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 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 : 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 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].

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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 . Let K, L ∈ H such that JK and JL R Proposition 5 be orthogonal to JNℜ . As we have DJZ g=0 for all Z ∈ H, and taking into account ∈ for all ∈ , we have J[ JZ ,] X JNℜ XNℜ On a Riemannian manifold (M, E), the space H Å ImR is involutive if

g(J[JK, L], JX)=0, g(J[K, JL], JX)=0, the rank of the space hNR is constant. that is to say, according to the nullity of the Nijenhuis tensor of J, we Proof: It is clear that for all X, Y∈ χ ( TM ), have [hX, hY]=h[hX,hY]+v[hX, hY]=h[hX, hY]+R(X,Y) [JK, JL]=J[JK, L]+J[K, JL] ∈ JN ℜ So [hX, hY] ∈ H Å ImR. This proves that Im ℜ is involutive. According to the proposition 2, it suffices to show that v[hX, R(Y,Z)] is

The space ImR is also involutive, this follows from the assertion 3) and orthogonal to JNR+{C}. We have from the following property: DgRYZChX ( ( , ), )−−= gDRYZC (hX ( , ), ) gRYZ ( ( , ), DChX ) 0. g([ JYJZ , ], C )= 0, forallJYJZ ,∈ Im R . The relation (2.3) gives Proposition 3 DRYZhX (, )= vhXRYZ [ , (, )]and DChX = [, hCX ] . Let X be a projectable . The following two relationships are The homogeneity of h leads to [h, C]=0, we get equivalent: g(v [ hX , R ( Y , Z )], C )= 0. (i) [hX, J]=0 Let T∈ hNR, then we can write (ii) [JX, h]=0 DhX gRYZ( ( , ), JT )−−= gDRYZ (hX ( , ), JT ) gRYZ ( ( , ), DhX JT ) 0. Proof: For a connection G=[J,S], the torsion is zero, that is, As we have g((,RY Z ),) JT= 0, DhX RY (, Z )= vhXRY [ , (, Z )]and [J,h](X,Y)=0 DhX JT= [, h JT ] X , we obtain for all X, Y∈ χ () TM . By developing the above relation, we have, gvhXRYZ( [ , ( , )], JT )+= gRYZ ( ( , ), vhJTX [ , ] ) 0. [JX, hY]+[hX, JY]+Jh[X,Y]+hJ[X,Y]-J[hX, Y]-J[X, hY]-h[JX,Y]-h[X, JY]=0 According to the proposition 4, we have v[h, JT]X ∈ JN , therefore as we have hJ=0 and, h[X,JY]=0, because X being projectable vector R fields by hypothesis, we have g( v [ hX , R ( Y , Z )], JT )= 0. [JX, h]Y+[hX, J]Y=J[X, h]Y Hence the result. The vector field X being projectable and Jh=J, the term J[X, h] is null. Proposition 6 We obtain On a Riemannian manifold (M, E), the space H Å ImR Å JNR is

[JX, h]Y+[hX, J]Y = 0 involutive if the rank of the space hNR is constant. Hence the equivalence of the relations (i) and (ii). Proof: The relation (2.4) is written: Proposition 4 ℜ(X, Y)Z=J[Z, R(X, Y)]-[JZ, R(X, Y)]+R([JZ, X], Y)+R(X, [JZ, Y]). If we have Z ∈ hN , we get (X, Y)Z=0 according to the relation If the rank of the horizontal nullity space hNR of the curvature R is R ℜ (4.5). As the elements of hN are generated by projectable vector fields constant, there is a local base of hNR verifying the proposition 3. R in hN , the relation above shows that [JZ, R(X,Y)] ∈ ImR Å JN , for all Proof: We will solve the equation R R X, Y∈ χ () TM . The proposition 5 shows that H Å ImR is involutive. We [hX, J]=0 will now show that v[JZ, hX] ∈ JNR. According to the proposition 4, we can find a base of hN verifying the proposition 3. This proves that v[JZ, hX] ∈ The vector fields which annihilate the tangent structure by the Lie R JN . derivative is well known [8]. The equation to be solved is written in natural R local coordinates on an open set U Ì M, (xi, yj) ∈ TU, i, j ∈ {1,…, n} Hence the result. ∂ ∂ ∂∂Xxj () ∂ Xxi()− Xx ij () G= Xx i () + y i Proposition 7 ∂xi i ∂ yj ∂ x i ∂∂ xy ij Let G be a linear connection without torsion satisfying the following that is to say: relationships: ∂Xxj () yi =−G Xxij() . 1) there is an energy function E which satisfies dE= 0; ∂xi i  R  2) the space H Å ImR is involutive; lj j G=Gjj As we have yG=Gil () xi (, xy ), and il li , the previous equation is equivalent to 3) the vertical space V decomposes into V = ImR Å {JNR +{C}} and j ∂Xx() the rank of JN is constant, then the 1-form dEv  is completely integrable =−GXxlj() . R ∂xi li Proof: The kernel of dEv  is H the horizontal space of G since we The condition for compatibility of the equation, according to the have vh = 0 , ImR by hypothesis, and the vertical vector fields such that

Frobenius theorem, is LEJX  = 0 . We will show that these vector fields are involutive.

