A geometric characterization of orientation reversing involutions

An existence and uniqueness result for orientation-reversing harmonic diffeomorphism from $\mathbb{H}_*^n$ to $\mathbb{R}_*^n$

In this paper, we prove an existence and uniqueness theorem for orientation-reversing harmonic diffeomorphisms from $\mathbb{H}_*^n$ to $\mathbb{R}_*^n$ with rotational symmetry, which is a generalization of the corresponding result for dimension $2$.

On symmetric Willmore surfaces in spheres II: the orientation reversing case

In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in $S^4$ and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (possibly branched) maps from $\mathbb{R} P^2$ to $S^4$. The Veronese sphere and several other examples of non-branched Willmore immersions from $\mathbb{R} P^2$ to $S^4$ are derived as an illustration of the general theory. The Willmore immersions of $\mathbb{R} P^2$, just mentio…

具有反向定向对称性的 MHD 平衡的闭合轨道

Closed orbits of MHD equilibria with orientation-reversing symmetry

本文研究具有非反射型反向定向对称性的 MHD 平衡，利用拓扑方法证明在压力函数具有环形嵌套水平集时环面上所有轨道均周期，并提出刻画微分同胚作用的指标。

Chern-Connes Character for the Invariant Dirac Operator in Odd Dimensions

In this paper we give a proof of the Lefschetz fixed point formula of Freed$^{\rm [1]}$ for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced in [2] and then we generalize this formula under the noncommutative geometry framework.

Seiberg-Witten invariants and real curves

On a compact oriented four-manifold with an orientation preserving involution c, we count solutions of Seiberg-Witten equations, which are moreover symmetrical in relation to c, to construct "real" Seiberg-Witten invariants. Using Taubes' results, we prove that on a symplectic almost complex manifold with an antisymplectic and antiholomorphic involution, this invariants are not all trivial, and that the canonical bundle is represented by a real holomorphic curve.

Isospectral hyperbolic surfaces have matching geodesics

We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces to Huber's theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.