By Luther Pfahler Eisenhart

Non-Riemannian Geometry bargains primarily with manifolds ruled by means of the geometry of paths co-developed by way of the prestigious mathematician Luther Pfahler Eisenhart, the writer of this article. He starts with a attention of uneven connections, after which proceeds to a contrasting survey of symmetric connections. Discusses projective geometry of paths and the geometry of sub-spaces. 1927 version.

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48 3. 9) classify global solutions, somewhat like in minimal surface theory. We know that F ∗ (u) is, in particular a set of ﬁnite perimeter and therefore almost every point, with respect to H n−1 F ∗ (u), is a diﬀerentiability point. That is, if x is one of those points, at x is well deﬁned a normal vector ν = ν(x) such that, if we set + Ω+ r = {y : r(y − x) ∈ Ω (u)} P + = P + (x, ν) = {y : y − x, ν > 0} π = π(x, ν) = {y : y − x, ν = 0} and let B = B(x) a (small) ball centered at x, then Per(Ω+ r ∩ B) converges in the sense of vector measures to Per(P + ∩ B), that is, for any continuous vector ﬁeld ϕ: r→0 ∂Ω+ r ∩B ϕ, ν d Per −−−→ ϕ, ν dH n−1 .

U Bε (xj ) therefore the quantities b) and c) are comparable. 3. STRONG RESULTS 47 since, for proper choices of c we can make Ncε (F (u)) ∩ BR ⊂ {0 < u < ε} ∩ BR or vice versa. It follows that the quantities a), b) and c) are all comparable to Rn−1 . Finally, let {Brj (xj )}, xj ∈ F (u), a ﬁnite covering of F (u)∩BR by balls of radius rj < ε, that approximates H n−1 (F (u) ∩ BR ). Let r < min rj and {Br (xkj )} a ﬁnite overlapping covering for F (u) ∩ Brj (xj ). Then, on one hand |∂Br (xkj )| ≤ cRn−1 k,j by the argument above with ε = r.

Let y ∈ Bε (x) and notice that if τ ∈ Γ( θ2 , en ) and τ¯ = τ − (y − x) τ − τ | = |x − y| ≤ |τ | sin θ2 . Also then α(τ, τ¯) ≤ 2θ , since |¯ |¯ τ | ≥ |τ | − |τ | sin since θ 2 1 θ ≥ |τ | 2 2 < π4 . 8, we deduce that inf B1/8 (x0 ) Dτ¯ u ≥ c0 ν, τ¯ |∇u(x0 )| ≥ c ν, τ¯ u(x0 ) τ | cos α(ν, τ¯) ≥ c1 |¯ sup u B1/8 (x0 ) ≥ bε sup u B1/8 (x0 ) where b = b(τ ) = C cos( θ2 + α(ν, τ )). 5 are satisﬁed. 6. 4, perhaps with a slightly diﬀerent enlarged cone, that we still denote by Γ(θ¯1 , ν¯1 ). 4. 3, with θ = θ/2, θ θ 1 + cμ cos + α(ν, τ ) 2 2 θ [1 + cμ sin E(τ )] = |τ | sin 2 θ +μ ¯E(τ ≡ ρ(τ ) ≥ |τ | sin 2 (1 + bμ)ε = |τ | sin with μ ¯ = μc θ20 .