Crofton Formula In 3 Dimensional Real Hyperbolic Space

Hyperbolic geometry is an important field in modern differential geometry, Crofton formulas in different spaces are important in integral geometry and have obtained many classical results. In chapter 3 , we obtain the Weierstrass model of 3 dimensional real hyperbolic space and verified the truth of the Hilbert axiom system. In chapter 4. we study the Crofton formula in 3 dimensional real hyperbolic space, obtain the Crofton formula of hyperbolic planes which meet an arbitrary rectifiable curve in 3 dimensional real hyperbolic space and the Crofton formula of n — 1 dimensional hyperbolic hyperplanes which meet an arbitrary rectifiable curve in n dimensional real hyperbolic space, improve Robertson's conclusion. The main results obtained in the thesis can be summarized as follows:Theorem 4.2.1(Crofton formula in H₊³(-1)) Let C be an arbitrary rectifiable curve which length is L, F_l² are hyperbolic plains, A = {l ∈IR₁⁴ = 1, F_l² ∩ C ≠ (?)}, then the measure of F_l² which meet C iswhere n(F_l²) is the number of times Ff meet C, dl is the volume element of the set of vertexes of l.Theorem 4.2.2(Crofton formula in H₊²(-1)) Let C be an arbitrary rectifiable curve which length is L, F_l¹ are hyperbolic lines, B = {l ∈IR₁³ = 1, F_l¹ ∩ C ≠ (?)}, then the measure of F_l¹ which meet C iswhere n(F_l¹) is the number of times F_l¹ meet C, dl is the volume element of the set of vertexes of l.Theorem 4.2.3(Crofton formula in H₊ⁿ(-1)) Let C be an arbitrary rectifiable curve which length is L, F_l^n-1 are n— 1 dimensional hyperbolic hyperplanes, Ω = {l ∈IR₁ⁿ⁺¹| (l,l) = 1, F_l^n-1 ∩ C ≠ (?)}, then the measure of F_l^n-1 which meet C iswhere n(F_l^n-1) is the number of times F_l^n-1 meet C, dl is the volume element of the set ofvertexes of l, k is constant.