Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Almost Isotropic Kaehler Manifolds   Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit
vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp
\rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) =
R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant
$\kappa_p \in \mathbf{R}$ such that $\mathcal{J}_v = \kappa_p \textit{
Id}_{v^\perp}$ for each unit vector $v \in T_pM$. If all points are isotropic,
then $M$ is said to be isotropic; it is a classical result of Schur that
isotropic manifolds of dimension at least 3 have constant sectional curvatures.
  In this paper we consider $\textit{almost isotropic manifolds}$, i.e.
manifolds having the property that for each $p \in M$, there exists a constant
$\kappa_p \in \mathbb{R}$, such that the Jacobi operators $\mathcal{J}_v$
satisfy $\text{rank}(\mathcal{J}_v - \kappa_p \textit{Id}_{v^\perp}) \leq 1$
for each unit vector $v \in T_pM$. Our main theorem classifies the almost
isotropic simply connected K\"ahler manifolds, proving that those of dimension
$d=2n \geq 4$ are either isometric to complex projective space or complex
hyperbolic space or are totally geodesically foliated by leaves isometric to
$\mathbf{C}^{n-1}.$

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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: eess