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despite its nearly a hundred - year - old age , einstein gravity continues to be a source of inspirations and surprises . besides the great success in describing physics in solar system , einstein gravity also predicts existence of various black holes and even extended objects like black strings @xcite and black rings @xcite in higher dimensions . not all aspects of these nontrivial solutions have been fully understood . even in the pure vacuum sector , einstein gravity has been shown to possess an unexpected richness in addition to the well known maximally symmetric vacua ( ds , ads , minkowski etc ) , einstein gravity also admits inhomogeneous vacua such as the anisotropic accelerating vacua @xcite and the massless topological black hole vacua @xcite and so on . in this article , we shall present a novel class of einstein vacua which possess accelerating horizons of nontrivial topology . concretely , the vacua we shall be discussing are accelerating vacua with horizons being conformally distorted sphere bundles over @xmath1 . after euclideanization , the whole vacuum metric becomes that of a conformally distorted sphere bundle over @xmath1 , which corresponds to a smooth , compact and inhomogeneous riemannian manifold . historically , the first known smooth , compact and inhomogeneous riemannian manifold of constant scalar curvature was the gravitational instanton discovered by don page @xcite . page s gravitational instanton describes a nontrivial metric of @xmath4 bundle over @xmath4 . such metrics were later generalized to higher dimensions @xcite , giving rise to constant curvature metrics of nontrivial @xmath5 bundle over @xmath4 . in differential geometry , sphere bundles over @xmath1 are more complicated than sphere bundles over @xmath4 , because the latter are simply connected and are known to be einstein manifolds , while the former are non - simply connected and are not einstein manifolds . the present work shows that , although @xmath2 bundles over @xmath1 are in general not einstein , there exists an einstein metric in the conformal class of such manifolds . we begin our study by presenting a novel einstein vacuum solution in five dimensions ( 5d ) . we start from 5d not because 5d is of any particular importance for the construction , but because we began studying this subject with the aim of finding black rings with cosmological constant . though we have not yet fulfilled our aim , the result presented here indeed bears some resemblance to black ring solutions , with the exception that instead of black ring horizon of topology @xmath6 , we now have a cosmological horizon of the same topology . the metric of the novel vacuum solution is given as follows , @xmath7 , \label{5dm}\end{aligned}\ ] ] where @xmath8 this metric is a generalized version of a 4-dimensional metric found in @xcite . it is straightforward to show that this metric is an exact solution to the vacuum einstein equation with a cosmological constant given by @xmath9 for @xmath10 and @xmath11 , this is an inhomogeneous de sitter spacetime with no singularities . for @xmath12 , the metric is still free of essential singularities , but the cosmological constant becomes negative , and there exist apparent singularities in the conformal factor which indicates non - compactness of the spacetime in this case . in the rest of this article , we assume @xmath11 , i.e. @xmath13 .    the singularity free nature of the metric is best manifested by the calculation of curvature invariants . for instances , we have @xmath14    physically , the metric describes an accelerating vacuum of einstein gravity with the accelerating horizon taking a nontrivial topology . the proper acceleration @xmath15 for the static observers in the spacetime has the norm @xmath16 this quantity has the finite value @xmath17 at @xmath18 and blows up to infinity at @xmath19 . so the @xmath20 hyper surface represents an accelerating horizon . notice that the coordinate @xmath21 is not the radial variable in polar coordinate system , @xmath18 corresponds not to the spacial origin but to a circle of radius @xmath22 . thus , unlike the usual de sitter spacetime , the acceleration horizon is a topologically nontrivial manifold . to understand the nontrivial topology of the acceleration horizon , we now look at the metric on the horizon surface . we have @xmath23 let us temporarily put the conformal factor aside . the 3d hyper surface with the metric @xmath24 has a very nice geometric interpretation . let us follow the treatment of @xcite of this geometry . consider a global embedding of a 3d hypersurface @xmath25 into 4d euclidean space @xmath26 . after introducing the toroidal coordinates @xcite @xmath27 on @xmath26 , where @xmath28 the 4d euclidean line element @xmath29 becomes @xmath30 . \end{aligned}\ ] ] the 3d hyper surface ( [ hs ] ) corresponds to constant @xmath31 , with @xmath32 the line element on this 3-surface is @xmath33.