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We can write Corollary [hX, hY]=h[hX, hY]+v[hX, hY]=h[hX, hY]+R(X, Y) Let M be a differentiable, paracompact, connected manifold of So dimension n ³ 2, G a linear connection without torsion such that the rank of the curvature is locally constant over M. Then, G is a canonical connection [hX, hY] ∈H Å ImR of a Riemannian manifold if and only if at the neighborhood of any point where the rank of the curvature R is constant; Let v be the vertical projector of the linear connection given by E, dEv 1) The distribution H Å ImR is involutive and V=ImR Å {JN +{C}} and dE are identical on the vertical space JH since vJ= v J = J . The R v  with hNR is generated by projectable vector fields in hNR difference between G linear connections coming from the energy function G Im h E and is in the horizontal space  and Imh, the two connections have 2) There is an energy function E such that dER  = 0. the same curvature. It remains to calculate v[JY, hX] for all X, Y∈ χ () TM . Proof: It follows from the theorem 1. Using a partition of the unit, we If JY Î ImR, v[JY, hX] Î ImR by hypothesis. If JY ∈ JN , according to the R glue together the local metrics to have a global metric. proposition 4, v[JY, hX] Î JNR and that H Å ImR Å JNR is also involutive The search for an energy function E of G leads to the search for an according to the proposition 6. The scalar 1-form dEv  is completely integrable like the scalar 1-form dE. energy function E such that dER  = 0. In natural local coordinates, the v   curvature R is written: Theorem 1 1 ∂ R= ylk R() x dx i ∧⊗ dx j , i , j , k , l ∈ {1,..., n }. Let G be a linear connection without torsion such that the rank of the 2 l, ij ∂yk curvature R is constant. The connection G comes from an energy function An energy function is written if and only if 1 E= g yyij. 1) the distribution H Å ImR is involutive and the vertical space  2 ij V=ImR Å {JN +{C}} with hN is generated by projectable vector fields in R R Thus, the relation dE= 0 is equivalent to the following system of hN R  R equations:

=  k 2) there is an energy function E such that dER  0 .  gRkl l, ij = 0  ϕ ()x kk=−≠ Then, there exist a constant j function on the bundles such that eE gkl Rr, ij g kr R l, ij withl r. is the energy function of G In the matrix form, the system is written:

Proof: A necessary and sufficient condition for a connection G to come 110 RR gg11 n 1 1,ij n , ij t from an energy function E is according to the proposition 1, − A    = A  d E=0 nn h ggRR1n nn1, ij n , ij  0 This results in Remark: Solving the inverse problem leads to the system of linear

dh dhE=0=2dRE equations, the other conditions are easy to verify, which is quite easy, in practice. In addition, the trace of Rk is zero. The conditions are therefore necessary according to the previous k, ij studies. Conversely, let E be an energy function such that dER  = 0 . We will show that with the hypothesis, there exist a constant j function on the On an Isotropic Finslerian Manifold [9] bundles such that

ϕ An isotropic finslerian manifold is a manifold [10] with a connection d( eE )= 0. h  such that The equation is equivalent to  =−⊗ 1 R2, EKJ KdJ E C dϕ = − dE. E h    R= iRKs , denotes the sectional curvature. The condition of integrability of such an equation is If K ¹ 0, the manifold is of regular curvature, that is to say, the 11 d() ∧+ dEhh ddE =0, dimension of ImR is equal to n-1. Our theorem remains valid. In the case of EE a finslerian manifold of dimension 2, the theorem is written. that is to say Theorem 2 [11] dE ddE= ∧ dE. hhE Let M be a connected paracompact differentiable manifold of dimension  2, G a connection without torsion with non-zero curvature over a dense According to the proposition 7, dEv  is completely integrable. We have, subset of  M=TM-{0}. according to the Frobenius theorem, G comes from a finslerian structure if and only if, at the neighborhood of any point where the curvature is non-zero, ddEvv∧= dE 0, i. H Å ImR is involutive By applying the inner product iC to the above equality, we obtain ii. The contracted of the curvature R is such dR is of maximum rank and dE ddEvv= ∧ dE,  the function iRS positive E that is to say To illustrate the results, we will give some examples. dE Example 1 ddE= ∧ dE. hhE  Let the manifolds M=3 have coordinates (x1, x2, x3) ∈ 3, (y1, y2, y3) on 3 This is the condition of integrability sought. the bundle of the tangent space T , G the connection [J, S] with

