Narrow Results

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus $g$ with one boundary component is connected and in the case of non-orientable Klein surfaces it has $\frac{g+1}{2}$ components, if $g$ is odd, and $\frac{g+2}{2}$ components for even $g$. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

In this paper, we investigate the exact time-dependent black hole solution on a warped five-dimensional Randall–Sundrum space–time in conformal dilaton gravity. The zeroes of the model are described by a meromorphic quintic polynomial, which has no essential singularities. The quintic equation can be reduced to the Brioschi form by means of the Weierstrass elliptic curve over $\mathbb{Q}$. The model fits the antipodal boundary condition, i.e. antipodal points in the projected space are identified using the embedding of a Klein surface in ${ℂ}^2$, using the ${\mathbb{Z}}_2$ symmetry on the two sides of the brane. If one writes ${}^{(5)}{g}_{\mu \nu }={\omega }^{4∕3}{}^{(5)}{\tilde{g}}_{\mu \nu },{}^{(5)}{\tilde{g}}_{\mu \nu }={}^{(4)}{\tilde{g}}_{\mu \nu }+n_{\mu }{n}_{\nu }$, ${}^{(4)}{\tilde{g}}_{\mu \nu }={\bar{\omega }}^2{}^{(4)}{ḡ}_{\mu \nu }$, with $n_{\mu }$ the normal to the brane and $\omega$ the dilaton field, then ${}^{(4)}{ḡ}_{\mu \nu }$ is conformally flat. It is the contribution from the bulk which determines the real pole on the effective four-dimensional space–time. There is no objection applying ’t Hooft’s back reaction method in constructing the unitary S-matrix for the Hawking radiation. Again, there is no “inside” of the black hole. The zeroes can also be analyzed by the icosahedron equation and by the Hopf-fibration of the Klein surface.

A new exact time-dependent Kerr-like black hole solution is found on a Randall–Sundrum brane world spacetime. The solution is also valid on the Wick-rotated Euclidean counterpart space, so represents equally well a gravitational instanton, i.e. a bump in spacetime. Since the r-dependent part is determined by a first-order differential equation, one conjectures that the solution is a self-dual solution comparable with the Eguchi–Hanson solution. The zeroes and poles of the spacetime are determined by a quintic polynomial. To describe the Hawking particles, one uses the antipodal boundary condition on a complex Klein-bottle hypersurface ${ℂ}^1\times {ℂ}^1$. We used the Hopf fibration to get $S^2$ as the black hole horizon, where the centrix is not in a torus but in the Klein bottle. The twist fits very well with the antipodal identification of the point on the horizon. No “cut-and-paste” is necessary, so the Hawing particles remain pure without instantaneous information transport. A local observer passing the horizon will not notice a central singularity in suitable coordinates. The black hole paradoxes are also revisited in our new black hole model. A connection is made with the geometric quantization of ${ℂ}^1\times {ℂ}^1\sim S^3$, by considering the symplectic 2-form. The model can be easily extended to the non-vacuum situation by including a scalar field. Both the dilaton and the scalar field can be treated as quantum fields when approaching the Planck area.

A compact bordered Klein surface of algebraic genus p ≥ 2 has at most 12(p-1) automorphisms. Automorphism groups which attain this bound are called M*-groups. In this paper, firstly, we define generalized M*-groups. Then, we show that there is a relationship between the extended Hecke groups and generalized M*-groups. Finally, we prove that a generalized M*-groups G is supersoluble if and only if |G| = 4 · q^r for q ≥ 3 prime number and for some positive integer r.

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by . It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with is finite. The value of is known when G belongs to certain families of groups. In particular, if o(G) < 32, is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.

The reconstruction of a Riemann surface starting from the meromorphic function field K, comes from Dedekind and Weber who developed an algebraic function theory in one variable over an algebraically closed field k. Alling and Greenleaf present a counterpart to this approach starting from a real algebraic curve. From this point of view, the residues theorem is a classical result which depends strongly on the algebraically closed character of the base field. In this paper, via the complex double, we translate this fact to the case where we start from a function field in one variable over R.