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HW8 210909

# HW8 210909 - Differential Geometry Homework 8 21.07.08 Hava...

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Differential Geometry Homework 8 - 21.07.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —[email protected] 1. In order to prove the theorem we must use Gauss’s Theorema Egregium Let X R 3 be a 2-dimensional submanifold and g the induced metric on X . For any x X and any basis { e 1 , e 2 } for T x ( X ) , the Gaussian curvature of X at x is given by K = R ( e 1 , e 2 , e 2 , e 2 ) | e 1 | 2 | e 2 | 2 - h e 1 , e 2 i 2 . Now let us prove that if dimX = 2 for any arbitrary vector field s, t, v, w one has R ( s, t, v, w ) = K ( h s, v i h t, w i - h s, w i h t, v i ) . Since both sides of this equation are tensors, we can compute them in terms of any basis. Let { e 1 , e 2 } be any orthonormal basis for T x ( X ) , and define a shorter symbole for the following components
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