By Toshiaki Adachi, Hideya Hashimoto, Milen J Hristov

This quantity comprises contributions by way of the most contributors of the 4th foreign Colloquium on Differential Geometry and its comparable Fields (ICDG2014). those articles hide contemporary advancements and are committed regularly to the learn of a few geometric constructions on manifolds and graphs. Readers will discover a wide evaluation of differential geometry and its courting to different fields in arithmetic and physics.

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**Additional info for Current Developments in Differential Geometry and its Related Fields: Proceedings of the 4th International Colloquium on Differential Geometry and its Related Fields**

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For a K¨ ahler magnetic field Bκ , we denote by ικ (p) the Bκ -injectivity radius at p ∈ M . It is defined as ικ (p) = sup r Bκ expp |Br (0p ) is injective , where Br (0p ) ⊂ Tp M denotes a Euclidean closed ball of radius r. On this manifold M , it is known that ικ (p) = ι−κ (p) and that a trajectory-ball Br (p; κ) = Bκ expp Br (0p ) coincides with Br (p; −κ) at each point p for an arbitrary r with 0 < r < ικ (p). Therefore on M we have Mr;κ = Mr;−κ for an arbitrary r with 0 < r < ικ (M ) = inf{ικ (p) | p ∈ M }.

3. We take a set Fig. 5. Petersen K¨ ahler graph Fig. 6. graph Complexified Petersen V = (i, j) i = 0, 1, 2, 3, j = 0, 1, 2, 3, 4 . We define E (p) as i) (i, j) ∼p (i, j + 1) for i = 0, 2, ii) (i, j) ∼p (i, j + 2) for i = 1, 3, iii) (0, j) ∼p (1, j) and (2, j) ∼p (3, j), and define E (a) as i) (i, j) ∼a (i, j + 1) for i = 1, 3, ii) (i, j) ∼a (i, j + 2) for i = 0, 2, iii) (1, j) ∼a (2, j) and (3, j) ∼a (0, j), where we consider the second index by modulo 5. We then get a regular (p) (a) K¨ ahler graph of dG = dG = 3 having 20 vertices.

For example, the Petersen K¨ ahler graph (Fig. 5) is obtained by this way. By the construction of G, if we put nI the cardinality of the set I, we find that (p) (p) dG (0, i) = dH (i) + nI , (p) (a) dG (1, i) = dH (i) + nI , dG (0, i) = dH (i) + nI , dG (1, i) = dH (i) + nI , (a) (a) (a) (p) page 41 August 27, 2015 9:16 Book Code: 9748 – Current Developments in Differential Geometry 42 ws-procs9x6˙ICDG2014 T. ADACHI (p) (a) hence find that dG (ǫ, i) = dG (ǫ, i) at each vertex (ǫ, i) ∈ V . If we take a map f : V → V given by (ǫ, i) → (ǫ + 1, i), where the first indices are considered by modulo 2, we find that it gives an isomorphism of GI (H) to its dual GI (H)∗ .