By Valentino Magnani

We receive an intrinsic Blow-up Theorem for normal hypersurfaces on graded nilpotent teams. This approach permits us to symbolize explicitly the Riemannian floor degree when it comes to the round Hausdorff degree with admire to an intrinsic distance of the crowd, particularly homogeneous distance. We practice this consequence to get a model of the Riemannian coarea forumula on sub-Riemannian teams, that may be expressed when it comes to arbitrary homogeneous distances.We introduce the average category of horizontal isometries in sub-Riemannian teams, giving examples of rotational invariant homogeneous distances and rotational teams, the place the coarea formulation takes a less complicated shape. via a similar Blow-up Theorem we receive an optimum estimate for the Hausdorff measurement of the attribute set relative to C1,1 hypersurfaces in 2-step teams and we end up that it has finite Q − 2 Hausdorff degree, the place Q is the homogeneous size of the gang.

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**Extra info for A Blow-up Theorem for regular hypersurfaces on nilpotent groups**

**Example text**

We then take as G the simplest case left and proceed as above. Let κ = κ1 pκ2 where dv[κ1 , p] = 1, (κ = α, β, γ, δ, , d, e). For convenience the groups are divided into three sets and each set is subdivided into eight cases. The sets are given by A: B: C: 2 2 2 = 0, β2 = 0, γ2 = 0, = 1, β2 = 0, γ2 = 0, = 2, β2 = 0, γ2 = 0. The subdivision into cases and results of the discussion are given in Table I. I. 1 2 3 4 5 6 7 8 δ2 1 0 1 1 0 0 1 0 e2 1 1 0 1 0 1 0 0 α2 1 1 1 0 1 0 0 0 A B A1 B1 A1 A3 A1 A3 A3 B1 B3 B1 B3 B3 C B1 B1 B3 B1 B3 B1 B3 B3 6.

2 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 2 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 e2 1 1 1 1 0 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 j2 1 1 1 1 1 0 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 II. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 B. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 × × 12 12 21 21 12 12 21 24 12 24 21 21 21 21 12 24 24 21 24 24 24 21 24 21 24 24 * 24 24 * 2 × 21 21 × 21 21 21 24 21 24 24 24 21 24 24 21 24 24 24 24 * 24 * 24 24 24 * * * * * * 3 × 31 31 × 4 A.

Equation (15) is replaced by m−4 g jp S −1 1 R S1 = R Q P (19) . From (16), (18), (19) and (12) S 1−p P S p1 = P 1+ and 2 ≡ 0 (mod p). Let 2 m−4 2p = p and (16) is replaced by S 1−1 P S1 = Rγ Qδ P 1+ (20) =P pm−4 . Transforming both sides of (1), (8) and (9) by S1 −1 −1 1+kp −1 S −1 S1 · S −1 1 Q 1 P S1 · S 1 QS1 = S 1 P −1 S −1 S1 1 R · S −1 1 P S1 · S −1 1 RS1 = β S −1 1 Q S1 m−4 · S1 , 1+αpm−4 S −1 S1 , 1 P m−4 −1 −1 −1 ap −1 S −1 S1 · S −1 1 R 1 QS1 · S 1 RS1 = S 1 QS1 · S 1 P S1 . Reducing these by (18), (19), (20) and (12) and rearranging 0, γ, δ + βc, 1 + c 2 m−4 + αc + k + acδ + aβ = [0, γ, δ, 1 + ( + k)p [0, γ, β + δ, 1 + {kg + + α + aδ − aγg}p − aγ pm−4 ].