Chart

curvilinear coordinate system, parametrization of a set $ M $

A one-to-one mapping

$$ x: M \rightarrow D,\ \ p \rightarrow x ( p) = ( x ^ {1} ( p) \dots x ^ {n} ( p)) , $$

of a set $ M $ onto an open subset $ D $ of the real vector space $ \mathbf R ^ {n} $. The integer $ n $ is called the dimension of the chart, and the components $ x ^ {i} ( p) $ of the vector $ x ( p) \in \mathbf R ^ {n} $ are called the coordinates of $ p \in M $ with respect to the chart $ x $.

An example of a chart is the Cartesian coordinate system in the plane and in space, introduced by P. Fermat and R. Descartes, and taken by them as the basis for analytic geometry. e root','../p/p074630.htm','Series','../s/s084670.htm','Stability of an elastic system','../s/s087010.htm','Siegel disc','../s/s110120.htm','Theta-function','../t/t092600.htm','Trigonometric series','../t/t094240.htm','Two-term congruence','../t/t094620.htm','Umbral calculus','../u/u095050.htm','Variation of constants','../v/v096160.htm','Variational calculus','../v/v096190.htm','Variational calculus, numerical methods of','../v/v096210.htm','Venn diagram','../v/v096550.htm','Zeta-function','../z/z099260.htm')" style="background-color:yellow;">L. Euler was the first to employ charts (curvilinear coordinates) on surfaces in geometric research. B. Riemann took up the notion of a chart as the basis for a new infinitesimal approach to the foundations of geometry (see [1]). In Riemann's view, the basic object of study in geometry is a manifold — a set $ M $ endowed with a chart. The modern concept of a manifold is a natural generalization of Riemann's definition.

A chart $ x: U \rightarrow D $ of some subset $ U $ of $ M $ is called a local chart of $ M $ with domain of definition $ U $. If $ M $ is endowed with the structure of a topological space, then it is further required that $ U $ be an open subset of $ M $ and that the mapping $ x $ be a homeomorphism. A chart can similarly be defined with values in $ F ^ { n } $, where $ F $ is any normed field, and more generally, a chart can take values in a topological vector space. Two local charts $ ( x, U) $, $ ( y, V) $ with domains of definition $ U, V $ in $ M $ are said to be compatible of class $ C ^ { l } $ if 1) their common domain of definition $ W = U \cap V $ is mapped by both charts onto an open set (that is, the sets $ x ( W) $ and $ y ( W) $ are open in $ \mathbf R ^ {n} $); and 2) the coordinates of a point of $ W $ with respect to one of these charts are $ l $ times continuously-differentiable functions of the coordinates of the same point with respect to the other chart, that is, the vector function

$$ y \circ x ^ {-} 1 : \ x ( W) \rightarrow y ( W) $$

is $ l $ times continuously differentiable. A family $ A = \{ ( x _ \alpha , U _ \alpha ) \} $ of pairwise-compatible local charts $ ( x _ \alpha , U _ \alpha ) $ of $ M $ that cover $ M $( that is, $ \cup _ \alpha U _ \alpha = M $) is called an atlas of $ M $. The specification of an atlas defines on $ M $ the structure of a differentiable manifold, and local charts that are compatible with all the charts of this atlas are said to be admissible (or $ C ^ { l } $- smooth).

The infinitesimal analogue of the notion of a chart is the concept of an infinitesimal chart of order $ k $( or a $ k $- jet (of a chart) or a co-frame of order $ k $). Two compatible local charts $ ( x, U) $, $ ( y, V) $ of a set $ M $ are said to be tangent to each other up to order $ k $ at a point $ p \in U \cap V $ if $ x ( p) = y ( p) $ and if all the partial derivatives up to order $ k $, inclusive, of the vector function $ y \circ x ^ {-} 1 : x \rightarrow y ( x) $ vanish at $ x ( p) $. The class $ j _ {p} ^ {k} ( x) $ of local charts tangent (up to order $ k $) at a point $ p \in U $ of an admissible local chart $ ( x, U) $ of a differentiable manifold $ M $ is called the infinitesimal chart of order $ k $ at $ p $, or $ k $- jet at $ p $.

The choice of a chart on a manifold $ M $ allows one to consider various field quantities on $ M $ as numerical functions and to apply to them the methods of analysis. In general, the value of a field quantity at a point depends on the choice of the chart. (Quantities which are independent of the choice of the chart are called scalars and are described by functions on $ M $.) However, for a wide and most important class of quantities (see Geometric objects, theory of), their value at a point depends only on the structure of the chart in the $ k $- th order infinitesimal neighbourhood of this point. Such quantities (examples of which are the tensor fields) are described by functions on the set of all co-frames of order $ k $ on $ M $. Along with these one studies the properties of quantities which do not depend on the choice of a chart. In this connection, the invariant coordinate-free approach to problems of differential geometry proves to be highly effective.

References

[1]	B. Riemann, "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)
[2]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[3]	R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Birkhäuser (1972)
[4]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[5]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)

Comments

For Riemann's view see, in particular, [1].

References

[a1]	O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1967)
[a2]	R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)