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HW7 140909

HW7 140909 - Differential Geometry Homework 7 10.07.08 Hava...

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Differential Geometry Homework 7 - 10.07.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —[email protected] 1. We need to prove that the deﬁned map satisfy the two conditions of a connection. (a) For a ﬁx function g C ( X ) we compute directly with the deﬁnitions ( g ) = ( g ) = f τ ( g ) , this computation proves the ﬁrst condition. (b) For a ﬁx function g C ( X ) we compute τ ( fg ) = τ ( fg ) = ( g ) + ( f ) = f τ ( g ) + τ ( f ) g, where the equality in the midle is true by the the Leibniz rule that was proven in HW3. 2. Let us denote := 1 - ∇ 2 (we chose an upper index since we usually write for a connection ( τ,s ) = τ ( v ) ).We now strive to prove that the connection is C -linear in all the arguments. is bilinear over R . Using the properties of a connection which are ( s ) = f τ ( s ) τ ( fs ) = f τ ( s ) + τ ( f ) s. We compute

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HW7 140909 - Differential Geometry Homework 7 10.07.08 Hava...

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