Mathematical Methods of Classical Mechanics by V. I. Arnold

Author:V. I. Arnold
Language: eng
Format: epub, pdf
Publisher: Springer-Verlag Wien 2012
Published: 2014-06-16T16:00:00+00:00

E The curvature tensor

We now consider, as in the two-dimensional case, parallel translation along small closed paths beginning and ending at a point of a riemannian manifold. Parallel translation along such a path returns vectors to the original tangent space. The map of the tangent space to itself thus obtained is a small rotation (an orthogonal transformation close to the identity). In the two-dimensional case we characterized this rotation by one number—the angle of rotation φ. In higher dimensions a skew-symmetric operator plays the role of φ. Namely, any orthogonal operator A which is close to the identity can be written in a natural way in the form

where Φ is a small skew-symmetric operator.

Problem. Compute Φ if A is a rotation of the plane through a small angle φ.

Answer.

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