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A concept of function which measures the distance to a point at infinity. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205501.png" /> be a [[Riemannian manifold|Riemannian manifold]]. The Riemannian metric induces a distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205502.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205503.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205504.png" /> be a ray in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205505.png" />, i.e., a unit-speed [[Geodesic line|geodesic line]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205506.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205507.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205508.png" />. The Busemann function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205509.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055010.png" /> is defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055011.png" /></td> </tr></table>
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Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055012.png" /> is bounded above by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055013.png" /> and is monotone non-decreasing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055014.png" />, the limit always exits. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055015.png" /> is a Lipschitz function with [[Lipschitz constant|Lipschitz constant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055016.png" />. The level surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055017.png" /> of a Busemann function are called horospheres. Busemann functions can also be defined on intrinsic (or length) metric spaces, in the same manner. Actually, H. Busemann [[#References|[a2]]] first introduced them on so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055020.png" />-spaces and used them to state the parallel axiom on straight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055021.png" />-spaces (cf. also [[Closed geodesic|Closed geodesic]]).
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A concept of function which measures the distance to a point at infinity. Let $M$ be a [[Riemannian manifold|Riemannian manifold]]. The Riemannian metric induces a distance function $d$ on $M$. Let $\gamma$ be a ray in $M$, i.e., a unit-speed [[Geodesic line|geodesic line]] $\gamma : [ 0 , \infty ) \rightarrow M$ such that $d ( \gamma ( t ) , \gamma ( 0 ) ) = t$ for all $t \geq 0$. The Busemann function $b _ { \gamma } : M \rightarrow \mathbf R$ with respect to $\gamma$ is defined by
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055022.png" /> has non-negative [[Sectional curvature|sectional curvature]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055023.png" /> is convex, see [[#References|[a3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055024.png" /> has non-negative [[Ricci curvature|Ricci curvature]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055025.png" /> is a [[Subharmonic function|subharmonic function]], see [[#References|[a4]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055026.png" /> is a [[Kähler manifold|Kähler manifold]] with non-negative holomorphic bisectional curvature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055027.png" /> is a [[Plurisubharmonic function|plurisubharmonic function]], see [[#References|[a7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055028.png" /> is a Hadamard manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055029.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055030.png" /> concave function, see [[#References|[a9]]], [[#References|[a2]]], and, moreover, the horospheres are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055031.png" />-hypersurfaces, see [[#References|[a9]]]. On the [[Poincaré model|Poincaré model]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055032.png" /> of the hyperbolic space, the horospheres coincide with the Euclidean spheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055033.png" /> which are tangent to the sphere at infinity. On Hadamard manifolds, it is more customary to call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055034.png" /> the Busemann function instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055035.png" />.
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\begin{equation*} b _ { \gamma } ( x ) = \operatorname { lim } _ { t \rightarrow \infty } ( t - d ( x , \gamma ( t ) ) ) , \quad x \in M. \end{equation*}
  
More recently, M. Gromov [[#References|[a1]]] introduced a generalization of the concept of Busemann function called the horofunction. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055036.png" /> be the set of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055037.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055038.png" /> the quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055039.png" /> modulo the constant functions. Use the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055040.png" /> induced from the [[Uniform convergence|uniform convergence]] on compact sets and its quotient topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055041.png" />. The embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055042.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055043.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055044.png" /> induces an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055045.png" />. The closure of the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055046.png" /> is a compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055047.png" /> (cf. also [[Compactification|Compactification]]). According to [[#References|[a1]]], [[#References|[a8]]], a horofunction is defined to be a class (or an element of a class) in the topological boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055050.png" />. Any Busemann function is a horofunction. For Hadamard manifolds, any horofunction can be represented as some Busemann function, see [[#References|[a1]]]. However, this is not necessarily true for non-Hadamard manifolds. Horofunctions have been defined not only for Riemannian manifolds but also for complete locally compact metric spaces.
