Metric

We treat several metrics simultaneously. A metric is represented by a symmetric matrix, which entries depend on coordinates. Coordinates are denoted x_1,x_2,.... The first metric we want to consider is induced by the standard Euclidean scalar product of ^4. Our manifold is S^3 with radius r, which we parametrize in polar coordinates below.

f = r {Cos[x_2] Cos[x_3] Sin[x_1], Cos[x_3] Sin[x_2], Sin[x_3], Cos[x_1] Cos[x_2] Cos[x_3]}

MF[ℊS3p = ΜPullback[f, 3, Id[4]]]

Det[ℊS3p]//S

{r Cos[x_2] Cos[x_3] Sin[x_1], r Cos[x_3] Sin[x_2], r Sin[x_3], r Cos[x_1] Cos[x_2] Cos[x_3]}

( {{r^2 Cos[x_2]^2 Cos[x_3]^2, 0, 0}, {0, r^2 Cos[x_3]^2, 0}, {0, 0, r^2}} )

r^6 Cos[x_2]^2 Cos[x_3]^4

Parametrizing S^3 differently, leads to another matrix g. However, the metric are equivalent.

f = {x_1, x_2, x_3, (r^2 - x_1^2 - x_2^2 - x_3^2)^(1/2)}

MF[ℊS3s = ΜPullback[f, 3, Id[4]]]

Det[ℊS3s]//S

{x_1, x_2, x_3, (r^2 - x_1^2 - x_2^2 - x_3^2)^(1/2)}

-r^2/(-r^2 + x_1^2 + x_2^2 + x_3^2)

The computations carried out, are simply the matrix products (df)^T.ℊ_^4.df. In the settings above, df is a 4×3 matrix and ℊ_^4=Id_4. For instance

Μd[f, 1, 3]//MF

( {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {-x_1/(r^2 - x_1^2 - x_2^2 - x_3^2)^(1/2), -x_2/(r^2 - x_1^2 - x_2^2 - x_3^2)^(1/2), -x_3/(r^2 - x_1^2 - x_2^2 - x_3^2)^(1/2)}} )

Metrics induced by the embedding of the Riemannian manifold ^n are intuitive. Another example is the metric of the geometric torus. Below, we define the domain of the parametrization, and the parametrization itself.

assum = {0<x_1<2π, 0<x_2<2π, 0<r<R} ;

f = {(R + r Cos[x_2]) Cos[x_1], (R + r Cos[x_2]) Sin[x_1], r Sin[x_2]} ;

pτ = ParametricPlot3D[Evaluate[f/.(inst = {R→2, r→1})], {x_1, 0, 2 π - 1.5}, {x_2, 0, 2π - 1}] ;

[Graphics:../HTMLFiles/index_78.gif]

Next, we present three ways to obtain the metric tensor over the particular domain.

MF[J = Μd[f, 1, 2]]

ShowMat[{ℊT2 = ΜPullback[f, 2, Id[3]], T[J] . Id[3] . J, Τα[Id[3], 0, J]}//S]

( {{-(R + r Cos[x_2]) Sin[x_1], -r Cos[x_1] Sin[x_2]}, {Cos[x_1] (R + r Cos[x_2]), -r Sin[x_1] Sin[x_2]}, {0, r Cos[x_2]}} )

{( {{(R + r Cos[x_2])^2, 0}, {0, r^2}} ), ( {{(R + r Cos[x_2])^2, 0}, {0, r^2}} ), ( {{(R + r Cos[x_2])^2, 0}, {0, r^2}} )}

Along the metric, we have notions such as the angle between two vectors, and the lengths of vectors, or curves. In general, having defined a volume form on the manifold, we can perform integration on subsets of the manifold. But first, lets continue with metrics of index ≥1. The pseudo sphere S_1^2 of radius r is defined as the points in ^3 which annihilate

n = 2 ;

imp = Id[n, 1] . Τ★[x, n + 1] . Τ★[x, n + 1] - r^2

Solve[imp == 0, x_ (n + 1)]

-r^2 + x_1^2 + x_2^2 - x_3^2

{{x_3→ -(-r^2 + x_1^2 + x_2^2)^(1/2)}, {x_3→ (-r^2 + x_1^2 + x_2^2)^(1/2)}}

We plot a part of the manifold for radius r=2.

τ = 5 ;

ContourPlot3D[imp/.r→2, {x_1, -τ, τ}, {x_2, -τ, τ}, {x_3, -τ, τ}, Axes→True] ;

[Graphics:../HTMLFiles/index_92.gif]

ℊpS21 = 1 ;

Hyperbolic metric

ΜℊHyperbolicPlane[4]//MF

( {{r^2/x_4^2, 0, 0, 0}, {0, r^2/x_4^2, 0, 0}, {0, 0, r^2/x_4^2, 0}, {0, 0, 0, r^2/x_4^2}} )

The following space (M,g) is geodesically complete. However, between two points there might not be a connecting geodesic.

assum = {-π/2<x_1<π/2} ;

ℊAdS = 1/Cos[x_1]^2 ({{-1, 0}, {0, 1}}) ;


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