Coordinate Charts on Differentiable Manifolds¶

The class DiffChart implements coordinate charts on a differentiable manifold over a topological field $K$ (in most applications, $K = \RR$ or $K = \CC$ ).

The subclass RealDiffChart is devoted to the case $K=\RR$ , for which the concept of coordinate range is meaningful. Moreover, RealDiffChart is endowed with some plotting capabilities (cf. method plot()).

Transition maps between charts are implemented via the class DiffCoordChange.

AUTHORS:

Eric Gourgoulhon, Michal Bejger (2013-2015) : initial version

REFERENCES:

Chap. 1 of [Lee13] J.M. Lee : Introduction to Smooth Manifolds, 2nd ed., Springer (New York) (2013)

class sage.manifolds.differentiable.chart.DiffChart(domain, coordinates='', names=None)¶

Bases: sage.manifolds.chart.Chart

Chart on a differentiable manifold.

Given a differentiable manifold $M$ of dimension $n$ over a topological field $K$ , a chart is a member $(U,\varphi)$ of the manifold’s differentiable atlas; $U$ is then an open subset of $M$ and $\varphi: U \rightarrow V \subset K^n$ is a homeomorphism from $U$ to an open subset $V$ of $K^n$ .

The components $(x^1,\ldots,x^n)$ of $\varphi$ , defined by $\varphi(p) = (x^1(p),\ldots,x^n(p))\in K^n$ for any point $p\in U$ , are called the coordinates of the chart $(U,\varphi)$ .

INPUT:

domain – open subset $U$ on which the chart is defined
coordinates – (default: ‘’ (empty string)) single string defining the coordinate symbols, with ‘ ‘ (whitespace) as a separator; each item has at most two fields, separated by ‘:’:
1. The coordinate symbol (a letter or a few letters)
2. (optional) The LaTeX spelling of the coordinate; if not provided the coordinate symbol given in the first field will be used.
If it contains any LaTeX expression, the string coordinates must be declared with the prefix ‘r’ (for “raw”) to allow for a proper treatment of LaTeX’s backslash character (see examples below). If no LaTeX spelling is to be set for any coordinate, the argument coordinates can be omitted when the shortcut operator <,> is used via Sage preparser (see examples below)
names – (default: None) unused argument, except if coordinates is not provided; it must then be a tuple containing the coordinate symbols (this is guaranteed if the shortcut operator <,> is used).

EXAMPLES:

A chart on a complex 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M', field='complex')
sage: X = M.chart('x y'); X
Chart (M, (x, y))
sage: latex(X)
\left(M,(x, y)\right)
sage: type(X)
<class 'sage.manifolds.differentiable.chart.DiffChart'>

To manipulate the coordinates $(x,y)$ as global variables, one has to set:

sage: x,y = X[:]

However, a shortcut is to use the declarator <x,y> in the left-hand side of the chart declaration (there is then no need to pass the string 'x y' to chart()):

sage: M = Manifold(2, 'M', field='complex')
sage: X.<x,y> = M.chart(); X
Chart (M, (x, y))

The coordinates are then immediately accessible:

sage: y
y
sage: x is X[0] and y is X[1]
True

The trick is performed by Sage preparser:

sage: preparse("X.<x,y> = M.chart()")
"X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)"

Note that x and y declared in <x,y> are mere Python variable names and do not have to coincide with the coordinate symbols; for instance, one may write:

sage: M = Manifold(2, 'M', field='complex')
sage: X.<x1,y1> = M.chart('x y'); X
Chart (M, (x, y))

Then y is not known as a global Python variable and the coordinate $y$ is accessible only through the global variable y1:

sage: y1
y
sage: latex(y1)
y
sage: y1 is X[1]
True

However, having the name of the Python variable coincide with the coordinate symbol is quite convenient; so it is recommended to declare:

sage: M = Manifold(2, 'M', field='complex')
sage: X.<x,y> = M.chart()

In the above example, the chart X covers entirely the manifold M:

sage: X.domain()
2-dimensional complex manifold M

Of course, one may declare a chart only on an open subset of M:

sage: U = M.open_subset('U')
sage: Y.<z1, z2> = U.chart(r'z1:\zeta_1 z2:\zeta_2'); Y
Chart (U, (z1, z2))
sage: Y.domain()
Open subset U of the 2-dimensional complex manifold M

In the above declaration, we have also specified some LaTeX writing of the coordinates different from the text one:

sage: latex(z1)
{\zeta_1}

Note the prefix r in front of the string r'z1:\zeta_1 z2:\zeta_2'; it makes sure that the backslash character is treated as an ordinary character, to be passed to the LaTeX interpreter.

