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F X (Y ) = f ∇X Y, 2. ∇(X1 +X2 ) Y = ∇X1 Y + ∇X2 Y, 3. ∇X (Y1 + Y2 ) = ∇X Y1 + ∇X Y2 , 4. ∇X f Y = X(f )Y + f ∇X Y, for all vector fields X, X1 , X2 , Y, Y1 , Y2 ∈ T (Rn ) and all smooth functions f . The definition states that the map ∇X is linear on X but behaves like a linear derivation on Y. For this reason, the quantity ∇X Y is called the covariant derivative of Y in the direction of X. 7 Proposition Let Y = f i ∂x , and let X another C ∞ vector field. 12) ∂x defines a Koszul connection. The proof just requires verification that the four properties above are satisfied, and it is left as an exercise.

2 We recall that the distance formula from a point x to a plane which contains x0 is just the scalar projection of (x − x0 ) onto the normal. Since the normal to the plane at x0 is the same as the unit normal to the surface and < xα , n >= 0, we find that the distance D is D = < x − x0 , n > 1 = < (x0 )αβ , n > duα duβ 2 1 = II0 2 The first fundamental form (or, rather, its determinant) also appears in calculus in the context of calculating the area of a parametrized surface. If one considers an infinitesimal parallelogram subtended by the vectors xu du and xv dv, then the differential of surface area is given by the length of the cross product of these two infinitesimal tangent vectors.

N − m)! m n−m ). 39) To describe the isomorphism between these two spaces, we will first need to introduce the totally antisymmetric Levi-Civita permutation symbol defined as follows:   +1 if(i1 , . . , im ) is an even permutation of(1, . . , m) −1 if(i1 , . . , im ) is an odd permutation of(1, . . 41) The permutation symbols are useful in the theory of determinants. 41), the reader can easily verify that detA = |A| = i 1 i2 i 3 i1 i2 i3 a1 a2 a3 . 42) This formula for determinants extends in an obvious manner to n × n matrices.

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A conjecture in arithmetic theory of differential equations by Katz N.M.

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