\end{aligned}\ ] ] after taking the coordinate transformation @xmath34 the above line element becomes ( [ dst ] ) . we can make the correspondence of the line element ( [ dst ] ) with the 3d hyper surface ( [ hs ] ) even more direct . to do this , we simply parametrize the 3-surface as @xmath35 in fact , the hypersurface equation ( [ hs ] ) describes an @xmath36 fibration over @xmath3 , with the circle @xmath3 parametrized by the angle @xmath37 . therefore , the horizon surface is nothing but a conformally distorted @xmath36 bundle over @xmath3 . note that the hypersurface ( [ hs ] ) is not of constant scalar curvature .      to understand how seriously the conformal factor distorts the geometry of @xmath36 fibration over @xmath3 , we need to study the behavior of the conformal factor . for any fixed @xmath38 ( which is the case when we study the horizon geometry ) , the square root of the conformal factor , @xmath39 , sweeps an ellipse if @xmath40 , a parabola if @xmath41 or a pair of hyperbola if @xmath42 . the value of @xmath43 corresponds to the eccentricity of these conics . since we shall be mainly interested in compact surfaces , we assume @xmath40 . under this condition , if the part of the horizon surface besides the conformal factor were a round sphere , then the effect of the conformal factor will simply be squashing the sphere into an ellipsoid @xcite . in the present case , however , the part of the horizon surface besides the conformal factor is an @xmath36 bundle over @xmath3 , so the full horizon surface is a conformally squashed @xmath36 bundle over @xmath3 .      to make thorough understanding of the structure of the metric ( [ 5 dm ] ) , we now study its causal structure . first , we introduce the eddington - finkelstein coordinates , @xmath44 where the tortoise coordinate @xmath45 is defined as @xmath46 and both @xmath47 and @xmath48 belong to the range @xmath49 . in this coordinate system , the metric ( [ 5 dm ] ) becomes @xmath50 , \label{metric3}\end{aligned}\ ] ] where @xmath51 and @xmath52 is the line element on the angular surface , whose geometry is very similar to that of the line element ( [ dst ] ) which represents an @xmath36 bundle over @xmath3 as mentioned in subsection [ hg ] . the kruskal coordinates are introduced as @xmath53 where the signs of @xmath54 and @xmath55 coincide if @xmath56 , and are opposite to each other if @xmath57 . so there are totally 4 different combinations , each of which corresponds to a causal patch in the conformal diagrams to be drawn below . in each cases , one finds that @xmath58 and eq.([metric3 ] ) becomes @xmath59,\end{aligned}\ ] ] where @xmath21 and @xmath60 are to be regarded as functions of @xmath54 and @xmath55 , @xmath61 finally , the carter - penrose coordinates can be introduced by the usual arctangent mappings of @xmath54 and @xmath55 @xmath62 in terms of which the metric becomes @xmath63.\end{aligned}\ ] ]    the values of the product @xmath64 at @xmath18 ( @xmath65 ) , @xmath66 , @xmath67 ( @xmath68 ) and @xmath69 are respectively @xmath70 the curves corresponding to @xmath18 , @xmath66 and @xmath67 can be easily depicted on the @xmath71 plane and they form various boundaries in the carter - penrose diagrams . however , from ( [ rho ] ) , it follows that @xmath69 will never be reachable if @xmath72 . so , only for @xmath73 , the @xmath69 curves on the @xmath71 plane can possibly play as boundaries in the carter - penrose diagram . when this the case , these curves will separate our spacetime into two patches : the ( + ) patch with @xmath74 and the @xmath75 patch with @xmath76 . below we shall consider 3 different sub cases in detail . patch , while the unshaded areas correspond to the ( + ) patch.,scaledwidth=100.0% ]    * @xmath77 ( @xmath78 ) . in this case , @xmath79 and there is no conformal infinity in the causal diagram . the carter - penrose diagram is consisted purely of the first 3 kinds of boundaries and is depicted in fig.[fig1 ] ( a ) . * @xmath80 ( @xmath81 ) . in this case , @xmath60 takes the fixed value @xmath82 , so is also no conformal infinities in the causal diagram . the carter - penrose diagram is depicted in fig.[fig1 ] ( b ) . * @xmath73 ( @xmath83 $ ] ) . in this case , @xmath84\cup(1 , \infty)$ ] . moreover , the @xmath85 and the @xmath86 curves coincide on the @xmath71 plane , which separate the spacetime into ( + ) and @xmath75 patches . it can be easily inferred from above that @xmath87 and its value is varying with @xmath88 . the corresponding carter - penrose diagram is depicted in fig.[fig1 ] ( c ) , in which the shaded areas correspond to the @xmath75 patch , while the unshaded area corresponds to the ( + ) patch . the variable @xmath60 is discontinuous in this case : it takes values in @xmath89 in the ( + ) patch and in @xmath90 $ ] in the @xmath75 patch . the metric depends on two parameters @xmath91 and @xmath22 , which can be chosen to take various limits . the first limit we can consider is taking @xmath92 while keeping @xmath22 finite . in this limit , both the conformal factor and the function @xmath93 goes to unity and the metric becomes @xmath94 which describes a ricci flat spacetime . the spacial slices of the spacetime are nothing but a trivial fibration of @xmath95 over @xmath3 . the corresponding spacetime has vanishing proper accelerations for any static observer and there is no