Coordinates are Sage symbolic variables (see sage.symbolic.expression):

sage: type(z1)
<type 'sage.symbolic.expression.Expression'>

In addition to the Python variable name provided in the operator <.,.>, the coordinates are accessible by their indices:

sage: Y[0], Y[1]
(z1, z2)

The index range is that declared during the creation of the manifold. By default, it starts at 0, but this can be changed via the parameter start_index:

sage: M1 = Manifold(2, 'M_1', field='complex', start_index=1)
sage: Z.<u,v> = M1.chart()
sage: Z[1], Z[2]
(u, v)

The full set of coordinates is obtained by means of the operator [:]:

sage: Y[:]
(z1, z2)

Each constructed chart is automatically added to the manifold’s user atlas:

sage: M.atlas()
[Chart (M, (x, y)), Chart (U, (z1, z2))]

and to the atlas of the chart’s domain:

sage: U.atlas()
[Chart (U, (z1, z2))]

Manifold subsets have a default chart, which, unless changed via the method set_default_chart(), is the first defined chart on the subset (or on a open subset of it):

sage: M.default_chart()
Chart (M, (x, y))
sage: U.default_chart()
Chart (U, (z1, z2))

The default charts are not privileged charts on the manifold, but rather charts whose name can be skipped in the argument list of functions having an optional chart= argument.

The action of the chart map $\varphi$ on a point is obtained by means of the call operator, i.e. the operator ():

sage: p = M.point((1+i, 2), chart=X); p
Point on the 2-dimensional complex manifold M
sage: X(p)
(I + 1, 2)
sage: X(p) == p.coord(X)
True

See also

RealDiffChart for charts on differentiable manifolds over $\RR$ .

transition_map(other, transformations, intersection_name=None, restrictions1=None, restrictions2=None)¶

Construct the transition map between the current chart, $(U,\varphi)$ say, and another one, $(V,\psi)$ say.

If $n$ is the manifold’s dimension, the transition map is the map

$\psi\circ\varphi^{-1}: \varphi(U\cap V) \subset K^n \rightarrow \psi(U\cap V) \subset K^n,$

where $K$ is the manifold’s base field. In other words, the transition map expresses the coordinates $(y^1,\ldots,y^n)$ of $(V,\psi)$ in terms of the coordinates $(x^1,\ldots,x^n)$ of $(U,\varphi)$ on the open subset where the two charts intersect, i.e. on $U\cap V$ .

By definition, the transition map $\psi\circ\varphi^{-1}$ must be of classe $C^k$ , where $k$ is the degree of differentiability of the manifold (cf. diff_degree()).

INPUT:

other – the chart $(V,\psi)$
transformations – tuple (or list) $(Y_1,\ldots,Y_2)$ , where $Y_i$ is the symbolic expression of the coordinate $y^i$ in terms of the coordinates $(x^1,\ldots,x^n)$
intersection_name – (default: None) name to be given to the subset $U\cap V$ if the latter differs from $U$ or $V$
restrictions1 – (default: None) list of conditions on the coordinates of the current chart that define $U\cap V$ if the latter differs from $U$ . restrictions1 must be a list of of symbolic equalities or inequalities involving the coordinates, such as x>y or x^2+y^2 != 0. The items of the list restrictions1 are combined with the and operator; if some restrictions are to be combined with the or operator instead, they have to be passed as a tuple in some single item of the list restrictions1. For example, restrictions1 = [x>y, (x!=0, y!=0), z^2<x] means (x>y) and ((x!=0) or (y!=0)) and (z^2<x). If the list restrictions1 contains only one item, this item can be passed as such, i.e. writing x>y instead of the single-element list [x>y].
restrictions2 – (default: None) list of conditions on the coordinates of the chart $(V,\psi)$ that define $U\cap V$ if the latter differs from $V$ (see restrictions1 for the syntax)

OUTPUT:

The transition map $\psi\circ\varphi^{-1}$ defined on $U\cap V$ , as an instance of DiffCoordChange.

