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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Differential Geometry Homework 8 - 21.07.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —hshabtai@campus.haifa.ac.il— 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 ih t,w i - h s,w ih 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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/21/2009 for the course MATH 106723 taught by Professor Michaelpolyak during the Spring '09 term at Technion.

Ask a homework question - tutors are online