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Register a new userNotes on the Jacobi equation
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Authors
Alexander Lytchak
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We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.
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We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi-Nijenhuis structures.
We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltonian and Jacobi equivalence.
In the paper Einstein metrics on compact simple Lie groups attached to standard triples, the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except the standard triple $(Sp(4),2Sp(2),4Sp(1))$. For the triple $(Sp(4),2Sp(2),4Sp(1))$, we find there exists an involution pair of $sp(4)$ such that $4sp(1)$ is the fixed point of the pair, and then give the decomposition of $sp(4)$ as a direct sum of irreducible $ad(4sp(1))$-modules. But $Sp(4)/4Sp(1)$ is not a generalized Wallach space. Furthermore we give left-invariant Einstein metrics on $Sp(4)$ which are non-naturally reductive and $Ad(4Sp(1))$-invariant. For the general case $(Sp(2n_1n_2),2Sp(n_1n_2),2n_2Sp(n_1))$, there exist $2n_2-1$ involutions of $sp(2n_1n_2)$ such that $2n_2sp(n_1))$ is the fixed point of these $2n_2-1$ involutions, and it follows the decomposition of $sp(2n_1n_2)$ as a direct sum of irreducible $ad(2n_2sp(n_1))$-modules. In order to give new non-naturally reductive and $Ad(2n_2Sp(n_1)))$-invariant Einstein metrics on $Sp(2n_1n_2)$, we prove a general result, i.e. $Sp(2k+l)$ admits at least two non-naturally reductive Einstein metrics which are $Ad(Sp(k)timesSp(k)timesSp(l))$-invariant if $k<l$. It implies that every compact simple Lie group $Sp(n)$ for $ngeq 4$ admits at least $2[frac{n-1}{3}]$ non-naturally reductive left-invariant Einstein metrics.
The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alternative proof of the splitting theorem of homogeneous Poisson structures.
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