By Linfan Mao

ISBN-10: 1931233926

ISBN-13: 9781931233927

A combinatorial map is a attached topological graph cellularly embedded in a floor. This monograph concentrates at the automorphism team of a map, that's concerning the automorphism teams of a Klein floor and a Smarandache manifold, additionally utilized to the enumeration of unrooted maps on orientable and non-orientable surfaces. a few effects for the automorphism teams of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are chanced on. An hassle-free class for the closed s-manifolds is located. Open difficulties with regards to the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries also are provided during this monograph for the additional functions of the combinatorial maps to the classical arithmetic.

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**Extra resources for Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Partially Post-Doctoral Research for the Chinese Academy of Sciences)**

**Sample text**

4. 2 The measures on a map On the classical geometry, its central question is to determine the measures on the objects, such as the distance, angle, area, volume, curvature, . .. For maps being a combinatorial model of Klein surfaces, we also wish to introduce various measures on a map and enlarge its application filed to other branch of mathematics.. 1 The angle on a map For a map M = (Xα,β , P), x ∈ Xα,β , the permutation pair {(x, Px), (αx, P −1 αx)} is called an angle incident with x, which is introduced by Tutte in [66].

By the previous voltage assignment, we also know that M l is a lifting of the quotient map M/G with the voltage assignment ϑ : Xα,β /G → G. Therefore, M ∼ = (M/G)ϑ . This completes the proof. 1, we get the following result for a group to be an automorphism group of a map. 2 If a group G, G |G|(χ(M/G) + AutM, is fixed-free on V (M), then (−1 + m∈O(F (M/G)) 1 )) = χ(M). 1, we know that there is a voltage assignment ϑ on the quotient map M/G such that M ∼ = (M/G)ϑ . 3, we know the Euler characteristic of the map M is χ(M) = |G|(χ(M/G) + (−1 + m∈O(F (M/G)) 1 )).

Case 1 M is orientable. Since χ(M/G) + 1 = (|Φv (g)| + |Φf (g)|) ≥ 0, we know that χ(M/G) ≥ −1. Whence, χ(M/G) = 0 or 2. Therefore, we have that |G| = 1 + χ(M/G) − χ(M) ≤ 3 − χ(M) = 2g(M) + 1. That is, Nmax ≤ 2g(M) + 1. Chapter 2 On the Automorphisms of a Klein Surface and a s-Manifold Case 2 56 M is non-orientable. In this case, since χ(M/G) ≥ −1, we know that χ(M/G) = −1, 0, 1 or 2. Whence, we have that |G| = 1 + χ(M/G) − χ(M) ≤ 3 − χ(M) = g(M) + 1. This completes the proof. 7, which is even more better than the results already known.

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