acceleration horizon . if , at this point , we require further @xmath96 , then the line element becomes that of the 5d minkowski spacetime , @xmath97 where @xmath98 is the metric of @xmath99 written in the standard hopf coordinates . it is amazing that the presence of the acceleration parameter produces both the horizon and its nontrivial topology . the next limit we can consider is a double scaling limit @xmath100 whilst @xmath101 is kept finite . in this limit , the spacetime becomes the standard spherically symmetric de sitter spacetime ds@xmath102 with radius @xmath103 . notice that we can not take the limit @xmath96 while keeping @xmath91 finite , because in that case the function @xmath104 becomes undefined . the third limit we can consider can be achieved by an overall rescaling of the metric . let us make a rescaling @xmath105 , @xmath106 , so that the resulting metric becomes @xmath107 where @xmath108 and @xmath109 notice that the cosmological constant can be expressed in terms of @xmath110 and @xmath111 as @xmath112 so we have @xmath113 for de sitter case , @xmath114 for flat case and @xmath115 for ads case . we can drop the overall constant factor @xmath110 in the line element , which just results in a rescaling of the cosmological constant . the resulting metric will then depend only on a single parameter @xmath111 . we can further take the limit @xmath116 , in which case the metric becomes that of the standard 5d de sitter of unit radius . note that this limit can not be achieved if we had not made the rescaling of the metric before . euclideanization of einstein vacua is an important subject of study from both physics and mathematical perspectives . the physics motivation for studying the euclideanized vacua is to study the vacuum transitions in euclidean quantum gravity . the known examples of @xcite and @xcite were found for this purpose . mathematically , euclideanized einstein metrics often gives explicit examples of smooth compact riemannian manifolds , which is an important class of manifold in differential geometry . in the following , we shall see that there exist a riemannian metric of constant scalar curvature in the conformal class of @xmath117 fibration over @xmath3 . the corresponding metric is nothing but the euclideanized version of our einstein vacuum solution . let us start from the line element ( [ lm2 ] ) . making a wick rotation @xmath118 and letting @xmath119 , we get @xmath120    in the absence of the conformal factor , this metric is just an @xmath117 fibration over @xmath3 which is not a constant curvature manifold ) without the conformal factor describes an @xmath117 fibration over @xmath3 can also be understood from an extrinsic geometric point of view , just like the way we understood the horizon geometry in section 2 . we put this extrinsic geometric description in the appendix . ] . the coordinate @xmath121 parametrizes the @xmath122 fiber , which can be regarded as a generalization of the standard hopf coordinates for @xmath99 . the ranges of these coordinates are given in ( [ range ] ) in the appendix . the conformal factor squashes the round @xmath117 fiber and makes the ricci scalar of the full space a constant . in other words , there is a constant scalar curvature manifold in the conformal class of @xmath117 fibration over @xmath3 . the study of conformal class of a given riemannian manifold is an important subject of study in differential geometry , because this subject is intimately related to the analysis of geometric flows . since the metric ( [ euc ] ) represents a conformally distorted @xmath117 fibration over @xmath3 , we shall refer to the corresponding geometry as a `` ring geometry '' , with the ring ( i.e. the @xmath3 factor ) parametrized by the angle @xmath37 . fibers of the ring geometry at a given @xmath37 is a conformal 4-sphere . the overall constant @xmath110 signifies the size of the ring surface , and the parameter @xmath111 represents the relative radius of the @xmath1 with respect to that of the @xmath122 . we have mentioned previously that when @xmath123 , the corresponding geometry is smooth , compact and inhomogeneous , with a positive constant scalar curvature . perhaps it is the first known example of such metrics on @xmath117 fibration over @xmath3 . the case @xmath124 corresponds to a degenerate case , i.e. a conformal @xmath125 . it is interesting to evaluate the volume of this compact space . without loss of generality , we set @xmath126 in the following calculations . we have , by direct calculation , @xmath127 using the coordinate range ( [ range ] ) , it is easy to evaluate the 5-volume of the space . the result is @xmath128 since the space is of constant scalar curvature , the corresponding einstein - hilbert action is proportional to the 5-volume , so its value is also zero . compact einstein metrics of zero volume are not rare . see @xcite for other examples . so far we have been restricting ourselves in five dimensions . as mentioned earlier , 5d is not of any particular importance in the construction . there is a higher dimensional cousin for the metric we studied in the previous sections . the dimensional metric can be written as @xmath129 where @xmath104 is as before , and @xmath130 is the line element of a round @xmath131-sphere . the associated cosmological constant is given by @xmath132 the physical and mathematical properties of this metric is extremely similar to the 5d case . for instance , the metric interpreted as einstein vacuum represents an accelerating vacuum in which the acceleration horizon has the topology of @xmath0 fibration over @xmath3 . the euclideanized version of the metric represents a conformally distorted metric on the full space of an @xmath2 fibration over @xmath3 and this metric is only compact for small value of @xmath111 , etc .    