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More generally, if g is C k, α (with k larger than one) and Ric(g) is C l, α relative to some coordinate charts, then the transition function to a harmonic coordinate chart will be C k + 1, α, and so Ric(g) will be C min(l, k), α in harmonic coordinate charts. So, by the previous result, g will be C min(l, k) + 2, α in harmonic coordinate ...
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions [clarification needed]: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate ...
Testo SE & Co. KGaA is a company from Lenzkirch, founded in 1957, with its headquarters in Titisee-Neustadt, Germany. The company has 35 subsidiary firms in China, Japan, Korea, USA, France, Spain, Italy, and other countries and employs around 3200 people. 1700 alone at its sites in Lenzkirch, Kirchzarten , and Titisee.
Flag manifold; Grassmann manifold; Stiefel manifold; Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.
The Whitney embedding theorem showed that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition, showing that the two concepts of manifold were equivalent. Due to this unification, it is said to be the first complete exposition of the modern concept of manifold.
This image is a derivative work of the following images: Image:Circle_manifold_chart_from_slope.png licensed with Cc-by-sa-2.5 . 2005-08-05T22:24:43Z KSmrq 1000x1000 (77431 Bytes) {{cc-by-sa-2.5}} PNG file created as SVG, rendered by Batik, and uploaded by author.
This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa. In general, computations with the maximal atlas of a manifold are rather unwieldy.
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