This paper is a continuation of the one with the same title([4]), in which we obtained a special solution for a system of differential equations on metric tensors on R4+ satisfying the Einstein condition for Case I generalizing the Ot-metric: in Theorem 1, and proved that there exist no solutions for Case II in Theorem 2. In this work, we shall show that we can obtain more general solutions for Case I which depend on the latitude parameter. We use the results in [4], so the section numbers start from 5. 1. Preliminaries and curvature tensor 2. Ricci tensor 3. Solutions for Case I 4. Analysis and a conclusion for Case II

This work is a continuation of the papers [3] and [4], in which we studied the metrics (1.1) and (1.2) on R 4 + . The metric (1.1) with a = 0: ds 2 = dx1dx1 + dx2dx2 + dx3dx3 − dx4dx4 x4x4 is analogous to the metric of the hyperbolic 4-space. We considered fundamentally metrics on R 4 + based on this hyperbolic type metric, not Euclidean or Minkowsky types.

We studied the geodesics of a spacetime with the pseudo-Riemannian metric: ds2=1x4x4{∑b,c=13(δbc−axbxc1+ar2)dxbdxc−11+ax4x4dx4dx4} on R3×R+, where r2=∑b=13xbxb and a= constant, which are plane quadratic curves (in [12]). In this paper, we shall determine all the Killing vector fields of this spacetime and choose special pairs out of them with interesting properties for the case a>0.

From a Minkowski-type metric on R4+ satisfying the Einstein condition, we derive a nonlinear partial differential equation. We obtained a solution for it under certain condition in the 4-dimensional case. Using this solution we shall make a model space on R4 with certain general connection which admits an interesting exposition for geodesics.

From a Minkowski-type metric on R+n satisfying the Einstein condition, we derived a nonlinear partial differential equation. We tried to get some numerical approximate solution with certain boundary conditions by the finite element method in [10]. Regarding the exact solution, we shall give a fundamental theorem, which tells us the above numerical approximate solution is not worth to using the word “approximate”.

A general connection Γ on a differentiable manifold M n is given as a geometrical object with components (Pi jTi jh) in local coordinates (ui) such that (P i j ) are the components of a tensor and (Ti j) satisfy the rules : Γ< _ dv’ f p k g V M0vT Jh duk 1 dvidvh + lmdvi dvh J ’ where T i j are the second components of Γ in local coordinates (vi). We call a point of M n is regular or singular with respect to Γ if det (P i j ) ≠ 0 or = 0, respectively. We say a curve γ(t) = (u i (t)) is a geodesic with respect to Γ if u i (t) satisfy the equations: ni d2v? „, du* duh „ and the parameter t is called its affine parameter.

where B=(n-l) n ~\(2) SsU, n)=(8n 2 -5);c 3 -2(87z 3 +20n 2 -15tt+20);c 2 and (3) S +3(12n 3 -42n 2 +37n-5)% 2 +3n(16n 2 -32n+9)A:+12?2 2 (2n-l).The present author proved the following facts: FACT 1. SO, ?2)>0 for Q^x^n, xφl, with n^2 (Proposition 4 in [1]); FACT 2. S(x, n) is decreasing in 0

On donne une metrique d'espace-temps R×(R 3 −{o}) avec la courbe r=0 comme trou noir et on rend lisse des connexions generales sur R 4 ayant le meme systeme de geodesiques avec celui de cette metrique pseudo-Riemann dans R×(R 3 −{o})

This is exactly a continuation of Part (VΠ) ([18]) with the same title written by the present author which proved the following conjecture is true for 5^n^9.7.We shall show that this conjecture is also true for 4.5rgn^5 in the present paper by the same method as in Part (VII) which is a little revised one and may be valid also for 3^n^4.5 (see the final remark of this paper), but perhaps useless for 2^n^3, because the constant b n , having carried out an important role in the proof of the conjecture, is defined for 2.5rgn