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Since $t - d ( x , \gamma ( t ) )$ is bounded above by $d ( x , \gamma ( 0 ) )$ and is monotone non-decreasing in $t$, the limit always exits. It follows that $ b _ { \gamma }$ is a Lipschitz function with [[Lipschitz constant|Lipschitz constant]] $1$. The level surfaces $b _ { \gamma } ^ { - 1 } ( t )$ of a Busemann function are called horospheres. Busemann functions can also be defined on intrinsic (or length) metric spaces, in the same manner. Actually, H. Busemann [[#References|[a2]]] first introduced them on so-called $G$-spaces and used them to state the parallel axiom on straight $G$-spaces (cf. also [[Closed geodesic|Closed geodesic]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055051.png" /> be a complete non-compact Riemannian manifold with non-negative sectional curvature, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055052.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055055.png" /> runs over all rays emanating from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055056.png" />. Then, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055057.png" /> is a convex exhaustion function, see [[#References|[a3]]], that is, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055060.png" /> is compact for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055061.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055062.png" /> plays an important role in the first step of the Cheeger–Gromoll structure theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055063.png" />, see [[#References|[a3]]]. Any Kähler manifold admitting a strictly plurisubharmonic exhaustion function is a [[Stein manifold|Stein manifold]], see [[#References|[a5]]]. This, together with the use of Busemann functions or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055064.png" />, yields various sufficient conditions for a Kähler manifold to be Stein; see, for example, [[#References|[a7]]], [[#References|[a17]]]. Some results for the exhaustion property of Busemann functions are known, see [[#References|[a14]]], [[#References|[a15]]], [[#References|[a12]]], [[#References|[a13]]]. For a generalization of the notion of a horofunction and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055065.png" />, see [[#References|[a18]]]. A general reference for Busemann function and its related topics is [[#References|[a16]]].
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If $M$ has non-negative [[Sectional curvature|sectional curvature]], $ b  _ { \gamma }$ is convex, see [[#References|[a3]]]. If $M$ has non-negative [[Ricci curvature|Ricci curvature]], $ b _ { \gamma }$ is a [[Subharmonic function|subharmonic function]], see [[#References|[a4]]]. If $M$ is a [[Kähler manifold|Kähler manifold]] with non-negative holomorphic bisectional curvature, $ b  _ { \gamma }$ is a [[Plurisubharmonic function|plurisubharmonic function]], see [[#References|[a7]]]. If $M$ is a Hadamard manifold, $ b  _ { \gamma }$ is a $C ^ { 2 }$ concave function, see [[#References|[a9]]], [[#References|[a2]]], and, moreover, the horospheres are $C ^ { 2 }$-hypersurfaces, see [[#References|[a9]]]. On the [[Poincaré model|Poincaré model]] $H ^ { 2 }$ of the hyperbolic space, the horospheres coincide with the Euclidean spheres in $H ^ { 2 }$ which are tangent to the sphere at infinity. On Hadamard manifolds, it is more customary to call $- b _ { \gamma }$ the Busemann function instead of $ b  _ { \gamma }$.
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More recently, M. Gromov [[#References|[a1]]] introduced a generalization of the concept of Busemann function called the horofunction. Let $C ( N )$ be the set of continuous functions on $M$ and let $C_{ * } ( M )$ the quotient space of $C ( N )$ modulo the constant functions. Use the topology on $C ( N )$ induced from the [[Uniform convergence|uniform convergence]] on compact sets and its quotient topology on $C_{ * } ( M )$. The embedding of $M$ into $C ( N )$ defined by $M \ni x \mapsto d ( x ,\, . ) \in C ( M )$ induces an embedding $\iota : M \rightarrow C_{*} ( M )$. The closure of the image $\iota ( M )$ is a compactification of $M$ (cf. also [[Compactification|Compactification]]). According to [[#References|[a1]]], [[#References|[a8]]], a horofunction is defined to be a class (or an element of a class) in the topological boundary $\partial \iota ( M )$ of $\iota ( M )$ in $C_{ * } ( M )$. Any Busemann function is a horofunction. For Hadamard manifolds, any horofunction can be represented as some Busemann function, see [[#References|[a1]]]. However, this is not necessarily true for non-Hadamard manifolds. Horofunctions have been defined not only for Riemannian manifolds but also for complete locally compact metric spaces.
