Remarks on the Distribution of Rays
Masao Maeda · Institutional Repositories DataBase (IRDB) · 1981
Let M bea 2-dimensional complete non-compact Riemannian manifold having non-negative Gaussian curvature K. Then from a well known theorem of Cohn-Vossen, the total curvature of M satisfies S.K dv:II2T, where dv is the volume element of M induced from the Riemannian metric of M, see [4].Obviously the total curvature of M is not a topological invariant when M is non-compact in contrast with compact case.And it seems for the auther that the total curvature of M is expressing a certain curvedness of M. From this point of view, in [5] we showed a fact as following manner;For a point qEM, T,(M) denotes the tangent space of M at q. Put Sg(M) := {veT,(M); norm of v=1}.Then form the Euclidean metric on T,(M), Sq(M) becomes a Riemannian submanifold of T,(M) which is the standard unit circle.Thus we can consider the Riemannian measure on S,(M).Let A(q)cS,(M) be the set defined as {vES,(M); geodesic r: [O, oo)-> M given by r(t)=exp, tv is a ray} Here exp,: T,(M) -> M is the exponential mapping of Mand geodesic r is called a ray when any subarc of r is a shortest connection between its end points.#A(q) denotes the number of elements of A(q).Under these notation, we haveFAcT.Let M be a 2--dimensional comPlete Riemannian manijold wz'th nonnegative Caussian curvature K; diffeomorphic to a Euclidean Plane.Then for any Point qEM such that #A(q)l2, measure A(q))-2z-i.Kdv･ Note that from non-compactness of M) for any point qEM, it holds #A(q) ;-})1.And for a certain point qEM, called a soul of M, it holds #A(q)}-ii2, see [4]., Note also that from classification by Cohn-Vossen, Mis isometric to a flat open M6bius band or flat cylinder or a one which is diffeomorphic to a Euclidean plane.The assumption in the above fact that #A(q)l;2 is not necessary, This was proved recently by K. Shiga in [6], And in [6], the above fact was generalized "