the euclideanized version of the higher dimensional metric is given as follows,@xmath133 this metric describes a conformally squashed @xmath2 fibration over @xmath1 . the cosmological constant can be written in terms of the parameters @xmath110 and @xmath111 as @xmath134    _ : _    the metric ( [ euc2 ] ) makes sense for all dimensions @xmath135 . we have already analyzed in detail the @xmath136 case . for @xmath137 , we just remove the @xmath138 term from the metric and recovers the metric given in @xcite . for @xmath139 , we remove the @xmath140 term altogether , and set @xmath141 everywhere else . we presented a novel class of accelerating einstein vacua with accelerating horizon bearing a nontrivial topology of @xmath0 bundle over @xmath1 . such solutions contain two parameters , one corresponds to the acceleration , the other corresponds to the relative radius of the @xmath1 base with respect to the @xmath0 fiber . there are various limiting cases for these parameters . among these , the zero acceleration limit corresponds to a ricci flat vacuum with no horizon . the double scaling limit gives rise to the standard de sitter vacua . upon euclideanization , the full space becomes a smooth compact inhomogeneous riemannian manifold with a positive constant scalar curvature . such euclidean manifolds can be regarded as analogues of page s gravitational instanton or its generalizations , but the topologies are now conformally distorted @xmath2 bundle over @xmath1 , rather than sphere bundles over @xmath4 .    since for generic values of the parameters , the vacua we obtained possess a positive cosmological constant and a ring - like acceleration horizon , we expect that such vacua should be the starting point to construct black rings with cosmological constant . asymptotically flat black ring solutions in einstein gravity were found over ten years ago , but so far no black ring solutions with cosmological constant were found . one of the major obstacle for constructing such solutions is the un - matching topologies : black ring solutions have horizons with @xmath6 topology , whilst the usual 5d de sitter spacetime has only an accelerating horizon of @xmath99 topology . the new vacua we found in this article do have the matching topology with black rings . to actually construct black rings which asymptote to our vacuum solutions , some complicated mathematical constructions are yet to be carried out . presumably the kerr - schild method which led to the discoveries of rotating black holes in higher dimensions with cosmological constants @xcite is a good starting point . we shall continue our explorations in this direction . consider the following surface @xmath142 in the 6d euclidean space with metric @xmath143 . it is straightforward to parametrize this surface as @xmath144 where the angular coordinates @xmath145 must be chosen such that these parametrization equations cover the surface ( [ s4xs1 ] ) exactly once . the ranges for these angular coordinates are given as follows : @xmath146 , & \theta\in [ 0,\pi],&\cr \chi\in[0,2\pi],&\psi\in[0,2\pi],&\phi\in[0,2\pi ] . \end{matrix }   \label{range}\end{aligned}\ ] ]    inserting the parametrization equations into the 6d euclidean line element we get the desired result @xmath147 which is identical to the line element ( [ euc ] ) without the conformal factor . the ricci scalar associated with the above metric reads @xmath148 which is not a constant . the surface ( [ s4xs1 ] ) clearly describes an @xmath122 fibration over @xmath1 . this work is supported in part by the national natural science foundation of china ( nsfc ) through grant no.10875059 . y.  hashimoto , m.  sakaguchi , and y.  yasui , `` new infinite series of einstein metrics on sphere bundles from ads black holes , '' commun.math.phys . 257 ( 2005 ) 273 - 285 [ arxiv : http://www.arxiv.org / abs / hep - th/0402199 [ hep - th/0402199 ] ] .          g.  w. gibbons , d.  n. page , and c.  n. pope , `` rotating black holes in higher dimensions with a cosmological constant , '' phys.rev.lett.93:171102,2004 [ arxiv : http://www.arxiv.org / abs / hep - th/0409155 [ hep - th/0409155 ] ] .
a novel class of einstein vacua is presented , which possess non - vanishing cosmological constant and accelerating horizon with the topology of @xmath0 fibration over @xmath1 . 
 after euclideanization , the solution describes a conformally distorted @xmath2 fibration over @xmath3 , which is smooth , compact and inhomogeneous , and can be regarded as analogue of don page s gravitational instanton .