Differential Geometry

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By Michael Spivak

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In Munich, Hermann Einstein established an electric equipment business with Albert’s uncle Jakob. However in time, this second business failed too. Extended family offered financial assistance if the Einstein family moved to Italy, but Albert’s parents did not want to disrupt his education. Therefore, arrangements were made to leave Albert with a relative in Munich, while his parents departed for Italy, confident that he would continue his education at the Luitpold Gymnasium. For its time, the Luitpold Gymnasium was considered a forward looking institution of the highest caliber.

This raises the possibility that the two groups are isomorphic. Two groups are said to be isomorphic if one can set up a one-to-one correspondence between the groups is such a way that if x in one group corresponds to xK in the second group and y corresponds to yK then x ı y corresponds to xK ı yK . For the cube and the octahedron, this is plausible because the numbers of fourfold, threefold, and twofold axes match up in the two groups. Nonetheless, it would 20 2 Clifford Algebra in Euclidean 3-Space a b Fig.

Marcel Grossmann directed his attention to Riemannian geometry, which was then a fairly obscure topic in mathematics. After many failed efforts, it was late in 1915 that Einstein published his General Theory of Relativity in its completed form. One virtue of the General Theory of Relativity was the fact that Einstein could use it to explain the deviation of the orbit of Mercury from the laws of Newton. This was a surprise. Although it was recognized that the orbit of Mercury had an odd behavior, it was thought that eventually the anomaly could be explained in Newtonian terms.

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