...ot clear which Riemann had in mind (see [3]), but today we would call it either elliptic or spherical geometry. All of these geometries (Euclidean, ... ...elliptic or spherical geometry. All of these geometries (Euclidean, hyperbolic, elliptic, and spherical) are homogeneous and isotropic. This is to sa... ...geometry sometimes refers specifically to one of the geometries just mentioned (elliptic or spherical), it is now most likely to be associated with ... ...re, geometry is studied through curvature, and the basic Euclidean, hyperbolic, elliptic, and spherical geometries are particular constant curvature... ... as those possible universes that inhabitants might mistake as being Euclidean, elliptic, or hyperbolic. Great insight comes from the realization th... ... make congruent vertical angles with the edge. A geodesic in a manifold is a curve that is as straight as possible. Lines in an s-manifold will b... ...m from the lines in the Euclidean plane. An s-line will be any piecewise linear curve that can be constructed from a line segment lying within one o... ...s-manifold, and in extending indefinitely, we may be traversing the same closed curve an infinite number of times. In addition, we will see that s-l... ...and flattened the surface. In the paper model shown in Figure 11, these s-lines curve, but only in a direction perpendicular to the surface. In othe...

.... Math. 45(1914), 170-186. 2. B.J. Birch, H.P.F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math. 212(1963), 7-25. 76 10 On the su... ...5(1914), 170-186. 2. B.J. Birch, H.P.F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math. 212(1963), 7-25. 76 10 On the sum of two ...

...........................104 5.3 Combinatorial construction of an algebraic curve of genus .................105 5.4 Classification of s-manifolds by map... ...troduced by Alling and Greenleaf [2] in 1971 concerned with real algebraic curves, correspondence with that of Riemann surface concerned with complex ... ...rrespondence with that of Riemann surface concerned with complex algebraic curves. For introducing this concept, it is need to enlarge analytic functi... ...ound them. A vertex around five, six or seven triangular disks is called an elliptic vertex, a Euclid vertex or a hyperbolic vertex, respectively. Ans-... ...nting a vertex u by a pointp(u),p(u) =p(v) ifu =v and an edge (u,v) by a curve connecting the points p(u) and p(v) on the plane. For example, the gr... ...n{5,6,7} as follows: Classical Type: (1) Δ 1 ={5−regular triangular maps} (elliptic); (2) Δ 2 ={6−regular triangular maps}(euclid); (3) Δ 3 ={7−regula... ...dache Type: (4) Δ 4 ={triangular maps with vertex valency 5 and 6} (euclid-elliptic); (5) Δ 5 ={triangular mapswithvertexvalency 5and 7} (elliptic-hyp... ...are formula, we know that χ(M) =ν(M)−ε(M)+φ(M) =( 3 k − 1 2 )φ(M). If M is elliptic, then k = 5. Whence, χ(M) = φ(M) 10 > 0. Therefore, if M is orient... ...roup 94 degree of a curve 105 dianalytic function 1 divisor 104 dual map 8 elliptic vertex 3 embedding of a graph 6 embedding sequence 24 equivalent e...