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Unformatted text preview: that satisFes the condition R 2 = x 2 + y 2 + z 2 + w 2 . Show that the metric on the 3-sphere is given by ds 2 = dr 2 + R 2 sin 2 ( r/R )[ d 2 + sin 2 d 2 ]. The rela-tion between x, y, z, w is x = R sin ( r/R ) sin cos , y = R sin ( r/R ) sin sin , z = R sin ( r/R ) cos , w = R cos ( r/R ). What are the natural ranges for r , and so that the 3-sphere is covered exactly once? (You might Fnd it helpful to review the natural ranges of coordinates on the 2-sphere.)...
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- Spring '10