Classification of generalized Yamabe solitons in Euclidean spaces
aa r X i v : . [ m a t h . DG ] S e p CLASSIFICATION OF GENERALIZED YAMABESOLITONS IN EUCLIDEAN SPACES
SHUNYA FUJII AND SHUN MAETA
Abstract.
In this paper, we consider generalized Yamabe soli-tons which include many notions, such as Yamabe solitons, almostYamabe solitons, h -almost Yamabe solitons, gradient k -Yamabesolitons and conformal gradient solitons. We completely classifythe generalized Yamabe solitons on hypersurfaces in Euclideanspaces arisen from the position vector field. Introduction An n -dimensional Riemannian manifold ( M, g ) is called a
Yamabesoliton , if there exist a complete vector field v and ρ ∈ R such that(1) ( R − ρ ) g = 12 L v g, where R is the scalar curvature of M and L v g is the Lie derivative of g . If v is the gradient of some smooth function f on M , then ( M, g, f )is called a gradient Yamabe soliton. If f is constant, then ( M, g, f ) iscalled trivial.Yamabe solitons are special solutions of the Yamabe flow introducedby R. Hamilton [17]. In the last decade, Yamabe solitons have devel-oped rapidly. To understand the Yamabe soliton, many generalizationsof it have been introduced:(1) Almost Yamabe solitons [2]:For a complete vector field v and a smooth function ρ on M ,( R − ρ ) g = 12 L v g. (2) Gradient k -Yamabe solitons [6]: Mathematics Subject Classification.
Key words and phrases.
Yamabe solitons; almost Yamabe solitons; k -Yamabesolitons; h -almost Yamabe solitons; Yamabe flow; concurrent vector fields; Hessianmanifolds.The second author is partially supported by the Grant-in-Aid for Young Scien-tists, No.19K14534, Japan Society for the Promotion of Science. For a smooth function f on M and ρ ∈ R ,2( n − σ k − ρ ) g = ∇∇ f, where σ k denotes the σ k -curvature of g , that is, σ k = σ k ( g − A ) = X i < ··· h <
0) on M ,( R − ρ ) g = h L v g. (4) Conformal gradient solitons [6]:For smooth functions f and ϕ on M , ϕg = ∇∇ f. Many examples of these solitons are known. A warped product man-ifold ( R × cosh t S n , dt + cosh tg S n , f = sinh t, ρ = sinh t + n ) is agradient almost Yamabe soliton, where ( S n , g S n ) is the standard sphere(cf. [2]). On the cylinder S n × R , gradient k -Yamabe solitons areconstructed (cf. [3]). For a nonzero real number m and a positiveconstant β , ( R n , g can , f = − m log( | x | + β ) , ρ = m | x | + β ) is a gradi-ent ( − m | x | + β )-almost Yamabe soliton, where ( R n , g can ) is the Euclideanspace (cf. [27]).Even though there are many examples of these solitons, some clas-sifications are known (cf. [2], [3], [5], [6], [16], [18], [19], [20], [21], [23]and [27]). In particular, there are many results for Yamabe solitons.Any compact Yamabe soliton has constant scalar curvature (cf. [7],[12] and [18]). P. Daskalopoulos and N. Sesum [16] showed that any lo-cally conformally flat complete gradient Yamabe soliton with positivesectional curvature has to be rotationally symmetric. G. Catino, C.Mantegazza and L. Mazzieri [6] classified nontrivial complete gradientYamabe solitons with nonnegative Ricci tensor. H.-D. Cao, X. Sun andY. Zhang [5] gave a useful classification. Recently, the second author[21] classified 3-dimensional complete gradient Yamabe solitons withdivergence-free Cotton tensor. ENERALIZED YAMABE SOLITONS 3
To consider all the generalized Yamabe solitons mentioned above, weconsider the following.
Definition 1.1.
A Riemannian manifold (
M, g ) is called a conformalsoliton if there exists a complete vector field v such that(2) ϕg = 12 L v g, for a smooth function ϕ : M → R . We denote the conformal soliton by(
M, g, v, ϕ ). If v ≡
0, then M is called trivial. Remark 1.2.
