how to visualise the orthonormal frame bundle? The orthonormal frame bundles $O(\Sigma)$ of $\Sigma$ is the set of pairs $(x,H)$, where x is a point of $\Sigma $ and H is an orthonormal frame of tangent vectors to $\Sigma $ at x. can it be visualised as the tangent space attached to each point x.
2026-05-04 12:22:39.1777897359
how to visualise orthonormal frame bundle?
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To visualize the orthonormal plane bundle of, say, the space in front of your face, hold up your right hand with the thumb, forefinger, and middle finger at right angles to each other.
Keeping your hand in the same position, rotate the hand rigidly through many different rotated positions, but keeping those three fingers at right angles to each other. As the hand rotates, you are moving through the fiber of the frame bundle over one point.
Next, move your hand until it is in a different position. Now, keeping it in that position, again rotate your hand rigidly through many different rotated positions. You are now moving through the fiber of the frame bundle over a different point.