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Let $M$ be a complete non-compact Riemannian manifold with non-negative sectional curvature, and for $p \in M$, let $b _ { p } ( x ) = \operatorname { sup } _ { \gamma } b _ { \gamma } ( x )$, $x \in M$, where $\gamma$ runs over all rays emanating from $p$. Then, $b _ { p }$ is a convex exhaustion function, see [[#References|[a3]]], that is, a function $f$ on $M$ such that $f ^ { - 1 } ( ( - \infty , t ] )$ is compact for any $t \in f ( M )$. The function $b _ { p }$ plays an important role in the first step of the Cheeger–Gromoll structure theory for $M$, see [[#References|[a3]]]. Any Kähler manifold admitting a strictly plurisubharmonic exhaustion function is a [[Stein manifold|Stein manifold]], see [[#References|[a5]]]. This, together with the use of Busemann functions or $b _ { p }$, yields various sufficient conditions for a Kähler manifold to be Stein; see, for example, [[#References|[a7]]], [[#References|[a17]]]. Some results for the exhaustion property of Busemann functions are known, see [[#References|[a14]]], [[#References|[a15]]], [[#References|[a12]]], [[#References|[a13]]]. For a generalization of the notion of a horofunction and of $b _ { p }$, see [[#References|[a18]]]. A general reference for Busemann function and its related topics is [[#References|[a16]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Ballmann,  M. Gromov,  V. Schroeder,  "Manifolds of nonpositive curvature" , ''Progr. Math.'' , '''61''' , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Busemann,  "The geometry of geodesics" , Acad. Press  (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Cheeger,  D. Gromoll,  "On the structure of complete manifolds of nonnegative curvature"  ''Ann. of Math. (2)'' , '''96'''  (1972)  pp. 413–443</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Cheeger,  D. Gromoll,  "The splitting theorem for manifolds of nonnegative Ricci curvature"  ''J. Diff. Geom.'' , '''6'''  (1971/72)  pp. 119–128</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Docquier,  H. Grauert,  "Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten"  ''Math. Ann.'' , '''140'''  (1960)  pp. 94–123</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Eberlein,  B. O'Neill,  "Visibility manifolds"  ''Pacific J. Math.'' , '''46'''  (1973)  pp. 45–109</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.E. Greene,  H. Wu,  "On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs"  ''Abh. Math. Sem. Univ. Hamburg'' , '''47'''  (1978)  pp. 171–185  (Special issue dedicated to the seventieth birthday of Erich Käler)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Gromov,  "Structures métriques pour les variétés riemanniennes" , ''Textes Mathématiques [Mathematical Texts]'' , '''1''' , CEDIC  (1981)  (Edited by J. Lafontaine and P. Pansu)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E. Heintze,  H.-C. Im Hof,  "Geometry of horospheres"  ''J. Diff. Geom.'' , '''12''' :  4  (1977)  pp. 481–491 (1978)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  N. Innami,  "On the terminal points of co-rays and rays"  ''Arch. Math. (Basel)'' , '''45''' :  5  (1985)  pp. 468–470</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  N. Innami,  "Differentiability of Busemann functions and total excess"  ''Math. Z.'' , '''180''' :  2  (1982)  pp. 235–247</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A. Kasue,  "A compactification of a manifold with asymptotically nonnegative curvature"  ''Ann. Sci. Ecole Norm. Sup. 4'' , '''21''' :  4  (1988)  pp. 593–622</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  Z. Shen,  "On complete manifolds of nonnegative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055066.png" />th-Ricci curvature"  ''Trans. Amer. Math. Soc.'' , '''338''' :  1  (1993)  pp. 