Researches On Curvatures Of Curves And Surfaces

H. Hopf was the first one who proved that the only constant mean curved surface with genus zero in Euclidean three space was the standard sphere. His idea was then extended and used by Calabi [4] and Chern [6] in the theory of minimal spheres. In this thesis, we note in particular the Alexandrov theorem, which is concerned with the constant mean curvature. We will reprove the theorem of Alexandrov.We will use the Ros' inequality to reprove the theorem of Alexandov: the only compact embedded surfaces of constant mean curvature in E³ are the standard spheres.Many other methods can be found to prove the theorem of Alexandrov. In the thesis, we will reprove the theorem of Alexandrov. We will use the Steiner symmetrization, the properties of symmetrization and the method of integral, etc. If we generalize the corresponding lemmas to higher dimensions, we can easily get the similar proof of Alexandrov theorem in Eⁿ.In the second part of the thesis, we investigate invariants of the intersection of submanifolds of E³.Let M^(k) (k = 1,2) be two compact submanifolds in the Euclidean space Eⁿ. Let G be the group of isometries in Eⁿ and dg the kinematic density. For g ∈ G, let I(M⁽¹⁾∩ gM⁽²⁾) be an invariant of M⁽¹⁾∩gM⁽²⁾(such as volume, area or integral of curvature). The integral ∫G I(M⁽¹⁾∩ gM⁽²⁾)dg is called the kinematic formula forI(M⁽¹⁾∩ gM⁽²⁾).It is expected that the integral I(M⁽¹⁾∩ gM⁽²⁾) be expressed by the invariants of M⁽¹⁾ and M⁽²⁾ and the "angle" between M⁽¹⁾ and M⁽²⁾, unfortunately, we know little about them. Kinematic formulas have many important applications in convexity, geometric inequalities and other mathematical branches (see [26], [38], [39], [40] for more details).Let M^(k) (k = 1,2) be two C² smooth surfaces in E³ and k_n^(k) be the normal curvatures of M^(k). For any g ∈ G₃, the rigid motion of E³, let k be the curvature of the intersection curve M⁽¹⁾∩gM⁽²⁾ and θ be the angle between M⁽¹⁾ and Then we haveAbove formula is due to Euler and has been reproved and applied to integral geometry by Zhang and Zhou ([35], [39]). Let kg be the geodesic curvature of the intersection curve (M⁽¹⁾∩ gM⁽²⁾) then the following analogue of the Euler formula is obtained by Chen, Zhao and Zhou ([7]).