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We investigate homogeneous geodesics in a class of homogeneous spaces called $M$-spaces, which are defined as follows. Let $G/K$ be a generalized flag manifold with $K=C(S)=S\times K_1$, where $S$ is a torus in a compact simple Lie group $G$ and $K_1$ is the semisimple part of $K$. Then the {\it associated $M$-space} is the homogeneous space $G/K_1$. These spaces were introduced and studied by H.C. Wang in 1954. We prove that for various classes of $M$-spaces the only g.o. metric is the standard metric. For other classes of $M$-spaces we give either necessary, or necessary and sufficient conditions, so that a $G$-invariant metric on $G/K_1$ is a g.o. metric. The analysis is based on properties of the isotropy representation $\mathfrak{m}=\mathfrak{m}_1\oplus \cdots\oplus \mathfrak{m}_s$ of the flag manifold $G/K$ (as Ad$(K)$-modules) and corresponding decomposition $\mathfrak{n}=\mathfrak{s}\oplus\mathfrak{m}_1\oplus \cdots\oplus \mathfrak{m}_s$ of the tangent space of the $M$-space $G/K_1$ (as Ad$(K_1)$-modules)., Comment: 28 pages

In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold., Comment: 11 pages, minor corrections, to appear in Differential Geometry and its Applications

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles., Comment: 17 pages. Any comments are welcome. arXiv admin note: text overlap with arXiv:1603.01975

We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the first author, we are able to provide an alternative proof for the classification theorem of the affine hypersurface with parallel Fubini-Pick forms, which has already been established by Z.J. Hu et al in a totally different way., Comment: 19 pages, already submitted in May, 2014. arXiv admin note: substantial text overlap with arXiv:1303.4278, arXiv:1401.0860

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/\mu_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan., Comment: 34pages

We prove that the orbifold quantum ring is preserved under singular symplectic flops. Hence we verify Ruan's conjecture for this case., Comment: 34pages

The paper was removed.

withdrawn by authors, Comment: 12 pages, latex

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point., Comment: LaTex, 14 pages