289–310</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  K. Shiohama,  "Busemann functions and total curvature"  ''Invent. Math.'' , '''53''' :  3  (1979)  pp. 281–297</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  K. Shiohama,  "The role of total curvature on complete noncompact Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055067.png" />-manifolds"  ''Illinois J. Math.'' , '''28''' :  4  (1984)  pp. 597–620</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  K. Shiohama,  "Topology of complete noncompact manifolds" , ''Geometry of Geodesics and Related Topics (Tokyo, 1982)'' , ''Adv. Stud. Pure Math.'' , '''3''' , North-Holland  (1984)  pp. 423–450</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  Y.T. Siu,  S.T. Yau,  "Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay"  ''Ann. of Math. (2)'' , '''105''' :  2  (1977)  pp. 225–264</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  H. Wu,  "An elementary method in the study of nonnegative curvature"  ''Acta Math.'' , '''142''' :  1–2  (1979)  pp. 57–78</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  W. Ballmann,  M. Gromov,  V. Schroeder,  "Manifolds of nonpositive curvature" , ''Progr. Math.'' , '''61''' , Birkhäuser  (1985)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H. Busemann,  "The geometry of geodesics" , Acad. Press  (1955)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Cheeger,  D. Gromoll,  "On the structure of complete manifolds of nonnegative curvature"  ''Ann. of Math. (2)'' , '''96'''  (1972)  pp. 413–443</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Cheeger,  D. Gromoll,  "The splitting theorem for manifolds of nonnegative Ricci curvature"  ''J. Diff. Geom.'' , '''6'''  (1971/72)  pp. 119–128</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  F. Docquier,  H. Grauert,  "Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten"  ''Math. Ann.'' , '''140'''  (1960)  pp. 94–123</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  P. Eberlein,  B. O'Neill,  "Visibility manifolds"  ''Pacific J. Math.'' , '''46'''  (1973)  pp. 45–109</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  R.E. Greene,  H. Wu,  "On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs"  ''Abh. Math. Sem. Univ. Hamburg'' , '''47'''  (1978)  pp. 171–185  (Special issue dedicated to the seventieth birthday of Erich Käler)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Gromov,  "Structures métriques pour les variétés riemanniennes" , ''Textes Mathématiques [Mathematical Texts]'' , '''1''' , CEDIC  (1981)  (Edited by J. Lafontaine and P. Pansu)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  E. Heintze,  H.-C. Im Hof,  "Geometry of horospheres"  ''J. Diff. Geom.'' , '''12''' :  4  (1977)  pp. 481–491 (1978)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  N. Innami,  "On the terminal points of co-rays and rays"  ''Arch. Math. (Basel)'' , '''45''' :  5  (1985)  pp. 468–470</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  N. Innami,  "Differentiability of Busemann functions and total excess"  ''Math. Z.'' , '''180''' :  2  (1982)  pp. 235–247</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  A. Kasue,  "A compactification of a manifold with asymptotically nonnegative curvature"  ''Ann. Sci. Ecole Norm. Sup. 4'' , '''21''' :  4  (1988)  pp. 593–622</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  Z. Shen,  "On complete manifolds of nonnegative $k$th-Ricci curvature"  ''Trans. Amer. Math. Soc.'' , '''338''' :  1  (1993)  pp. 289–310</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  K. Shiohama,  "Busemann functions and total curvature"  ''Invent. Math.'' , '''53''' :  3  (1979)  pp. 281–297</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  K. Shiohama,  "The role of total curvature on complete noncompact Riemannian $2$-manifolds"  ''Illinois J. Math.'' , '''28''' :  4  (1984)  pp. 597–620</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  K. Shiohama,  "Topology of complete noncompact manifolds" , ''Geometry of Geodesics and Related Topics (Tokyo, 1982)'' , ''Adv. Stud. Pure Math.'' , '''3''' , North-Holland  (1984)  pp. 423–450</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  Y.T. Siu,  S.T. Yau,  "Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay"  ''Ann. of Math. (2)'' , '''105''' :  2  (1977)  pp. 225–264</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  H. Wu,  "An elementary method in the study of nonnegative curvature"  ''Acta Math.'' , '''142''' :  1–2  (1979)  pp. 57–78</td></tr></table>

Latest revision as of 17:02, 1 July 2020

A concept of function which measures the distance to a point at infinity. Let $M$ be a Riemannian manifold. The Riemannian metric induces a distance function $d$ on $M$. Let $\gamma$ be a ray in $M$, i.e., a unit-speed geodesic line $\gamma : [ 0 , \infty ) \rightarrow M$ such that $d ( \gamma ( t ) , \gamma ( 0 ) ) = t$ for all $t \geq 0$. The Busemann function $b _ { \gamma } : M \rightarrow \mathbf R$ with respect to $\gamma$ is defined by

\begin{equation*} b _ { \gamma } ( x ) = \operatorname { lim } _ { t \rightarrow \infty } ( t - d ( x , \gamma ( t ) ) ) , \quad x \in M. \end{equation*}

Since $t - d ( x , \gamma ( t ) )$ is bounded above by $d ( x , \gamma ( 0 ) )$ and is monotone non-decreasing in $t$, the limit always exits. It follows that $ b _ { \gamma }$ is a Lipschitz function with Lipschitz constant $1$. The level surfaces $b _ { \gamma } ^ { - 1 } ( t )$ of a Busemann function are called horospheres. Busemann functions can also be defined on intrinsic (or length) metric spaces, in the same manner. Actually, H. Busemann [a2] first introduced them on so-called $G$-spaces and used them to state the parallel axiom on straight $G$-spaces (cf. also Closed geodesic).

If $M$ has non-negative sectional curvature, $ b _ { \gamma }$ is convex, see [a3]. If $M$ has non-negative Ricci curvature, $ b _ { \gamma }$ is a subharmonic function, see [a4]. If $M$ is a Kähler manifold with non-negative holomorphic bisectional curvature, $ b _ { \gamma }$ is a plurisubharmonic function, see [a7]. If $M$ is a Hadamard manifold, $ b _ { \gamma }$ is a $C ^ { 2 }$ concave function, see [a9], [a2], and, moreover, the horospheres are $C ^ { 2 }$-hypersurfaces, see [a9]. On the Poincaré model $H ^ { 2 }$ of the hyperbolic space, the horospheres coincide with the Euclidean spheres in $H ^ { 2 }$ which are tangent to the sphere at infinity. On Hadamard manifolds, it is more customary to call $- b _ { \gamma }$ the Busemann function instead of $ b _ { \gamma }$.

More recently, M. Gromov [a1] introduced a generalization of the concept of Busemann function called the horofunction. Let $C ( N )$ be the set of continuous functions on $M$ and let $C_{ * } ( M )$ the quotient space of $C ( N )$ modulo the constant functions. Use the topology on $C ( N )$ induced from the uniform convergence on compact sets and its quotient topology on $C_{ * } ( M )$. The embedding of $M$ into $C ( N )$ defined by $M \ni x \mapsto d ( x ,\, . ) \in C ( M )$ induces an embedding $\iota : M \rightarrow C_{*} ( M )$. The closure of the image $\iota ( M )$ is a compactification of $M$ (cf. also Compactification). According to [a1], [a8], a horofunction is defined to be a class (or an element of a class) in the topological boundary $\partial \iota ( M )$ of $\iota ( M )$ in $C_{ * } ( M )$. Any Busemann function is a horofunction. For Hadamard manifolds, any horofunction can be represented as some Busemann function, see [a1]. However, this is not necessarily true for non-Hadamard manifolds. Horofunctions have been defined not only for Riemannian manifolds but also for complete locally compact metric spaces.

Let $M$ be a complete non-compact Riemannian manifold with non-negative sectional curvature, and for $p \in M$, let $b _ { p } ( x ) = \operatorname { sup } _ { \gamma } b _ { \gamma } ( x )$, $x \in M$, where $\gamma$ runs over all rays emanating from $p$. Then, $b _ { p }$ is a convex exhaustion function, see [a3], that is, a function $f$ on $M$ such that $f ^ { - 1 } ( ( - \infty , t ] )$ is compact for any $t \in f ( M )$. The function $b _ { p }$ plays an important role in the first step of the Cheeger–Gromoll structure theory for $M$, see [a3]. Any Kähler manifold admitting a strictly plurisubharmonic exhaustion function is a Stein manifold, see [a5]. This, together with the use of Busemann functions or $b _ { p }$, yields various sufficient conditions for a Kähler manifold to be Stein; see, for example, [a7], [a17]. Some results for the exhaustion property of Busemann functions are known, see [a14], [a15], [a12], [a13]. For a generalization of the notion of a horofunction and of $b _ { p }$, see [a18]. A general reference for Busemann function and its related topics is [a16].

References

[a1] W. Ballmann, M. Gromov, V. Schroeder, "Manifolds of nonpositive curvature" , Progr. Math. , 61 , Birkhäuser (1985)
[a2] H. Busemann, "The geometry of geodesics" , Acad. Press (1955)
[a3] J. Cheeger, D. Gromoll, "On the structure of complete manifolds of nonnegative curvature" Ann. of Math. (2) , 96 (1972) pp. 413–443
[a4] J. Cheeger, D. Gromoll, "The splitting theorem for manifolds of nonnegative Ricci curvature" J. Diff. Geom. , 6 (1971/72) pp. 119–128
[a5] F. Docquier, H. Grauert, "Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten" Math. Ann. , 140 (1960) pp. 94–123
[a6] P. Eberlein, B. O'Neill, "Visibility manifolds" Pacific J. Math. , 46 (1973) pp. 45–109
[a7] R.E. Greene, H. Wu, "On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs" Abh. Math. Sem. Univ. Hamburg , 47 (1978) pp. 171–185 (Special issue dedicated to the seventieth birthday of Erich Käler)
[a8] M. Gromov, "Structures métriques pour les variétés riemanniennes" , Textes Mathématiques [Mathematical Texts] , 1 , CEDIC (1981) (Edited by J. Lafontaine and P. Pansu)
[a9] E. Heintze, H.-C. Im Hof, "Geometry of horospheres" J. Diff. Geom. , 12 : 4 (1977) pp. 481–491 (1978)
[a10] N. Innami, "On the terminal points of co-rays and rays" Arch. Math. (Basel) , 45 : 5 (1985) pp. 468–470
[a11] N. Innami, "Differentiability of Busemann functions and total excess" Math. Z. , 180 : 2 (1982) pp. 235–247
[a12] A. Kasue, "A compactification of a manifold with asymptotically nonnegative curvature" Ann. Sci. Ecole Norm. Sup. 4 , 21 : 4 (1988) pp. 593–622
[a13] Z. Shen, "On complete manifolds of nonnegative $k$th-Ricci curvature" Trans. Amer. Math. Soc. , 338 : 1 (1993) pp. 289–310
[a14] K. Shiohama, "Busemann functions and total curvature" Invent. Math. , 53 : 3 (1979) pp. 281–297
[a15] K. Shiohama, "The role of total curvature on complete noncompact Riemannian $2$-manifolds" Illinois J. Math. , 28 : 4 (1984) pp. 597–620
[a16] K. Shiohama, "Topology of complete noncompact manifolds" , Geometry of Geodesics and Related Topics (Tokyo, 1982) , Adv. Stud. Pure Math. , 3 , North-Holland (1984) pp. 423–450
[a17] Y.T. Siu, S.T. Yau, "Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay" Ann. of Math. (2) , 105 : 2 (1977) pp. 225–264
[a18] H. Wu, "An elementary method in the study of nonnegative curvature" Acta Math. , 142 : 1–2 (1979) pp. 57–78
How to Cite This Entry:
Busemann function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Busemann_function&oldid=50439
This article was adapted from an original article by T. Shioya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article