By Hervé M. Pajot

In accordance with a graduate path given via the writer at Yale collage this e-book bargains with advanced research (analytic capacity), geometric degree thought (rectifiable and uniformly rectifiable units) and harmonic research (boundedness of singular fundamental operators on Ahlfors-regular sets). particularly, those notes include an outline of Peter Jones' geometric touring salesman theorem, the facts of the equivalence among uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular units, the total proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, purely the Ahlfors-regular case) and a dialogue of X. Tolsa's answer of the Painlevé challenge.

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15 is complete. 5 E. Cartan structural equations of a connection Given an aﬃne connection ∇, we put T (X, Y ) = ∇Y X − ∇X Y + [X, Y ]. 52) The mapping (σ, X, Y ) ∈ Γ 1 (Q) × X 2 (Q) → σ(T (X, Y )) ∈ D(Q) is a mixed tensor ﬁeld of type (1,2) called the torsion tensor ﬁeld of ∇. 54) where X, Y ∈ X (Q). 26) by (∇X ω)(Y ) = X(ω(Y )) − ω(∇X Y ). 54) imply dω(X, Y ) = (∇X ω)(Y ) − (∇Y ω)(X) − ω(T (X, Y )). 55) Let p ∈ Q and (X1 , X2 , . . , Xn ) a basis for the vector ﬁelds in some neighborhood Np of p, that is, any vector ﬁeld X on Np can be written as n X = i=1 fi Xi where fi ∈ D(Np ).

R , Y1 , . . , Ys ) = ∇Φ(σ 1 , . . , σ r , Y1 , . . , Ys , W ). 7. Covariant derivative ∇W and covariant diﬀerential ∇ of a mixed tensor ﬁeld, commute with both contraction and type changing operations. 3) tensor ﬁeld. 26), so ¯ (∇W R)(σ, X, Y, Z) = W (σ(RX,Y Z)) − − σ (RX,∇W Y Z + R∇W X,Y Z + RX,Y (∇W Z)) − (∇W σ)(RX,Y Z). 24) we identify σ with the vector ﬁeld V given by σ(·) = V, · , then ¯ = ∇W V, X ¯ for all X ¯ ∈ X (Q). We can give an interpretation (∇W σ)(X) to the last equality in the following way: for ﬁxed W, X, Y ∈ X (Q), to each Z ∈ X (Q) one associates the vector ﬁeld (∇W R)(X, Y )Z deﬁned by V, (∇W R)(X, Y )Z = W ( V, RX,Y Z ) − − V, R∇W X,Y Z + RX,∇W Y Z + RX,Y (∇W Z) − ∇W V, RXY Z for all V ∈ X (Q).

It is clear that h(0) = c(0, q, v) = q and that h(0) = ac(0, ˙ q, v) = av. Moreover, h is a geodesic because Dh˙ d c(at, ˙ q, v) = 0 = ∇ d c(at,q,v) c(at, q, v) = a2 ∇c(at,q,v) ˙ dt dt dt d where in ∇, dt c(at, q, v) represents an extension of h˙ to a neighborhood of c(at, q, v), in Q. The uniqueness of geodesics gives ﬁnally: h(t) = c(at, q, v) = c(t, q, av) for t ∈ (−δ/a, +δ/a). 22) called the exponential map in U, which is a diﬀerentiable map. If we ﬁx q ∈ Q, one may consider Bε˜(0) ⊂ U ∩ Tq Q where Bε˜(0) is a ball centered at 0 ∈ Tq Q with a suitable radius ε˜ > 0, and deﬁne expq : Bε˜(0) −→ Q by expq (v) = exp(q, v), v ∈ Bε˜(0).