## Tits geometry associated with 4-dimensional closed real-analytic manifolds of nonpositive curvature.(English)Zbl 0920.53021

Let $$X$$ denote a complete, simply connected Riemannian manifold of nonpositive sectional curvature, and let $$X(\infty)$$ denote the boundary sphere consisting of equivalence classes of asymptotic geodesics of $$X$$. The Tits pseudometric Td is defined on $$X$$ as follows. Fix a basepoint $$p$$ in $$X$$, and let points $$x$$ and $$y$$ in $$X(\infty)$$ be given. If $$\gamma_{px}$$ and $$\gamma_{py}$$ denote the unit speed geodesics starting at $$p$$ that belong to $$x$$ and $$y$$, then $$\text{Td}(x,y) =\lim_{t\to\infty} d_S(\gamma_{px}(t), \gamma_{py} (t))/t$$, where $$d_S(\gamma_{px} (t),\gamma_{py}(t))$$ denotes the distance between $$\gamma_{px}(t)$$ and $$\gamma_{py}(t)$$ in the sphere $$S$$ of radius $$t$$ centered at $$p$$. The value of $$\text{Td}(x,y)$$ does not depend on the basepoint $$p$$, and in general it may equal $$+\infty$$. For example, the Tits distance between any two distinct points of $$X(\infty)$$ is $$+\infty$$ if the sectional curvature is bounded above by a negative constant. The pseudometric Td is always a finite metric if $$X$$ is a symmetric space of rank at least two. Define an equivalence relation on $$X( \infty)$$ in which two points $$x$$ and $$y$$ are equivalent if $$\text{Td}(x,y)$$ is finite. An equivalence class of points in $$X (\infty)$$ will be called a component of $$(X(\infty),\text{Td})$$. Any two points in a component of $$(X(\infty),\text{Td})$$ can be joined by a Tits geodesic, which is unique if $$\text{Td}(x,y)<\pi$$. The collection of components of $$(X (\infty), \text{Td})$$ that contain more than one point is denoted $$\partial_TX$$.
In this article, the authors consider the case that $$X$$ is 4-dimensional and real analytic, and they investigate the components of $$(X(\infty), \text{Td})$$, which they classify into standard and nonstandard types. $$Y$$ is a called a higher rank submanifold of $$X$$ if $$Y$$ is a totally geodesic submanifold that is isometric to a 2- or 3-dimensional Euclidean space (a 2- or 3-flat) or a Riemannian product $$Q\times \mathbb{R}$$, where $$Q$$ is a 2-dimensional totally geodesic submanifold that is not flat. The Tits pseudometric Td is finite on $$Y (\infty)$$ and the component of $$(X(\infty),\text{Td})$$ that contains $$Y(\infty)$$ is called a standard component. The main result of this article is Theorem 1. Let $$X$$ be the universal Riemannian cover of a compact, real analytic of nonpositive sectional curvature. Then:
(1) Each point in a nonstandard component determines an equivalence class of sequences of maximal higher rank submanifolds, where two sequences are equivalent if they agree beyond some finite point.
(2) Every nonstandard component of $$\partial_TX$$ is an interval of length $$\pi-\delta$$, where $$\delta$$ depends only on $$X$$.
(3) The Tits boundary $$\partial_TX$$ contains nonstandard components if and only if $$X$$ contains distinct 3-dimensional higher rank submanifolds $$W_1$$ and $$W_2$$ whose intersection is nonempty.
The authors also obtain some results about quasi-flats, whose definition we omit. Theorem 2. Let $$X$$ be as in Theorem 1 and assume furthermore that $$X$$ contains no 3-flat. Then every $$(L, C)$$-quasi 2-flat in $$X$$ lies at a finite distance from a conical quasi 2-flat and hence at a finite distance from a finite number $$N=N(L)$$ of 2-flats in $$X$$. If $$L$$ is sufficiently close to 1, then $$N=1$$. Corollary. Let $$f:X_1\to X_2$$ be a quasi-isometry, where $$X_1$$ and $$X_2$$ satisfy the hypothesis of Theorem 2. Then the image under $$f$$ of a 2-flat in $$X_1$$ lies at a finite distance from a 2-flat in $$X_2$$.

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 57M05 Fundamental group, presentations, free differential calculus
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