Research Article Journal of Volume 14:3, 2020 DOI: 10.37421/glta.2020.14.306 Generalized Lie Theory and Applications ISSN: 1736-4337 Open Access On the Existence of a Riemannian Manifolds at a Given Connection Manelo Anona* and Ratovoarimanana Hasina Department of Mathematics and Computer Science, University of Antananarivo, Antananarivo 101, PB 906, Madagascar Abstract We give necessary and sufficient conditions for a linear connection without torsion to come from a Riemannian metric. The nullity space and the image space of the curvature are involved in the formulation of the results. Mathematics Subject Classification (2010) 53XX, 53C08, 53B05. Keywords: Riemannian manifolds • Linear connection • Spray • Nijenhuis tensor • Nullity space of connection Introduction eigenvalue +1, v the vertical projector corresponding to the eigenvalue-1. The curvature of G is defined by 1 A Riemannian manifold defines a unique linear connection without R= [,] hh (2.2) torsion and conservative. By studying the properties of the space of nullity 2 and images of the curvatures, we reduce the resolution of the inverse The vector 2-form R is also called the Nijenhuis tensor of h. The scalar problem to a system of linear equations with some properties of the 2-form allows to define a metric g on the vertical bundle by curvatures quite simple to verify. g(JX, JY) = Ω (JX, Y) The formalism used is that of [1] and [2] and the formulas are those of [3], the method following [4]. for all X, Y∈ χ () TM , where χ()TM denotes the set of vector fields on TM. Canonical Connection of a Riemannian There exists, [2], one and only one metric lift D of the canonical connection such that: Manifold 1. DJ=0; Let M be a n dimensional differentiable paracompact manifold and of 2. DC=v, (C being the Liouville canonical field on TM); class ∞ , J is the natural tangent structure of the tangent bundle: TM → M. 3. DG=0; The derivation dJ [5] is defined by dJ=[iJ, d]. 4. Dg=0. Definition 1 The D connection is called the Cartan connection. We have We call the Riemannian manifold, [1], the couple (M, E): DJY= [,], JJYX DJY= [,]. hJYX (2.3) • M a differentiable manifold JX hX + ∞ With the linear connection D, we associate a curvature • E a function of =TM −{0} in , with E(0)=0, on , 2 on the null section, and homogeneous of degree two such that ddJE has ℜ=(,)XYZ DDJZhX hY − DDJZ hY hX − D[,]hX hY JZ, a maximum rank for all X,, Y Z∈ χ ( TM ). The relation between the curvatures ℜ and The application E is called energy function, the fundamental scalar R is [2]: 2-form Ω=dd E defines a spray S [6]: J ℜ= − + + (XYZ , ) JZRXY [ , ( , )] [ JZRXY , ( , )] R ([ JZX , ], Y ) RX ( ,[ JZY , ]). iSJ dd E= − dE, (2.1) (2.4) the derivation is being the inner product with respect to S. The vector In particular, 1-form G=[J, S] is called canonical connection. The connection G thus defined is an almost product structure: G2=I, I is the vector 1-form identity. ℜ=−(,)XYS RXY (,). (2.5) By asking In natural local coordinates on an open set U of M, (xi, yi ) ∈ TU, i, j ∈ 11 { 1,...,n}. The energy function is written [1] p.330 h=() I +G and v = () I −G , 22 1 E= g( x1 ,..., xn ) yy ij h is the horizontal projector, projector of the proper subspace at the 2 ij 1 n where gij (x ,…, x ) are symmetric positive functions such that the matrix 1 n i dd E= − dE *Address for Correspondence: Manelo Anona, Department of Mathematics (gij (x ,…, x )i,j) is invertible. And the relation SJ gives the spray S and Computer Science, Faculty of Sciences, University of Antananarivo, ∂∂ Antananarivo 101, PB 906, Madagascar, E-mail: [email protected]. S= yi − 2 Gx in (11 ,..., x , y ,..., y n ) ∂∂xyii Copyright: © 2020 Manelo A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits with unrestricted use, distribution, and reproduction in any medium, provided the 1 ij original author and source are credited. G= yyγ k 2 ikj Received 30 July, 2020; Accepted 08 August, 2020; Published 25 August, 2020 Manelo A, et al. J Generalized Lie Theory Appl, Volume 14:3, 2020 where g(ℜ (XYZJT , ) , ) =−ℜ g ( ( XYTJZ , ) , ); (3.1) ∂∂ 1 ggkj ∂gik ij γ ikj =( +−). ℜ(,)XYZ +ℜ (,) YZX +ℜ (,) ZXY = 0; (3.2) 2 ∂∂∂xxxi jk By asking g(ℜ (, TXYJZ ), ) +ℜ g ( (,), TYZJX ) +ℜ g ( (, TZXJY ) , ) = 0;(3.3) k kl g((,),ℜ=ℜ XYZJT ) g ((,), ZTXJY ). (3.4) γγij= g ilj i, j, k, l∈ {1,…,n}, we have Properties of the Horizontal Nullity 1 Gk= yy i jkγ . Space of Curvatures 2 ij j lj From the properties of the curvatures given above and by posing: We note G=Gi (,xy ) yil () x, the horizontal projector is written: N={ X ∈χχ () TM suchthat R (,)0,()}, X Y= ∀∈ Y TM ∂∂ ∂ ∂ R = −Gj = hh()iii j, ( j ) 0. =∈χχ ℜ =∀∈ ∂∂xx ∂ y ∂ y Nℜ { X () TM suchthat (,)0,, X Y Z Y Z ()}; TM The vertical projector becomes we have ∂j ∂ ∂∂ ⊥ ⇔∈ = ∈χ (4.1) vv( )=G= ,( ) . JX ImR X KerR ( R iRS ), X ( TM ); ∂xii ∂ y j ∂∂ yy ii JX ⊥ ImR ⇔R ( S, X) Y = 0, X ∈χχ( TM), ∀∈ Y ( TM ); 1 The curvature R= 2 [h, h] is then (4.2) 1 ki j ∂ R= R dx ∧⊗ dx X∈⇔ NR JX ⊥ R ( S, Y) Z , X ∈χχ( TM), ∀ Y, Z ∈( TM ); ij ∂ k 2 y (4.3) with kkkk ∂G ∂G X∈⇔ Nℜ JX ⊥ R ( Y, Z) T , X ∈χχ( TM), ∀ Y, Z , T ∈( TM ); k∂Giijjll∂G Rij = − +Gi −Gj ,i , jkl , ,∈ {1,..., n }. (4.4) ∂∂xxji ∂ y l ∂ y l Proposition 1 X∈⇔ Nℜ R ( YZ,) X =∀∈ 0, YZ, χ ( TM). (4.5) Let E be an energy function, G a connection such that G=[J,S]. The Properties of the Nullity Space and the following two relationships are equivalent: Curvature Image Space i. iSJ dd E= − dE ; . Let H=Imh the horizontal space, V=Imv the vertical space and ImR ii. dEh = 0 (respectively Imℜ ) the module on F(TM) generated by the image of the curvature R (respectivelyℜ ). Proof: We notice that if we have the relation (i), we get Proposition 2 iSS i dd J E=0. = LS E (2.6) On a Riemannian manifold (M, E) the following properties are verified: We can write successively, taking into account that E is homogeneous ℜ⊕ = of degree 2 and that JS=C, 1) Im ℜ and JNℜ are orthogonal and Im JNℜ V ; 2) ImR and JNR+{C} are orthogonal; ImR Å {JNR+{C}}=V; iS dd J E=−() L S di S d J E = L SJ d E −=− 2 dE dE 3) = or NNR ℜ 4) At the neighborhood of a point z of TM, the subspaces LSJ d E= dE. (2.7) NR∩∩ HN,ℜℜ HN ,,,R N hN RR ⊕ JN , ImR and Im ℜ are involutive From the formula Ld−= dL d , we have SJ JS[S, J ] Proof: The property 1) follows from the relation (4.4). LdE= d E + dLE =(2 dE −+ dE ) dLE . SJ[S, J ] JS h JS According to the relation (4.2), JX ^ ImR if and only if According to the two relations (2.6) and (2.7), we find ℜ(S , X ) Y = 0, ∀∈ Yχ ( TM ). From the relation (2.4), we have dEh = 0. ℜ=(SXY , ) JYRX [ , ( )] − [ JYRX , ( )] + RJYSX ([ , ], ) + R ([ JYX , ]) = 0. [,]JS+ I Conversely, if we have dEh = 0 , by definition h = , we can Taking into account the relation (4.1), the relation 2 write J[ JY ,] S= JY , ∀∈ Yχ ( TM )and that the curvature R is semi-basic, we d[,]JS E+= dE 0. get RXY( , )= R [ JYX , ], ∀∈ Yχ ( TM ) (5.1) By noting that hS= S and LES = 0 , we can go up the calculation and one obtains As R is alternating bilinear on the vector fields of χ()TM considered as (TM)-module, the relation is not possible unless X=S or, if X ∈ N , and i dd E= − dE R SJ if X is generated by projectable vector fields in the nullity space of R. The converse is immediate according to the relations (4.2), (4.3) and (3.3). Properties of the Curvature R of a ℜ The property NR =NR follows from the link between and R given Riemannian Manifold by the relation (2.4) and from the above remark that NR is generated by projectable vector fields in NR. At the neighborhood of a point z of TM, the From the properties of the connection stated above, we obtain the ∩ subspaces NR Ç H, NHℜ , are involutive. These are well known results following classic results: for all X,,, Y Z T∈ χ ( TM ) see [7]. Page 2 of 6 Manelo A, et al. J Generalized Lie Theory Appl, Volume 14:3, 2020 Let us now show that hN Å JN is involutive. You just have to check jj R R ∂G ∂G on the generators of hN and JN . Since hN is generated by projectable Xl( li− lk +Gsj G −G s G j) = 0,i , jkls , , ,∈ {1,..., n }. R R R ∂∂xxkili sk lk si vector fields in hNR and, hNR is involutive, then JNR is generated by the commutative generators in JN . JN is therefore involutive. It is the same R R This condition is none other than X ∈ hNR. Hence the result. for [X, JY] with X∈ hNR andY∈ JN .