Complete Riemannian Metric

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/CompleteRiemannianMetric.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CompleteRiemannianMetric.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Original notebook by
Todd Rowland
July 10, 2000

Initialization

<<Graphics`ComplexMap`
<<Graphics`Colors`

Figure 1

A geodesic which can only go a finite distance because it  reaches the hole in the punctured plane, implying that the punctured plane with the usual metric is not complete.

Show[Graphics[{PointSize[.03],Point[{0,0}],Line[{{0,0},{2,1}}]}],AspectRatio->Automatic,PlotRange->All]

Figure 2

The second graphic is meant to be the punctured plane with another metric which is complete.  The path in this picture has infinite length in the given metric.

PolarMap[(1/#1)&,{1,10},{0,2Pi},
PlotStyle->Red,
Ticks->False,PlotRange->{{-1,1},{-1,1}},AspectRatio->Automatic,
Epilog->{PointSize[.1],Point[{0,0}]}]