On the Maslov index of symplectic paths that are not transversal to the Maslov cycle. Semi-Riemannian index theorems in the degenerate case
R. Giambo, P. Piccione, A. Portaluri
备注：
55 pages, LaTeX2e amsart class, 2 figures. In this revised version of October 26th, 2004, a short note has been added in the title page and some references have been updated. The note provides information on the contribution of each author to the development of the theory exposed in the paper
我们使用广义签名的概念，在一个平滑曲线的对称双线性型的奇异性，以确定一个公式计算的Maslov指数的情况下，一个真正的解析路径有可能非横向交叉。我们讨论了一些应用的理论，特别强调的研究雅可比方程沿半黎曼测地线。本文的研究工作源于第三作者提出的将谱流与部分特征联系起来的建议。他指出了几个参考文献，并制定了声明，在可逆端点的情况下，频谱流的一个真正的分析路径的自伴弗雷德霍姆算子是由总和的奇数部分签名在每个退化的时刻。在本文的当前版本中，这一陈述是命题2.9的一部分;总的来说，第三作者对理论的贡献包括定义2.3、公式（2.1）和（2.2）、命题2.4的第一个陈述、注释2.5的部分、定义2.6以及命题2.9的部分陈述和部分证明。除了这一原始贡献，本文所载的材料完全是由前两位作者在Universita' di Camerino（意大利），圣保罗大学（巴西）开发和编写的。
英文摘要：
We use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a real-analytic path having possibly non transversal intersections. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semi-Riemannian geodesic.
The research work exposed in this paper originated from a suggestion given by the third author relating the spectral flow with the partial signatures. He pointed out several references and formulated the statement that in the invertible endpoints case, the spectral flow of a real analytic path of self-adjoint Fredholm operators is given by the sum of the odd partial signatures at each degeneracy instant. In the present version of the article, this statement is part of Proposition 2.9; totally, the contribution of the third author to the theory consists in Definition 2.3, formulas (2.1) and (2.2), the first statement of Proposition 2.4, parts of Remark 2.5, Definition 2.6, and parts of the statement and parts of the proof of Proposition 2.9. Apart from this original contribution, the material contained in this paper was entirely developed and written by the first two authors at the Universita' di Camerino (Italy), Universidade de Sao Paulo (Brazil).
We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the curve's Maslov index by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the …
We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.
In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along closed or non-closed paths therein. In especially, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on $L$, hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown tha…
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In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic…
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The imploded cross-section of a symplectic manifold is a stratified space allowing for an abelianization of its symplectic reduction. After recalling symplectic and Poisson reduction and reviewing the basics of symplectic implosion, we prove a cross-section theorem for Poisson manifolds, generalizing the Guillemin-Sternberg theorem for symplectic manifolds, which constitutes a first step towards Poisson implosion. On our way, we find and fix a mistake in the proof of Guillemin-Sternberg's theorem, and we identify Poisson transversals as the right analogue to symplectic submanifolds in this con…
On a compact Kähler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$ as $\hbar \to 0^+$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localiza…
Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated …
