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**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 \(X_2\).

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 \(X_2\).

Reviewer: P.Eberlein (Chapel Hill)