Conformal solitons include Yamabe solitons, almost Yam-abe solitons, gradient k -Yamabe solitons, h -almost Yamabe solitons andconformal gradient solitons. Therefore, all the results in this papercan be applied to all these solitons. v is called a conformal vector field. We can construct many examples of conformal solitons. For example,for a smooth function f = log( e x + e y +1) on R with coordinate system { x, y } , g = ∇∇ f is a Riemannian metric on R . Thus ( R , g, f, ϕ = 1)is a conformal (gradient) soliton.To understand conformal solitons, we consider them as hypersurfacesof some Riemannian manifold. As the metric of M is induced from theambient space, it seems natural to take a soliton vector field v fromthe ambient space. Let V be some vector field of the ambient space.Then we can decompose V as the tangential component V T and thenormal component V ⊥ . Therefore, if the ambient space has a (natural)vector field V , it is natural to take v = V T . The Euclidean space E n +1 is the most basic and natural one, because it has the position vectorfield V . Some solitons on hypersurfaces in Euclidean spaces or moregeneral manifolds have been studied (cf. [9], [10], [13], [14], [15] and[23]).In this paper, we completely classify conformal solitons on a hyper-surface in the Euclidean space E n +1 arisen from the position vectorfield. Theorem 1.3.
Any conformal soliton ( M, g, V T , ϕ ) on a hypersurfacein the Euclidean space E n +1 is contained in a hyperplane, a conic hy-persurface or a hypersphere. A submanifold M n in the Euclidean space E m is called a conic sub-manifold if it is an open portion of a cone with vertex at the origin(cf. [8]). Here we remark that by the definition, n -dimensional planesthrough the origin are included in n -dimensional conic submanifolds.The following fact is used later. SHUNYA FUJII AND SHUN MAETA
Proposition 1.4 ([8]) . Let f : M → E m be an isometric immersionof an n -dimensional Riemannian manifold into the m -dimensional Eu-clidean space E m . Then V = V T holds identically if and only if M isa conic submanifold. Preliminaries
Let ( N, ˜ g ) be an m -dimensional Riemannian manifold and ( M, g ) bean n -dimensional submanifold in ( N, ˜ g ). All manifolds in this paperare assumed to be smooth, orientable and connected. We denote Levi-Civita connections on ( M, g ) and ( N, ˜ g ) by ∇ and ˜ ∇ , respectively. TheLie derivative of g is defined by L X g ( Y, Z ) = X ( g ( Y, Z )) − g ([ X, Y ] , Z ) − g ( Y, [ X, Z ]) , for any vector fields X, Y, Z on M .For any vector fields X, Y tangent to M and η normal to M , theformula of Gauss is given by˜ ∇ X Y = ∇ X Y + h ( X, Y ) , where ∇ X Y and h ( X, Y ) are the tangential and the normal componentsof ˜ ∇ X Y . The formula of Weingarten is given by˜ ∇ X η = − A η ( X ) + D X η, where − A η ( X ) and D X η are the tangential and the normal componentsof ˜ ∇ X η . A η ( X ) and h ( X, Y ) are related by g ( A η ( X ) , Y ) = ˜ g ( h ( X, Y ) , η ) . The mean curvature vector H of M in N is given by H = 1 n trace h. For any vector fields
X, Y, Z, W tangent to M , the equation of Gaussis given by ˜ g ( ˜ Rm ( X, Y ) Z, W ) = g ( Rm ( X, Y ) Z, W )+˜ g ( h ( X, Z ) , h ( Y, W )) − ˜ g ( h ( X, W ) , h ( Y, Z )) , where Rm and ˜ Rm are Riemannian curvature tensors of M and N ,respectively. The equation of Codazzi is given by( ˜ Rm ( X, Y ) Z ) ⊥ = ( ¯ ∇ X h )( Y, Z ) − ( ¯ ∇ Y h )( X, Z ) , ENERALIZED YAMABE SOLITONS 5 where ( ˜ Rm ( X, Y ) Z ) ⊥ is the normal component of ˜ Rm ( X, Y ) Z and¯ ∇ X h is defined by( ¯ ∇ X h )( Y, Z ) = D X h ( Y, Z ) − h ( ∇ X Y, Z ) − h ( Y, ∇ X Z ) . If N is a space of constant curvature, then the equation of Codazzireduces to 0 = ( ¯ ∇ X h )( Y, Z ) − ( ¯ ∇ Y h )( X, Z ) . Conformal solitons with a concurrent vector field
The position vector field V on Euclidean spaces satisfies ∇ X V = X, for any vector field X . We consider one of the generalizations of theposition vector field, namely, a concurrent vector field. Definition 3.1.
A vector field V on M is called a concurrent vectorfield if it satisfies ∇ X V = X, for any vector field X on M .There are several studies of concurrent vector fields (see for example[4], [25] and [26]).In this section, we consider a conformal soliton with a concurrentvector field.Firstly, we show a useful formula for study of conformal solitons. Lemma 3.2.
