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In this thread, I plan to try to explain (with some apropos ctensor examples) in a simple and concrete context some basic techniques and notions about Riemannian two-manifolds which also apply to general Riemannian/Lorentzian manifolds.
Suppose we have a euclidean surface given by a C^2 (continuously twice differentiable) height function z=f(x,y) where x,y,z are Cartesian coordinates in euclidean-three space. In fact, to be safe we can assume f is C^\infty ("smooth", continuously differentiable to any order). In examples, f will often be real analytic (possess a two-variable McLaurin series convergent in some open neighborhood).
It is useful to write the parameterized surface in the form
<br /> \left[ \begin{array}{c} f \\ x \\ y \end{array} \right] <br />
Then two linearly independent tangent vectors are obtained by differentiating this vector function of x,y wrt the two independent variables:
<br /> \partial_x = \left[ \begin{array}{c} f_x \\ 1 \\ 0 \end{array} \right], \; \;<br /> \partial_y = \left[ \begin{array}{c} f_y \\ 0 \\ 1 \end{array} \right]<br />
Now taking dot products in euclidean three space by the usual vector calculus rules gives the line element written in the so-called Monge chart
<br /> ds^2 = (1 + f_x^2) \; dx^2 + 2 \; f_x \; f_y \; dx \; dy + (1 + f_y^2) \; dy^2<br />
or more properly
<br /> ds^2 = (1 + f_x^2) \; dx \otimes dx + 2 \, f_x \, f_y \; dx \otimes dy + (1 + f_y^2) \; dy \otimes dy<br />
In plain language: the components of the metric tensor wrt the coordinate basis are
<br /> \left[ \begin{array}{cc} 1 + f_x^2 & f_x \; f_y \\ f_x \; f_y & 1 + f_y^2 \end{array} \right]<br />
Next we must choose a coframe, i.e. two orthogonal unit covectors
<br /> \sigma^1 = a_{1x} \, dx + a_{1y} \, dy, \; \sigma^2 = a_{2x} \, dx + a_{2y} \, dy<br />
where we have four undetermined functions of x,y, such that
<br /> g = \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2<br />
(I like to use integers to index frame and coframe components, and the coordinate names to index coordinate components, when I am simply reporting the results of computation of the components of some tensor.)
That is, the metric should arise from using our covectors in place of the coordinate covectors dx,dy in the standard euclidean metric. Expanding the tensor products, we find
<br /> a_{1x}^2 + a_{2x}^2 = g_{xx}<br />
and so forth. So the choice amounts to solving some high school algebra equations--- there will be infinitely many solutions, so we can pick any convenient one. I chose to put a_{1y} = 0 which gives
<br /> \begin{array}{rcl}<br /> \sigma^1 & = & \sqrt{\frac{1+f_x^2 + f_y^2}{1+f_y^2}} \; dx \\<br /> \sigma^2 & = & \frac{f_x \; f_y}{\sqrt{1+f_y^2}} \; dx + \sqrt{1+f_y^2} \; dy<br /> \end{array}<br />
Now we can start to compute interesting quantities, beginning with the Gaussian curvature R_{1212} (components wrt the coframe, not the coordinate covectors!). I'll explain later Cartan's method for doing this working directly with the coframe; for now I just want to urge those of you who have installed Maxima to run this ctensor file in batch mode to see the results of some computations:
These computations show that the area form is
<br /> (1 + f_x^2 + f_y^2) \; dx \wedge dy<br />
and the Gaussian curvature is
<br /> K = R_{1212} = \frac{ f_{xx} \; f_{yy} - f_{xy}^2 }{ \left( 1 + f_x^2 + f_y^2 \right)^2}<br />
There are no further components of the Riemann tensor which are algebraically independent of this one, in two dimensions.
Incidentally, the only algebraically independent component of the Riemann tensor expanded wrt the coordinate basis is
R_{xyxy} = \frac{ f_{xx} \; f_{yy} - f_{xy}^2 }{ \left( 1 + f_x^2 + f_y^2 \right)}<br />
which is not the Gaussian curvature. (I should caution that some authors adopt sign conventions which ensure that R_{1212} = -K.)
An excellent reference for surface theory (and abstract Riemannian manifolds) a la Cartan is Flanders, Differential Forms with Applications to the Physical Sciences, Dover reprint.
Suppose we have a euclidean surface given by a C^2 (continuously twice differentiable) height function z=f(x,y) where x,y,z are Cartesian coordinates in euclidean-three space. In fact, to be safe we can assume f is C^\infty ("smooth", continuously differentiable to any order). In examples, f will often be real analytic (possess a two-variable McLaurin series convergent in some open neighborhood).
