HW6 130909 - Differential Geometry Homework 6 03.07.08 Hava...

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Differential Geometry Homework 6 - 03.07.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —hshabtai@campus.haifa.ac.il— 1. Let φ : U -→ X be the char described in Theorem 1.11 ϕ ( u ) = ( u,f ( u )) = ( u 1 ,...,u n ,f ( u 1 ,...,u n )) = = ( u 1 ,...,u n ,f 1 ( u ) ,...,f m ( u )) . This is a smooth parametrization of X , and the induced metric on X is given in graph coordinates by ϕ * g = ϕ * ( ( dx 1 ) 2 + ··· + ( dx N ) 2 ) = = ( du 1 ) 2 + ··· + ( du n ) 2 + ( df 1 ) 2 + ··· + ( df m ) 2 2. The first embedding is not isometry and the second embedding is an isometry. (a) Let us take the geodesic line on T 2 between the points (0 , 0) and (0 ,π/ 2) , its length is π/ 2 as we can easily compute, however the end point of the image of the geodesic line are i (0 , 0) = (0 , 0 , 0) and i (0 ,π/ 2) = (0 , 2 , 0) hence the length of the image of the geodesic line is 2, since isometry must preserve the distance between two points this map is not an isometry. (b) Let us compute
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This note was uploaded on 09/21/2009 for the course MATH 106723 taught by Professor Michaelpolyak during the Spring '09 term at Technion.

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HW6 130909 - Differential Geometry Homework 6 03.07.08 Hava...

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