Geodesics, Exponential Map

Cosider the hyperbolic metric on R×^+ defined for r>0 as

MF[ℊH2 = ΜℊHyperbolicPlane[2]]

( {{r^2/x_2^2, 0}, {0, r^2/x_2^2}} )

The parameter r does not change the geodesics on R×^+ qualitatively, for r does not appear in the geodesic equations below. Although the expressions for the geodesics are simple, Mathematica has trouble solving the differential equations. Maybe in the next version...

DSolve[E0[ΧGeodesic[ℊH2]], Χ★[2], ]

DSolve[E0[ΧGeodesic[ℊH2], Χ0[0, {0, 1}], Χ0[1, {0, 1}]], Χ★[2], ]

For illustration purposes numerial approximation is good enough. We set r=1, but only for normalization of the initial vector.

p = {.8, 1} ;

sol = With[{v = {Cos[#1], Sin[#1]}}, Χγ[ℊH2/.r→1, p, ΜNormalize[ℊH2◦p/.r→1, v], {-2, 2}]] &/@Range[-π/4, π/2, .13] ;

ParametricPlot[Evaluate[sol], {, -2, 2}, AspectRatio→Automatic] ;

[Graphics:../HTMLFiles/index_189.gif]

We visualize the function exp_p:T_pM→M for p∈M for a particular point p. The lines that appear correspond to the canonical coordinate lines of T_pM.

pτ = ParametricPlot3D[{x_1, x_2, 0} ◦Evaluate[Χ[ℊH2/.r→1, {0, 1}, {v_1, v_2}]], {v_1, -1, 2}, {v_2, -2, 2}, PlotPoints→ {11, 11}] ;

[Graphics:../HTMLFiles/index_194.gif]

Another example: The following space (M,g) is geodesically complete. However, between two points there might not be a connecting geodesic. M=(-π/2,π/2R equipped with

assum = {-π/2<x_1<π/2} ;

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

Note that the metric does not depend on the coordinate x_2.

geo[{0, 0}] ;

geo[{.4, .2}] ;

[Graphics:../HTMLFiles/index_203.gif]

[Graphics:../HTMLFiles/index_204.gif]

The space is heavily curved near the "boundary"?

Showℛ[Μℛ[ℊAdS]]

Plot[Sec[x_1]^2, {x_1, -π/2, π/2}] ;

ℛ_ (1, 2, j)^i
( {{0, Sec[x_1]^2}, {Sec[x_1]^2, 0}} )

[Graphics:../HTMLFiles/index_209.gif]

1/Sec[x_1]

Cos[x_1]

Not really. The sectional curvature is constant =1 for non-degenerated planes.

ΜK[ℊAdS, {a, b}, {c, d}]

Divide @@ ΜK[ℊAdS, {a, b}, {c, d}]

{-(b c - a d)^2 Sec[x_1]^4, -(b c - a d)^2 Sec[x_1]^4}

1

Finally, the image of the exponential map exp_p for p=(.4,.2) with coordinate lines.

pτ = ParametricPlot3D[{x_1, x_2, 0} ◦Evaluate[Χ[ℊAdS, {.4, .2}, {v_1, v_2}]], {v_1, -4, 4}, {v_2, -4, 4}, PlotPoints→ {11, 11}] ;

[Graphics:../HTMLFiles/index_218.gif]

Next, we consider M=<^2,g> with a special Lorentzian metric g. The semi-Riemannian manifold M is not complete, i.e. not all geodesics are defined on the entire real line.

ℊR2L = ({{Cos[x_2]^4 - 1, -1}, {-1, 0}}) ;

DSolve[E0[ΧGeodesic[ℊR2L]], Χ★[2], ]

We know, that γ(t)={1/-t,ArcTan[t]} is a solution. Although, lets check:

eqs = ΧGeodesic[ℊR2L]/.Ft[{γ = Χ★[2] → {1/ - , ArcTan[]}, ∂_γ, ∂_∂_γ}]

E0[eqs]

{True}

Everybody can tell what happens for t→0.

ParametricPlot[{1/ - , ArcTan[]}, {, .01, 5}] ;

[Graphics:../HTMLFiles/index_229.gif]

Our code does not compensate for this incident.

geo[{1, 1}] ;

NDSolve :: ndsz : At  == -0.727384, step size is effectively zero; singularity or stiff system suspected.  More…

InterpolatingFunction :: dmval : Input value  {-1.} lies outside the range of data in the interpolating function. Extrapolation will be used. More…

InterpolatingFunction :: dmval : Input value  {-1.} lies outside the range of data in the interpolating function. Extrapolation will be used. More…

InterpolatingFunction :: dmval : Input value  {-1.} lies outside the range of data in the interpolating function. Extrapolation will be used. More…

General :: stop : Further output of InterpolatingFunction :: dmval will be suppressed during this calculation. More…

[Graphics:../HTMLFiles/index_237.gif]


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