Titlebook: Geometry VI; Riemannian Geometry M. M. Postnikov Textbook 2001 Springer-Verlag Berlin Heidelberg 2001 Lie groups.Minimal surface.Riemannian

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Structural Equations. Local Symmetries,As we know (see Chap. 36), instead of the curvature tensor, it is convenient to consider the ..

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Lie Functor,The main goal of this chapter is to present the procedure for reconstructing a Lie group from its Lie algebra. Moreover, incidentally, we here present certain general mathematical concepts that were already mentioned repeatedly in passing.

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Affine Fields and Related Topics,As Exercise 5.5 shows, Lie groups are a particular case of symmetric spaces. This gives us an idea to generalize the construction of the Lie algebra of a Lie group to symmetric spaces. This can be done, but instead of Lie algebras, we obtain more general algebraic objects, as should be expected.

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Cartan Theorem,The Lie ternary . constructed in the previous chapter depends on the choice of the point .0∈., i.e., it is a function of the pair (.0). Such pairs are called .. A .: (.0) → (.0) of punctured spaces is a morphism . → . such that .(.0) = .0. It is clear that all punctured symmetric spaces and their morphisms form a category.

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Metric Properties of Geodesics,For a Riemannian (but not a pseudo-Riemannian) space . along with the energy Lagrangian, we can also consider the Lagrangian. which is expressed in local coordinates by

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Minimal Surfaces,We can replace the real coordinates . and . on a surface . with one complex coordinate . = . + .. In the case where the coordinates . and . are isothermal, the coordinate . is called a . on the surface. (Certain authors also apply this name to the coordinates . and ..)