EXAMPLES:

Transition map between two stereographic charts on the circle $S^1$ :

sage: M = Manifold(1, 'S^1')
sage: U = M.open_subset('U') # Complement of the North pole
sage: cU.<x> = U.chart() # Stereographic chart from the North pole
sage: V = M.open_subset('V') # Complement of the South pole
sage: cV.<y> = V.chart() # Stereographic chart from the South pole
sage: M.declare_union(U,V)   # S^1 is the union of U and V
sage: trans = cU.transition_map(cV, 1/x, intersection_name='W',
....:                           restrictions1= x!=0, restrictions2 = y!=0)
sage: trans
Change of coordinates from Chart (W, (x,)) to Chart (W, (y,))
sage: trans.display()
y = 1/x

The subset $W$ , intersection of $U$ and $V$ , has been created by transition_map():

sage: M.list_of_subsets()
[1-dimensional differentiable manifold S^1,
 Open subset U of the 1-dimensional differentiable manifold S^1,
 Open subset V of the 1-dimensional differentiable manifold S^1,
 Open subset W of the 1-dimensional differentiable manifold S^1]
sage: W = M.list_of_subsets()[3]
sage: W is U.intersection(V)
True
sage: M.atlas()
[Chart (U, (x,)), Chart (V, (y,)), Chart (W, (x,)), Chart (W, (y,))]

Transition map between the polar chart and the Cartesian one on $\RR^2$ :

sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart()
sage: U = M.open_subset('U') # the complement of the half line {y=0, x >= 0}
sage: c_spher.<r,phi> = U.chart(r'r:(0,+oo) phi:(0,2*pi):\phi')
sage: trans = c_spher.transition_map(c_cart, (r*cos(phi), r*sin(phi)),
....:                                restrictions2=(y!=0, x<0))
sage: trans
Change of coordinates from Chart (U, (r, phi)) to Chart (U, (x, y))
sage: trans.display()
x = r*cos(phi)
y = r*sin(phi)

In this case, no new subset has been created since $U\cap M = U$ :

sage: M.list_of_subsets()
[2-dimensional differentiable manifold R^2,
 Open subset U of the 2-dimensional differentiable manifold R^2]

but a new chart has been created: $(U, (x, y))$ :

sage: M.atlas()
[Chart (R^2, (x, y)), Chart (U, (r, phi)), Chart (U, (x, y))]

class sage.manifolds.differentiable.chart.DiffCoordChange(chart1, chart2, *transformations)¶

Bases: sage.manifolds.chart.CoordChange

Transition map between two charts of a differentiable manifold.

Giving two coordinate charts $(U,\varphi)$ and $(V,\psi)$ on a differentiable manifold $M$ of dimension $n$ over a topological field $K$ , the transition map from $(U,\varphi)$ to $(V,\psi)$ is the map

$\psi\circ\varphi^{-1}: \varphi(U\cap V) \subset K^n \rightarrow \psi(U\cap V) \subset K^n,$

In other words, the transition map $\psi\circ\varphi^{-1}$ expresses the coordinates $(y^1,\ldots,y^n)$ of $(V,\psi)$ in terms of the coordinates $(x^1,\ldots,x^n)$ of $(U,\varphi)$ on the open subset where the two charts intersect, i.e. on $U\cap V$ .

By definition, the transition map $\psi\circ\varphi^{-1}$ must be of classe $C^k$ , where $k$ is the degree of differentiability of the manifold (cf. diff_degree()).

