...ia Hao, Professor Weili He and Erling Wei for their kindly helps and often discussing problems in mathe- matics altogether. Of course, I am responsibl... ...per half planeH ={z∈C|Imz > 0} with{(U 1 =H,φ 1 = 1 H )} and the open unit disc D = {z ∈ C||z| < 1} with {(U 1 = d,φ 1 = 1 D )} in C are two Klein sur... ... the Smarandache manifolds (abbreviateds-manifolds) are the central object discussed in the Smarandache geometries today. More results for theSmaranda... ...p with respect to the composition operation and AutH =PGL(2,R). Let Γ be a discrete subgroup of AutH. We say that Γ is a non-euclidean crystallographi... ...uclidean crystallographic group( shortly NEC group) if the quotient H/Γ is compact. More results can be seen in [11]. Typical results for automorphism... ...phisms of a Klein surface S are as follows. Theorem 1.1.2([11]) Let S be a compact Klein surface, g = g(S) and k = k(S), then (i) there existsanNEC gr... ...tition of a surface gotten by T.Rad´ o in 1925. Theorem 1.2.1([52])For any compact surfaceS, there exist a triangulation{T i ,i≥ 1} on S. This theorem... ... of complex algebraic curves are the same as the automorphism groupsof the compact Riemann surfaces. The latter can be combinatorially dealt with the ... ...ered the order of an automorphism of a Riemann surface of genus p≥ 2 and a compact non-orientable Klein surface without boundary of genus q ≥ 3. Their...