Fixing orientation by fixing a frame in a tangent space

In summary, the conversation discusses the equivalence between fixing the orientation of a k-manifold smooth connected S in R^n and fixing a frame for one of its tangent spaces. It also mentions that different orientations correspond to orienting atlases with overlapping domains that cannot be consistent with each other. The conversation then addresses the statement made about fixing a frame anywhere on S to determine its orientation, and clarifies that this only applies to simply connected manifolds, as explained in Claudio Gorodski's answer on a related topic.
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I would like to show that fixing the orientation of k-manifold smooth connected ##S## in ##\mathbb {R} ^ n ## is equivalent to fixing a frame for one of its tangent spaces.

What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with maps in other orienting atlases of other equivalence classes, when their domains of action overlap.

How could we pass from this fact, to the fact that it is enough to fix a frame anywhere ##x_0 \in S ## to determine the orientation of ## S ##?
 
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FAQ: Fixing orientation by fixing a frame in a tangent space

1. What is meant by "fixing orientation by fixing a frame in a tangent space"?

Fixing orientation refers to setting a consistent direction or orientation for a given system. In the context of a tangent space, this means choosing a specific frame or set of axes to use for measuring and describing the system's orientation.

2. Why is fixing orientation important in scientific research?

Fixing orientation allows for consistent and accurate measurements and descriptions of a system. It also helps to avoid confusion and discrepancies when comparing data from different experiments or studies.

3. How is a frame fixed in a tangent space?

A frame can be fixed in a tangent space by choosing a specific set of axes or vectors to represent the orientation of the system. This can be done by using physical markers or reference points, or by using mathematical calculations to determine the frame's orientation.

4. Can orientation be fixed in any type of space?

Yes, orientation can be fixed in any type of space, including tangent spaces. However, the method for fixing orientation may vary depending on the type of space and the specific system being studied.

5. Are there any limitations to fixing orientation by fixing a frame in a tangent space?

One limitation is that fixing orientation in a tangent space may not accurately represent the orientation of the system in other spaces. Additionally, if the frame is not fixed correctly, it can lead to errors in measurements and data analysis.

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