It is useful to write the parameterized surface in the form
<br /> \left[ \begin{array}{c} f \\ x \\ y \end{array} \right] <br />
Then two linearly independent tangent vectors are obtained by differentiating this vector function of x,y wrt the two independent variables:
<br /> \partial_x = \left[ \begin{array}{c} f_x \\ 1 \\ 0 \end{array} \right], \; \;<br /> \partial_y = \left[ \begin{array}{c} f_y \\ 0 \\ 1 \end{array} \right]<br />
Now taking dot products in euclidean three space by the usual vector calculus rules gives the line element written in the so-called Monge chart
<br /> ds^2 = (1 + f_x^2) \; dx^2 + 2 \; f_x \; f_y \; dx \; dy + (1 + f_y^2) \; dy^2<br />
or more properly
<br /> ds^2 = (1 + f_x^2) \; dx \otimes dx + 2 \, f_x \, f_y \; dx \otimes dy + (1 + f_y^2) \; dy \otimes dy<br />
In plain language: the components of the metric tensor wrt the coordinate basis are
<br /> \left[ \begin{array}{cc} 1 + f_x^2 & f_x \; f_y \\ f_x \; f_y & 1 + f_y^2 \end{array} \right]<br />
Next we must choose a coframe, i.e. two orthogonal unit covectors
<br /> \sigma^1 = a_{1x} \, dx + a_{1y} \, dy, \; \sigma^2 = a_{2x} \, dx + a_{2y} \, dy<br />
where we have four undetermined functions of x,y, such that
<br /> g = \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2<br />
(I like to use integers to index frame and coframe components, and the coordinate names to index coordinate components, when I am simply reporting the results of computation of the components of some tensor.)
That is, the metric should arise from using our covectors in place of the coordinate covectors dx,dy in the standard euclidean metric. Expanding the tensor products, we find
<br /> a_{1x}^2 + a_{2x}^2 = g_{xx}<br />
and so forth. So the choice amounts to solving some high school algebra equations--- there will be infinitely many solutions, so we can pick any convenient one. I chose to put a_{1y} = 0 which gives
<br /> \begin{array}{rcl}<br /> \sigma^1 & = & \sqrt{\frac{1+f_x^2 + f_y^2}{1+f_y^2}} \; dx \\<br /> \sigma^2 & = & \frac{f_x \; f_y}{\sqrt{1+f_y^2}} \; dx + \sqrt{1+f_y^2} \; dy<br /> \end{array}<br />
Now we can start to compute interesting quantities, beginning with the Gaussian curvature R_{1212} (components wrt the coframe, not the coordinate covectors!). I'll explain later Cartan's method for doing this working directly with the coframe; for now I just want to urge those of you who have installed Maxima to run this ctensor file in batch mode to see the results of some computations:
Code:
/*
General E^2 manifold; cartesian Monge chart
Chart covers at most
-infty < x,y < infty
We assume our surface is given by an embedding in E^3
z = f(x,y)
*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 2;
/* list the coordinates */
ct_coords: [x,y];
/* define variables */
depends(f,[x,y]);
/* define background metric */
lfg: ident(2);
/* define the coframe */
fri: zeromatrix(2,2);
fri[1,1]: sqrt(1+diff(f,x)^2+diff(f,y)^2)/sqrt(1+diff(f,y)^2);
fri[2,1]: diff(f,x)*diff(f,y)/sqrt(1+diff(f,y)^2);
fri[2,2]: sqrt(1+diff(f,y)^2);
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
factor(fr);
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: factor(invert(lg));
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
factor(lriem[2,2,1,1]);<br /> (1 + f_x^2 + f_y^2) \; dx \wedge dy<br />
and the Gaussian curvature is
<br /> K = R_{1212} = \frac{ f_{xx} \; f_{yy} - f_{xy}^2 }{ \left( 1 + f_x^2 + f_y^2 \right)^2}<br />
There are no further components of the Riemann tensor which are algebraically independent of this one, in two dimensions.
Incidentally, the only algebraically independent component of the Riemann tensor expanded wrt the coordinate basis is
R_{xyxy} = \frac{ f_{xx} \; f_{yy} - f_{xy}^2 }{ \left( 1 + f_x^2 + f_y^2 \right)}<br />
which is not the Gaussian curvature. (I should caution that some authors adopt sign conventions which ensure that R_{1212} = -K.)
An excellent reference for surface theory (and abstract Riemannian manifolds) a la Cartan is Flanders, Differential Forms with Applications to the Physical Sciences, Dover reprint.
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