Two Problems In The Differential Geometry And Integral Geometry

In this thesis we mainly solve three problems, One is that given two surfaces ∑_k(k = i,j) in the Euclidean space E³, we get an intersection curve Γ of the two surfaces, the curve Γ has curvature k in E³, and the normal curvature K_n^k(k = i, j) and the geodesic curvature K_g^k(k = i,j) on the surfaces. Euler has found the relation between the curvature k and the normal curvature K_n^k(k = i,j), respectively, and the angle 9 between the two surfaces (see theorem [3]). We get an analogue Euler formula that is the geodesic curvature k_g^k (k = i, j) and the θ also satisfy the relation similar to the Euler formula (see theorem [4]).Isoperimetric inequality is an old problem, it says that given a domain D in the Euclidean plane, if the area is fixed then the circle has the minimum length; or if the length is fixed, the circle has the maximum area. And expressed in mathematic language is that: the area A and the length L of any domain D in the plane then satisfy the inequality L² — 4Ï€A ≥ 0; with the equality hold if and only if D is a disc. Many mathematician has been interested in obtain sufficient condition to guarantee that a given domain D\ of surface area A₁, bounded by a simple piecewise smooth boundary (?)D₁, of volume V₁ may be contained in another given domain D₀ of surface area A₀, bounded by a simple piecewise smooth boundary (?)D₀, of volume V₀. The type of condition sought is meaningful if it just depends on volumes V₁, V₀, surface areas A₁, A₀, curvature integrals of boundaries (?)D₁ and (?)D₀ of the two domains involved. Hadwiger [26, 30] was the first to use the method of integral geometry to obtain some sufficient conditions for domains in a 2 dimensional Euclidean plane R² by estimating the kinematic measure of one domain moving into another under the rigid motions in R². Forty-five years later, Delhi Ren [11, 26] obtained other sufficient conditions in R². Bonnesen get a stronger inequality by using the containing problem. In this thesis another problem we solved is that we use the method of integral geometry and the condition of one domain to contain another domain in constant plane, get the analogue of Bonnesen-type inequality.Furthermore, the upper limits of isoperimetric deficit is also an interesting problem, Delin Ren has obtained the upper limits by the method of choosing a suitable circle that can't containing or contained in a fixed domain in Euclidean space. In thesis thesis we use the same method and obtain the upper limits of...