Introduction.As is shown in [6] and [8], the following nonlinear differential equation:where n(> 1) is a real constant, is the equation for the support function x(t) of a geodesic in the 2-dimensional Riemannian manifold Cξ with the metric:(0.1)zV(n -z){z{nz) n ι -

\S 0. Introduction.As is shown in [6], the nonlinear differential equation (E)$nx(1-x^{2})\frac{d^{2}x}{dt^{2}}+(\frac{dx}{dt})^{2}+(1-x^{2})(nx^{2}-1)=0$ , where $n$ is an integer $\geqq 2$ , is the equation for the support function $x(t)$ of a geodesic in the 2-dimensional Riemannian manifold $O_{n}^{2}$ with the metric:(0.1)in the unit disk: $u^{2}+v^{2}<1$ .Another geometric meaning of (E) is given in [4].Any non constant solution $x(t)$ of (E) such thatis periodic and its period $T$ is given by the improper integral:whereis the integral constant of (E) and $0\pi$ , (ii) $\varliminf_{(0}T=\pi$ and $\lim_{c\sim A}T=\sqrt{2}\pi$ .By means of a numerical analysis and observation about (E) in [5] and[7], M. Urabe conjectures the inequality Bound for peri0ds of solutions of a certain nonlinear differential equation (U)$ T<\sqrt{2}\pi$ .The author however wanted originally to have the inequality (0.4)from the standpoint of a geometrical problem related with the existence of compact minimal hypersurfaces of a certain type in the spheres.S. Furuyagave firstly an answer to it by proving the inequality (0.5)in [2] and the author proved a little sharper inequality (0.6) ) is true by (0.5) or (0.6) when $n=2$ and S. Furuya proved also that (U) is true when $n=3$ .The equation (E) however may be considered for any real number $n\geqq 2$ .In the present paper the author will prove (U) for any real number $n\geqq 3$ .$\varphi^{\prime}(x)=\frac{1-x}{x(n-x)}\varphi(x)$ , $\varphi^{\prime\prime}(x)=-\frac{n-1}{x^{2}(n-x)^{2}}\varphi(x)$ , $\varphi^{\prime\prime}(x)=\frac{(n-1)(2n-1-3x)}{x^{3}(n-x)^{3}}\varphi(x)$ and $\varphi^{(4)}(x)=-\frac{(n-1)\{(3n-1)(2n-1)-8(2n-1)x+12x^{2}\}}{x^{4}(n-x)^{4}}\varphi(x)$ .PROOF.We get easily $\varphi^{\prime}(x),$ $\varphi^{\prime\prime}(x)$ and $\varphi^{\prime\prime\prime}(x)$ , from which 208 T.

For a submanifold M in a Riemannian manifold M, the minimal index (m-index ;i at a point of M is by definition the dimension of the linear space of all the 2nd fundamental forms with vanishing trace. The geodesic codimension (g-codim) of M in M is defined by the minimum of codimensions of M in totally geodesic submanif olds of M containing M. In [8] and [9], the author investigated minimal submanif olds with m-index 2 everywhere in Riemannian manifolds of constant curvature and gave some typical examples of such submanif olds with g-codim 3 and g-codim 4 in the space forms of Euclidean, elliptic and hyperbolic types. Each example is the locus of points on a moving totally geodesic submanifold intersecting orthogonally a surface at a point. This surface is called the base surface. This situation is quite analogous to the case of the right helicoid in E3 generated by a moving straight line along a base helix. When the ambient space is Euclidean, the base surface of the example in case of g-codim 4 is a minimal surface in a 6-sphere, whose equations are analogous to those of the so-called Veronese surface which is a minimal surface in a 4-sphere with m-index 2 and g-codim 2. In [2], T. Itoh gave a minimal surface of the same sort in an 8-sphere. In the present paper, the author will give some examples of minimal submanifolds with m-index 2 and g-codim of any integer ? 2 in the space forms of Euclidean, elliptic and hyperbolic types. The base surfaces corresponding to the minimal submanifolds with m-index 2 and even geodesic codimension in Euclidean spaces will be called generalized Veronese surfaces.