# Orientation of a Submanifold, given an Orientation of Ambient Manifold

• WWGD
In summary, the condition for orientability of a submanifold S of an oriented manifold M is that the normal bundle to S is orientable, which can be determined by checking if the first Stiefel-Whitney class of the normal bundle is zero. If S is orientable, a positively oriented basis for the tangent space of S can be obtained by extending a positively oriented basis of the normal bundle with tangent vectors. If S is not orientable, its normal bundle cannot be orientable.
WWGD
Gold Member
Hi, All:

Say S is a submanifold of an ambient, oriented manifold M; M is embedded in some R^k;
let ## w_m ## be an orientation form for M.

I'm trying to see under what conditions I can orient S , by contracting ## w_m ## , i.e., by
using the interior product with the "right" vector field X. Clearly this is not always possible, since
not every submanifold is orientable, e.g., even-dimensional projective spaces embedded in
odd-dimensional projective spaces. Just curious as to what vector field we can use, and how
to determine when this can or cannot be done, i.e., how this process could fail when S is not
orientable.

Thanks.

I think the condition you are looking for is that the normal bundle to the submanifold,S,is orientable(not necessarily trivial). Then S will be orientable if M is.

At each point of S choose a positively orientated basis for the normal bundle. Extend this basis with vectors tangent to S to a positively oriented basis of M. These tangent vector will give you a positively oriented basis for the tangent space of S.

If the normal bundle to S is trivial, then one gets a set of linearly independent normal vector fields along S. Choose some ordering of these vector fields and contract the orientation form of M with them. This will give you an orientation form for S.

One sees from this that if S is not orientable then its normal bundle can not be orientable.

Form an Algebraic Topology point of view, if S is non-orientable then its first Stiefel-Whitney class is not zero. But since M is orientable the Whitney sum of the tangent and normal bundles to S must have zero first Stiefel Whitney class. This can only happen of the first Stiefel Whitey class of the normal bundle equals the first Stiefel Whitney class of S ( by the Whitney sum formula.)

## 1. What is the meaning of "orientation" in the context of submanifolds?

In mathematics, the concept of orientation refers to the direction or ordering of a set of basis vectors in a vector space or manifold. For submanifolds, this refers to the direction in which the submanifold is embedded within the ambient manifold.

## 2. How is the orientation of a submanifold determined given an orientation of the ambient manifold?

The orientation of a submanifold can be determined by examining the orientation of the basis vectors of the tangent space of the submanifold. If the basis vectors align with the orientation of the ambient manifold, then the submanifold is said to have the same orientation. If they are opposite, then the submanifold is said to have the opposite orientation.

## 3. Can a submanifold have a different orientation from the ambient manifold?

Yes, a submanifold can have the opposite orientation from the ambient manifold. This means that the basis vectors of the tangent space of the submanifold are aligned in the opposite direction from the basis vectors of the tangent space of the ambient manifold.

## 4. How does the orientation of a submanifold affect its properties?

The orientation of a submanifold can affect its properties in various ways. For example, the orientation can determine whether a submanifold is orientable or non-orientable. It can also impact calculations involving the submanifold, such as integration and curvature calculations.

## 5. Why is it important to consider the orientation of a submanifold?

Considering the orientation of a submanifold is important because it affects how the submanifold is embedded and how its properties are defined. It also helps to distinguish between different submanifolds that may have the same shape but different orientations, such as a Mobius strip and a cylinder.

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