Let ( M, g, f, ϕ ) be a conformal gradient soliton. Then,we have (3) ( n − ϕ + 12 g ( ∇ R, ∇ f ) + Rϕ = 0 . Proof.
Since ∆ ∇ i f = ∇ i ∆ f + R ij ∇ j f, ∆ ∇ i f = ∇ k ∇ k ∇ i f = ∇ k ( ϕg ki ) = ∇ i ϕ, and ∇ i ∆ f = ∇ i ( nϕ ) = n ∇ i ϕ, we have(4) ( n − ∇ i ϕ + R ij ∇ j f = 0 , where R ij is the Ricci tensor of M . By applying ∇ l to the both sidesof (4), we obtain(5) ( n − ∇ l ∇ i ϕ + ∇ l R ij · ∇ j f + R ij ∇ l ∇ j f = 0 . Taking the trace, we obtain (3). (cid:3)
SHUNYA FUJII AND SHUN MAETA
Proposition 3.3.
Any conformal soliton ( M, g, v, ϕ ) which has a con-current vector field v is a conformal gradient soliton with ϕ = 1 .Proof. Since v is a concurrent vector field, we have(6) g ( v, X ) = g ( v, ∇ X v ) = X ( 12 g ( v, v )) , and(7) L v g ( X, Y ) = vg ( X, Y ) − g ([ v, X ] , Y ) − g ( X, [ v, Y ])= vg ( X, Y ) − vg ( X, Y ) + g ( ∇ X v, Y ) + g ( X, ∇ Y v )= 2 g ( X, Y ) , for any vector fields X , Y on M . By putting f = g ( v, v ) on theequation (6), we obtain v = ∇ f . Substituting (7) into (2), we have ϕ = 1 . (cid:3) From Proposition 3.3, the equation of the conformal soliton with aconcurrent vector field is as follows: g = ∇∇ f. Therefore g should be a Hessian metric. The Hessian metric is animportant notion on Geometry and Physics (cf. [1], [11], [22] and [24]). Example 1.
A Hessian manifold (
M, g ) is a conformal gradient solitonwith ϕ = 1.If M is compact, then there exist no non trivial conformal solitonswith a concurrent vector field. Corollary 3.4.
Any compact conformal soliton ( M, g, v, ϕ ) with a con-current vector field v is trivial.Proof. By Proposition 3 . f = n . By applyingmaximum principle, we get f is constant. (cid:3) Conformal solitons on submanifolds
In this section, we assume that ( N, ˜ g ) is a Riemannian manifoldendowed with a concurrent vector field V and ( M, g ) is a submanifoldin ( N, ˜ g ). V T and V ⊥ denote the tangential and the normal componentsof V , respectively.Firstly, we show the following lemma which will be used later for thepurpose of classification of the conformal solitons. ENERALIZED YAMABE SOLITONS 7
Lemma 4.1.
Any conformal soliton ( M, g, V T , ϕ ) on a submanifold M in N satisfies (8) ( ϕ − g ( X, Y ) = g ( A V ⊥ ( X ) , Y ) , for any vector fields X, Y on M .Proof. From the definition of the Lie derivative, we have(9)( L V T g )( X, Y ) = V T g ( X, Y ) − g ( ∇ V T X − ∇ X V T , Y ) − g ( X, ∇ V T Y − ∇ Y V T )= g ( ∇ X V T , Y ) + g ( X, ∇ Y V T )= ˜ g ( ˜ ∇ X V − ˜ ∇ X V ⊥ , Y ) + ˜ g ( X, ˜ ∇ Y V − ˜ ∇ Y V ⊥ )= 2 g ( X, Y ) + 2 g ( A V ⊥ ( X ) , Y ),for any vector fields X, Y on M . Combining (9) with (2), we obtain(8). (cid:3) Proposition 4.2.
Any conformal soliton ( M, g, V T , ϕ ) on a submani-fold M in N is a conformal gradient soliton.Proof. Set f = 12 ˜ g ( V, V ) . By the proof of Proposition 3 . g ( V, X ) = g ( V T , X ) , for any vector field X on M , we obtain V T = ∇ f. (cid:3) Proposition 4.3.
If a conformal soliton ( M, g, V T , ϕ ) on a submani-fold M in N is minimal, then ϕ = 1 .Proof. Let { e , · · · , e n } be an orthonormal frame on M . By Lemma 4 . ϕ − g ij = g ( A V ⊥ ( e i ) , e j ) = ˜ g ( h ( e i , e j ) , V ⊥ ) . Since M is minimal and taking the trace, we obtain n ( ϕ −
1) = n ˜ g ( H, V ⊥ ) = 0 . Therefore, we conclude that ϕ = 1 . (cid:3) SHUNYA FUJII AND SHUN MAETA
Corollary 4.4.
