By Lugo G.

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17 Definition The torsion of a connection ∇ is the operator T such that ∀X, Y, T (X, Y ) = ∇X Y − ∇Y X − [X, Y ]. 29) A connection is called torsion-free if T (X, Y ) = 0. In this case, ∇X Y − ∇Y X = [X, Y ]. We will say more later about the torsion and the importance of torsion-free connections. For the time being, it suffices to assume that for the rest of this section, all connections are torsion-free. Using this assumption, it is possible to prove the following important theorem. 18 Theorem The Weingarten map is a self-adjoint operator on T M.

Therefore, LX is orthogonal to N ; hence, it lies in T (M). In the previous section, we gave two equivalent definitions < dx, dx >, and < X, Y >) of the first fundamental form. We will now do the same for the second fundamental form. 15 Definition The second fundamental form is the bilinear map II(X, Y ) =< LX, Y > . 16 Remark It should be noted that the two definitions of the second fundamental form are consistent. This is easy to see if one chooses X to have components xα and Y to have components xβ .

Jq ∧ (dxj1 ∧ . . ∧ dxjq )] = dα ∧ β + (−1)p α ∧ dβ. jp through p 1-forms of the type dxi , one must perform p transpositions. 23 Example Let α = P (x, y)dx + Q(x, y)dβ. Then, dα ∂P ∂P ∂Q ∂Q dx + ) ∧ dx + ( dx + ) ∧ dy ∂x ∂y ∂x ∂y ∂P ∂Q = dy ∧ dx + dx ∧ dy ∂y ∂x ∂Q ∂P = ( − )dx ∧ dy. 4. THE HODGE- ∗ OPERATOR 25 This example is related to Green’s theorem in R2 . 24 Example Let α = M (x, y)dx + N (x, y)dy, and suppose that dα = 0. Then, by the previous example, ∂N ∂M dα = ( − )dx ∧ dy. ∂x ∂y Thus, dα = 0 iff Nx = My , which implies that N = fy and Mx for some function f (x, y).