[Paper Review] A Kähler Structure on Cartan Spaces

E. Peyghan, Akbar Tayebi|arXiv (Cornell University)|Mar 12, 2010

Advanced Differential Geometry Research12 references3 citations

TL;DR

This paper introduces a new Riemannian metric $ G $ on the cotangent bundle $ T^*M_0 $ of a Cartan manifold $ M $, endowing it with an almost complex structure $ J $, and proves that $ (T^*M_0, G, J) $ becomes a Kähler manifold when $ M $ has constant negative flag curvature. For positive curvature, a Kähler structure exists on a tubular neighborhood of the zero section. Crucially, the authors show that $ (T^*M_0, G, J) $ cannot be Einstein or locally symmetric unless $ M $ is Riemannian, thus proving that non-Riemannian Cartan structures do not admit these geometric properties under this construction.

ABSTRACT

In this paper, we define a new metric on Cartan manifolds and obtain a Kähler structure on their cotangent bundles. We prove that on a Cartan manifold M of negative constant flag curvature, (T* M_0, G, J) has a Käahlerian structure. For Cartan manifolds of positive constant flag curvature, we show that the tube around the zero section has a Käahlerian structure. Finally by computing the Levi-Civita connection and components of curvature related to this metric, we show that there is no non- Riemannian Cartan structure such that (T* M_0, G, J) became a Einstein manifold or locally symmetric manifold.

Motivation & Objective

• To define a new Riemannian metric $ G $ on the cotangent bundle $ T^*M_0 $ of a Cartan manifold $ M $, enabling the study of Kähler geometry on its cotangent bundle.
• To investigate under what conditions the almost complex structure $ J $ on $ T^*M_0 $ becomes integrable, leading to a Kähler structure.
• To determine whether $ (T^*M_0, G, J) $ can be an Einstein or locally symmetric manifold, and if so, under what geometric constraints on $ M $.
• To analyze curvature components and Levi-Civita connections associated with the metric $ G $, particularly for Cartan spaces of constant flag curvature.
• To establish that non-Riemannian Cartan structures cannot yield Einstein or locally symmetric Kähler manifolds under this construction, thereby characterizing the rigidity of the geometry.

Proposed method

• A new Riemannian metric $ G $ is defined on $ T^*M_0 $ using a symmetric tensor field $ G_{ij} = \frac{1}{\beta}g_{ij} + \frac{v(\tau)}{\alpha\beta}p_i p_j $, where $ v $ is a smooth function on $ [0,\infty) $, and $ \alpha, \beta $ are constants.
• An almost complex structure $ J $ is defined via $ J(\delta_i) = G_{ik}\dot{\partial}^k $ and $ J(\dot{\partial}^i) = -G^{ik}\delta_k $, and its integrability is analyzed via the Nijenski bracket.
• The integrability of $ J $ is shown to hold if and only if $ M $ has constant scalar curvature $ c $ and $ v = -c\alpha\beta^2 $, which leads to a Kähler structure on $ T^*M_0 $ for $ c < 0 $.
• For $ c > 0 $, a Kähler structure is constructed on a tubular neighborhood $ T_\beta^*M_0 $ defined by $ 2\tau < \frac{1}{c\beta^2} $, ensuring the metric remains well-defined and non-degenerate.
• The Levi-Civita connection $ \nabla $ of $ G $ is computed explicitly, and all components of the curvature tensor are derived using adapted local frames $ (\delta_i, \dot{\partial}^i) $.
• The curvature and Ricci tensor components are used to analyze whether $ (T^*M_0, G, J) $ can be Einstein or locally symmetric, leading to contradiction arguments when assuming non-Riemannian structures.

Experimental results

Research questions

• RQ1 Under what conditions does the almost complex structure $ J $ on $ T^*M_0 $ become integrable, leading to a Kähler structure?
• RQ2 Can $ (T^*M_0, G, J) $ be a Kähler manifold for Cartan manifolds of negative constant flag curvature?
• RQ3 Does a Kähler structure exist on a tubular neighborhood of the zero section for Cartan manifolds of positive constant flag curvature?
• RQ4 Can $ (T^*M_0, G, J) $ be an Einstein manifold if $ M $ is non-Riemannian?
• RQ5 Can $ (T^*M_0, G, J) $ be a locally symmetric Kähler manifold if $ M $ is non-Riemannian?

Key findings

• For Cartan manifolds of negative constant flag curvature $ c < 0 $, the cotangent bundle $ T^*M_0 $ equipped with the metric $ G $ and almost complex structure $ J $ admits a Kähler structure.
• For Cartan manifolds of positive constant flag curvature $ c > 0 $, a Kähler structure exists on the tubular neighborhood $ T_\beta^*M_0 $ defined by $ 2\tau < \frac{1}{c\beta^2} $.
• The metric $ G $ on $ T^*M_0 $ cannot make $ (T^*M_0, G, J) $ an Einstein manifold unless $ M $ is Riemannian, as shown by contradiction in the Ricci curvature computation.
• Similarly, $ (T^*M_0, G, J) $ cannot be a locally symmetric Kähler manifold unless $ M $ is Riemannian, as the vanishing of the covariant derivative of the curvature tensor forces the Cartan structure to be Riemannian.
• The computation of curvature components using the Levi-Civita connection shows that non-vanishing $ C^{ijk} $ (the Cartan tensor) leads to contradictions in the Einstein and locally symmetric conditions, proving rigidity of the geometry.
• The results imply that non-Riemannian Cartan structures do not support the Kähler-Einstein or locally symmetric Kähler geometry under this metric construction.