Any compact conformal soliton ( M, g, V T , ϕ ) on aminimal submanifold in N is trivial.Proof. By Proposition 4 . f = n . By applyingmaximum principle, we get f is constant. (cid:3) Proof of Theorem . V by the position vector field of E n +1 . In thissection, we give the proof of Theorem 1 . Theorem 5.1.
Any conformal soliton ( M, g, V T , ϕ ) on a hypersurfacein the Euclidean space E n +1 is contained in a hyperplane, a conic hy-persurface or a hypersphere.Proof. Let α be a mean curvature and λ be a support function of M ,i.e., H = αN and λ = ˜ g ( N, V ) with a unit normal vector field N . FromLemma 4 .
1, we have( ϕ − g ij = ˜ g ( h ( e i , e j ) , V ⊥ ) = ˜ g ( κ i g ij N, V ) = κ i g ij λ, where A N ( e i ) = κ i e i , ( i = 1 , · · · , n ). Hence we have(10) ϕ − λκ i . Case 1. λ = 0 on M : By taking the summation, we obtain(11) ϕ − λα. Comparing (10) and (11), we have κ i = α. Thus M is totally umbilical with A N ( e i ) = αe i and h satisfies h ( X, Y ) = αg ( X, Y ) N . Since N is a unit normal vector field, we have0 = ˜ ∇ X (˜ g ( N, N )) = 2˜ g ( ˜ ∇ X N, N ) = 2˜ g ( D X N, N ) . Therefore, D X N = 0. Hence we obtain( ¯ ∇ X h )( Y, Z ) = D X h ( Y, Z ) − h ( ∇ X Y, Z ) − h ( Y, ∇ X Z )= X ( α ) g ( Y, Z ) N, for any vector fields X, Y, Z on M . From the equation of Codazzi, wehave X ( α ) Y = Y ( α ) X. Since we can assume that X and Y are linearly independent, we con-clude that α is constant.If α = 0, then by ˜ ∇ X N = 0 , N , restricted to M , is a constant vectorfield in E n +1 and we have˜ ∇ X (˜ g ( V, N )) = ˜ g ( ˜ ∇ X V, N ) + ˜ g ( V, ˜ ∇ X N ) = ˜ g ( X, N ) = 0 . ENERALIZED YAMABE SOLITONS 9
This shows that λ = ˜ g ( V, N ) is constant when V and N are restrictedto M . Therefore, M is contained in a hyperplane normal to N whichdoes not through the origin and ϕ = 1.If α = 0, then we have˜ ∇ X ( V + α − N ) = X + α − ˜ ∇ X N = X + α − ( − A N ( X )) = 0 . This shows that the vector field V + α − N , restricted to M , is a constantone in E n +1 . Therefore, M is contained in a hypersphere.Case 2. λ = 0 on M :We have V = V T . By Proposition 1.4, we obtain that M is containedin a conic hypersurface.Case 3. Others:Set U = { x ∈ M | λ = 0 } . By Proposition 1.4, U is a conic sub-manifold. Take p ∈ M \ U , that is, λ = 0 on some open set Ω ∋ p . Bythe same argument as in Case 1, we have Ω is an open portion of ahyperplane or a hypersphere.We consider the case that Ω is an open portion of a hyperplane.Without loss of generality, we can take Ω which is the maximum con-nected component including p on M \ U . On Ω, λ = ˜ g ( V, N )( = 0) isconstant, say λ Ω . Since M is connected, if Ω is closed, then Ω = M ,which is a contradiction. If Ω is not closed, then we have ∂ Ω = ∅ and ∂ Ω ∩ Ω = ∅ . Take q ∈ ∂ Ω. Since λ is continuous, we have λ ( q ) = λ Ω .Thus we can take open neighborhood U q of q such that λ = 0 on U q .Since Ω is the maximum connected component, we have a contradic-tion. Hence, we have that Ω is an open portion of a hypersphere.Note that M is a smooth and connected hypersurface in E n +1 . Ifdim U = n , since a conic hypersurface is flat and a hypersphere hasa positive constant curvature, M must not be any sum of them. Ifdim U ≤ n −
1, then U must be some set of points which are includedin a hypersphere. The other case cannot happen, because M is asmooth and connected hypersurface.Therefore, M is contained in a hypersphere. (cid:3) References
1. S. Amari and J. Armstrong,
Curvature of Hessian manifolds , Diff. Geom. Appl. (2014), 1-12.2. E. Barbosa and E. Ribeiro, On conformal solutions of the Yamabe flow , Arch.Math. (2013), 79-89.3. L. Bo, P. T. Ho and W. Sheng,
The k -Yamabe solitons and the quotient Yamabesolitons , Nonlinear Anal. (2018), 181-195.4. F. Brickell and K. Yano, Concurrent vector fields and Minkowski structures ,Kodai Math. Sem. rep. (1974), 22-28.