INPUT:

chart1 – chart $(U,\varphi)$
chart2 – chart $(V,\psi)$
transformations – tuple (or list) $(Y_1,\ldots,Y_2)$ , where $Y_i$ is the symbolic expression of the coordinate $y^i$ in terms of the coordinates $(x^1,\ldots,x^n)$

EXAMPLES:

Transition map on a 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: Y.<u,v> = M.chart()
sage: X_to_Y = X.transition_map(Y, [x+y, x-y])
sage: X_to_Y
Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))
sage: type(X_to_Y)
<class 'sage.manifolds.differentiable.chart.DiffCoordChange'>
sage: X_to_Y.display()
u = x + y
v = x - y

class sage.manifolds.differentiable.chart.RealDiffChart(domain, coordinates='', names=None)¶

Bases: sage.manifolds.differentiable.chart.DiffChart, sage.manifolds.chart.RealChart

Chart on a differentiable manifold over $\RR$ .

Given a differentiable manifold $M$ of dimension $n$ over $\RR$ , a chart is a member $(U,\varphi)$ of the manifold’s differentiable atlas; $U$ is then an open subset of $M$ and $\varphi: U \rightarrow V \subset \RR^n$ is a homeomorphism from $U$ to an open subset $V$ of $\RR^n$ .

The components $(x^1,\ldots,x^n)$ of $\varphi$ , defined by $\varphi(p) = (x^1(p),\ldots,x^n(p))\in \RR^n$ for any point $p\in U$ , are called the coordinates of the chart $(U,\varphi)$ .

INPUT:

domain – open subset $U$ on which the chart is defined
coordinates – (default: ‘’ (empty string)) single string defining the coordinate symbols and ranges, with ‘ ‘ (whitespace) as a separator; each item has at most three fields, separated by ‘:’:
1. The coordinate symbol (a letter or a few letters)
2. (optional) The interval $I$ defining the coordinate range: if not provided, the coordinate is assumed to span all $\RR$ ; otherwise $I$ must be provided in the form (a,b) (or equivalently ]a,b[). The bounds a and b can be +/-Infinity, Inf, infinity, inf or oo. For singular coordinates, non-open intervals such as [a,b] and (a,b] (or equivalently ]a,b]) are allowed. Note that the interval declaration must not contain any whitespace.
3. (optional) The LaTeX spelling of the coordinate; if not provided the coordinate symbol given in the first field will be used.
The order of the fields 2 and 3 does not matter and each of them can be omitted. If it contains any LaTeX expression, the string coordinates must be declared with the prefix ‘r’ (for “raw”) to allow for a proper treatment of LaTeX backslash characters (see examples below). If no interval range and no LaTeX spelling is to be set for any coordinate, the argument coordinates can be omitted when the shortcut operator <,> is used via Sage preparser (see examples below)
names – (default: None) unused argument, except if coordinates is not provided; it must then be a tuple containing the coordinate symbols (this is guaranteed if the shortcut operator <,> is used).

EXAMPLES:

Cartesian coordinates on $\RR^3$ :

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart = M.chart('x y z'); c_cart
Chart (R^3, (x, y, z))
sage: type(c_cart)
<class 'sage.manifolds.differentiable.chart.RealDiffChart'>

To have the coordinates accessible as global variables, one has to set:

sage: (x,y,z) = c_cart[:]

However, a shortcut is to use the declarator <x,y,z> in the left-hand side of the chart declaration (there is then no need to pass the string 'x y z' to chart()):

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x,y,z> = M.chart(); c_cart
Chart (R^3, (x, y, z))

The coordinates are then immediately accessible:

sage: y
y
sage: y is c_cart[2]
True

The trick is performed by Sage preparser:

sage: preparse("c_cart.<x,y,z> = M.chart()")
"c_cart = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = c_cart._first_ngens(3)"

Note that x, y, z declared in <x,y,z> are mere Python variable names and do not have to coincide with the coordinate symbols; for instance, one may write:

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x1,y1,z1> = M.chart('x y z'); c_cart
Chart (R^3, (x, y, z))

Then y is not known as a global variable and the coordinate $y$ is accessible only through the global variable y1:

sage: y1
y
sage: y1 is c_cart[2]
True

However, having the name of the Python variable coincide with the coordinate symbol is quite convenient; so it is recommended to declare:

sage: forget()   # for doctests only
sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x,y,z> = M.chart()

Spherical coordinates on the subset $U$ of $\RR^3$ that is the complement of the half-plane $\{y=0, x\geq 0\}$ :

sage: U = M.open_subset('U')
sage: c_spher.<r,th,ph> = U.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: c_spher
Chart (U, (r, th, ph))

Note the prefix ‘r’ for the string defining the coordinates in the arguments of chart.

