is something that can be dome with the Thom class of the normal bundle to the submanifold.
 
  • #5
I think you can Just contract the volume element of the ambient manifold by a local orthonormal basis for the normal bundle over each coordinate chart. Over each chart you get a local volume form for the submanifold and since the coordinate transformations are in SO(n) they should piece together to give you a global orientation for on the submanifold. Is this right?
 
  • #6
BTW: it follows that if the normal bundle is orientable then the submanifold must be orientable as well.
 
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  • #7
I think this is correct. Well done lavinia!
 
  • #8
lavinia said:
BTW: it follows that if the normal bundle is orientable then the submanifold must be orientable as well.

More generally, I think if a vector bundle E splits as a direct sum of two subbundle F and G, then 2 of them are orientable iff the third one is.
 

1. What is the concept of orientation forms in different codimension?

Orientation forms refer to a mathematical concept used in differential geometry to define a consistent direction or orientation for a given manifold. The codimension of a manifold is the difference between the dimension of the manifold and the dimension of the ambient space in which it is embedded.

2. How are orientation forms used in differential geometry?

In differential geometry, orientation forms are used to define a consistent orientation for a given manifold, which is important for calculating various geometric quantities such as volume, surface area, and curvature. They also play a crucial role in defining the orientation of tangent spaces and differential forms on a manifold.

3. What is the difference between orientation forms in different codimension?

The main difference between orientation forms in different codimension lies in their definition and properties. Orientation forms in higher codimension have more complex structures and require a different mathematical approach, while orientation forms in lower codimension are simpler and easier to understand.

4. Can orientation forms in different codimension be compared?

Yes, orientation forms in different codimension can be compared by using the concept of pullback. This is a mathematical operation that allows us to compare geometric objects defined on different manifolds. By pulling back an orientation form from a higher codimension manifold to a lower codimension one, we can compare their orientations and determine if they are the same or different.

5. What are some real-world applications of orientation forms in different codimension?

Orientation forms in different codimension have various applications in physics, engineering, and computer graphics. They are used to define the direction of magnetic fields, calculate the moment of inertia of rigid bodies, and generate realistic 3D models of surfaces and shapes. They also play a crucial role in understanding the topology of higher-dimensional spaces.

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