5. H.-D. Cao, X. Sun and Y. Zhang,
On the structure of gradient Yamabe solitons,
Math. Res. Lett. (2012), 767–774.6. G. Catino, C. Mantegazza and L. Mazzieri, On the global structure of conformalgradient solitons with nonnegative Ricci tensor , Comm. Contempt. Math. (2012), 12pp.7. L. F. D. Cerbo and M. M. Disconzi, Yamabe Solitons, Determinant of theLaplacian and the Uniformization Theorem for Riemann Surfaces , Lett. Math.Phys. (2008), 13–18.8. B. Y. Chen Differential geometry of rectifying submanifolds , Int. Electron. J.Geom. (2016), 1-8.9. B. Y. Chen and S. Deshmukh, Ricci solitons and concurrent vector fields,
Balkan J. Geom. Appl. (2015), no. 1, 14-25.10. B. Y. Chen and S. Deshmukh, Yamabe and quasi-Yamabe solitons on Euclideansubmanifolds , Mediterr. J. Math. (2018), 9 pp.11. S. Y. Cheng, S. T. Yau, The real Monge-Amp´ere equation and affine flat struc-tures , in: Proceedings of the 1980 Beijing Symposium on Differential Geometryand Differential Equations, Vol.1-3, Beijing, (1980), 339-370.12. B. Chow, P. Lu and L. Ni,
Hamilton’s Ricci Flow , Graduate Studies in Math-ematics , Amer. Math. Soc., (2006).13. J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complexspace form , Tohˆoku Math. J. (2009), 205-212.14. J. T. Cho and M. Kimura, Ricci solitons of compact real hypersurfaces in K¨ahlermanifolds , Math. Nachr. (2011), 1385-1393.15. J. T. Cho and M. Kimura,
Ricci solitons on locally conformally flat hypersur-faces in space forms , J. Geom. Phys. (2012), 1882-1891.16. P. Daskalopoulos, and N. Sesum, The classification of locally conformally flatYamabe solitons,
Adv. Math. (2013), 346–369.17. R. Hamilton,
Lectures on geometric flows , (1989), unpublished.18. S. Y. Hsu,
A note on compact gradient Yamabe solitons,
J. Math. Anal. Appl. (2) (2012) 725–726.19. L. Ma and V. Miquel,
Remarks on scalar curvature of Yamabe solitons,
Ann.Glob. Anal. Geom. (2012), 195-205 DOI 10.1007/s10455-011-9308-7.20. S. Maeta, Complete Yamabe solitons with finite total scalar curvature , Diff.Geom. Appl. (2019), 75-81.21. S. Maeta, Three-dimensional complete gradient Yamabe solitons withdivergence-free Cotton tensor , Ann. Glob. Anal. Geom. (2020), 227-237.22. M. Mirghafouri and F. Malek, Long-time existence of a geometric flow on closedHessian manifolds , J. Geom. Phys. (2017), 54-65.23. T. Seko and S. Maeta,
Classification of almost Yamabe solitons in Euclideanspaces , J. Geom. Phys. (2019), 97-103.24. H. Shima and K. Yagi,
Geometry of Hessian manifolds , Diff. Geom. Appl. (1997), 277-290.25. K. Yano, Sur le parall ´ e lisme et la concourance dans l’espace de Riemann, Proc.Imp. Acad. Tokyo, (1943), 189-197.26. K. Yano and B. Y. Chen, On the concurrent vector fields of immersed manifolds,
Kodai Math. Sem. Rep. (1971), 343-350.27. F. Zeng, On the h-almost Yamabe soliton
J. Math. Study (to appear).
ENERALIZED YAMABE SOLITONS 11
Department of Mathematics, Shimane University, Nishikawatsu 1060Matsue, 690-8504, Japan.
E-mail address : [email protected] Department of Mathematics, Shimane University, Nishikawatsu 1060Matsue, 690-8504, Japan.
E-mail address : [email protected] oror