Lesson plan
Exponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include: * exponential map (Riemannian geometry) for a manifold with a Riemannian metric, * exponential map (Lie theory) from a Lie algebra to a Lie group, * More generally, in a manifold with an affine connection, X \mapsto \gamma_X(1) , where \gamma_X is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections. * Euler's formula forming the unit circle in the complex plane.
Article summary
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include: * exponential map (Riemannian geometry) for a manifold with a Riemannian metric, * exponential map (Lie theory) from a Lie algebra to a Lie group, * More generally, in a manifold with an affine connection, X \mapsto \gamma_X(1) , where \gamma_X is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections. * Euler's formula forming the unit circle in the complex plane.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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