Coordinates are Sage symbolic variables (see sage.symbolic.expression):

sage: type(th)
<type 'sage.symbolic.expression.Expression'>
sage: latex(th)
{\theta}
sage: assumptions(th)
[th is real, th > 0, th < pi]

Coordinate are also accessible by their indices:

sage: x1 = c_spher[1]; x2 = c_spher[2]; x3 = c_spher[3]
sage: [x1, x2, x3]
[r, th, ph]
sage: (x1, x2, x3) == (r, th, ph)
True

The full set of coordinates is obtained by means of the operator [:]:

sage: c_cart[:]
(x, y, z)
sage: c_spher[:]
(r, th, ph)

Let us check that the declared coordinate ranges have been taken into account:

sage: c_cart.coord_range()
x: (-oo, +oo); y: (-oo, +oo); z: (-oo, +oo)
sage: c_spher.coord_range()
r: (0, +oo); th: (0, pi); ph: (0, 2*pi)
sage: bool(th>0 and th<pi)
True
sage: assumptions()  # list all current symbolic assumptions
[x is real, y is real, z is real, r is real, r > 0, th is real,
 th > 0, th < pi, ph is real, ph > 0, ph < 2*pi]

The coordinate ranges are used for simplifications:

sage: simplify(abs(r)) # r has been declared to lie in the interval (0,+oo)
r
sage: simplify(abs(x)) # no positive range has been declared for x
abs(x)

Each constructed chart is automatically added to the manifold’s user atlas:

sage: M.atlas()
[Chart (R^3, (x, y, z)), Chart (U, (r, th, ph))]

and to the atlas of its domain:

sage: U.atlas()
[Chart (U, (r, th, ph))]

Manifold subsets have a default chart, which, unless changed via the method set_default_chart(), is the first defined chart on the subset (or on a open subset of it):

sage: M.default_chart()
Chart (R^3, (x, y, z))
sage: U.default_chart()
Chart (U, (r, th, ph))

The default charts are not privileged charts on the manifold, but rather charts whose name can be skipped in the argument list of functions having an optional chart= argument.

The action of the chart map $\varphi$ on a point is obtained by means of the call operator, i.e. the operator ():

sage: p = M.point((1,0,-2)); p
Point on the 3-dimensional differentiable manifold R^3
sage: c_cart(p)
(1, 0, -2)
sage: c_cart(p) == p.coord(c_cart)
True
sage: q = M.point((2,pi/2,pi/3), chart=c_spher) # point defined by its spherical coordinates
sage: c_spher(q)
(2, 1/2*pi, 1/3*pi)
sage: c_spher(q) == q.coord(c_spher)
True
sage: a = U.point((1,pi/2,pi)) # the default coordinates on U are the spherical ones
sage: c_spher(a)
(1, 1/2*pi, pi)
sage: c_spher(a) == a.coord(c_spher)
True

Cartesian coordinates on $U$ as an example of chart construction with coordinate restrictions: since $U$ is the complement of the half-plane $\{y=0, x\geq 0\}$ , we must have $y\not=0$ or $x<0$ on U. Accordingly, we set:

sage: c_cartU.<x,y,z> = U.chart()
sage: c_cartU.add_restrictions((y!=0, x<0)) # the tuple (y!=0, x<0) means y!=0 or x<0
sage: # c_cartU.add_restrictions([y!=0, x<0]) would have meant y!=0 AND x<0
sage: U.atlas()
[Chart (U, (r, th, ph)), Chart (U, (x, y, z))]
sage: M.atlas()
[Chart (R^3, (x, y, z)), Chart (U, (r, th, ph)), Chart (U, (x, y, z))]
sage: c_cartU.valid_coordinates(-1,0,2)
True
sage: c_cartU.valid_coordinates(1,0,2)
False
sage: c_cart.valid_coordinates(1,0,2)
True

Chart grids can be drawn in 2D or 3D graphics thanks to the method plot().

Coordinate Charts on Differentiable Manifolds¶

Previous topic

Next topic

This Page