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Example 3

1∂∂∂ 2 3 12x3 2 3 ∂ S=++− y y y2(( y ) e + yy ) . We take M=4 and the energy function is written: ∂∂∂xxx123 ∂y1 j 1 xx4412 22 x 3 32 42 j ∂ = +++ The coefficients G of G being j G , the non-zero coefficients are E( ey () ey () ey () ()). y i G=i 2 ∂yi 1 1x3 1 31 2 The canonical spray of E is written: G=12,ye G= 23 y , G= y . 41 ∂∂∂∂ ∂ ∂1 ∂eyx (( )2 + ( y 22 ) ) ∂ S=+++− y1 y 2 y 3 y 4 yy 1 4 − yy 2 4 −() y32 + . A base of the horizontal space of G is written ∂∂∂∂xxxx1234 ∂ y 1 ∂ y 222 ∂ y 3 ∂y4

∂3 ∂∂ ∂∂ ∂ −2,ye1x −− y 32 , y The non-zero coefficients of G are ∂x1 ∂∂ yx12 ∂∂ yx 13 ∂ y 1

xx4412 The curvature R is written: 1411242233411111 ey 4 ey G=142yyyyy,,,,, G= G= G= 4 G= 31 G=− , G=− 2 . 3 ∂ 22222 2 2 R= ex dx1 ∧−(( y 1 y 2 ) dx 3 − y 32 dx ) ⊗ . ∂y1 The horizontal fields are generated by

The horizontal nullity space is generated by 44 ∂11 ∂eyxx12 ∂∂ ∂ey ∂∂ 1 ∂ −+yy44, −+ ,, − y 3 ∂∂ ∂ 1 1 42 2 43 3 ()yy12− + y 2 − [()()]. y 22123 +− yyy ∂x22 ∂ y ∂∂ yx 2 ∂ y 2 ∂∂ yx 2 ∂ y ∂∂xx23 ∂y1 ∂∂∂11 The horizontal nullity space is not generated by projectable vector −−yy12. ∂∂∂41 2 fields in hNR, NR is not involutive. This linear connection according to the xyy22 proposition 2 cannot come from an energy function. The horizontal nullity space is generated by Example 2 ∂∂1 − y3 . We take M = 4 and the energy function is written: ∂∂xy332

1 xx1212 22 x 3 32 2 x 1 42 The basic element of the nullity space verifying the proposition 3 is: E=(() ey +++ ey () ey () e ()). y 2 3 −x ∂∂1 The canonical spray of E is written: ey2 ( − 3 ). ∂∂xy332 ∂∂∂∂11 ∂ 11 ∂ ∂ Sy=+++−1 y 2 y 3 y 4 (() y12 − () ye 42 x ) − () y22 − () y32 . ∂∂∂∂xxxx12342∂y1 22 ∂ y 2 ∂ y 3 The non-zero elements of the curvature are: G The non-zero coefficients of are xx44 1ee 121 11 11 R2,12 =−=−=, RR4,14 ,,1,12 G=1y 11, G=− ye 4x , G= 2 y 23 , G= y 34 , G= y 44 ,. G= y 1 4 44 12 4 2 22 3 14 44 1 eexx The horizontal fields are generated by R2=−==,, RR 44 4,24 441,14 2,24 4

∂1 ∂ ∂∂ 11 ∂∂ ∂∂1 ∂ ∂ −−y14 y ,,, −y2 −y 3 + ye 4x − y 1 And ImR generated by ∂x12 ∂ y 1 ∂∂ yx 42 22 ∂∂ yx 23 ∂∂ yx 34 ∂ y 1 ∂ y 4

4∂∂ ∂44 ∂ ∂ ∂ eyyx(−+21), −+ yyeyye 41xx, −+ 4 2 The horizontal nullity space is generated by ∂∂yy12 ∂ y 1 ∂ y 4 ∂ y 2 ∂ y 4

4 ∂1123 ∂∂ ∂ (T ). −−yy,.   ∂x222 ∂∂ yx 23 ∂ y 3 We see that H Å ImR is involutive. The two basic elements of the nullity space verifying the proposition 3 are: The matrix is written  xx44 2  −−xx23 ee x4 0−− 00 0 00 ∂1123 ∂∂∂ e 000 e22( −− ye), ( y).  44  2 23 3 4 ∂∂∂∂xyxy22 0ex 00 xx442  (gR )(×=i ) ee= ij j,12 3  000 0 00, 00ex 044  The non-zero elements of the curvature are: 0 000 0 0 00 0 001    0 000 0 0 00 x1 14e 1 x4 RR4,14 =−=,.1,14 1 e x4 0 00− − 22 e 000  0 00 4 4 x4 i 0e 00 0 000 0 0 00 1 ∂∂ (gRij )(×= j,14 ) = =−+41x 4 x3 0 000 0 0 00 , and Im R ye14 y (T ). 00e 0  4 ∂∂yy x x4 0 001 e e 000 0 00 . 4  We see that H Å ImR is involutive. 4 0000 0 00 0 x4 The matrix is written    4 e 000 1 ex 1 1  x 2x 4 0 00− e x 0 00− x1 e 4  e 000 000− 0 00−  i 0e 00 4 (gRij )(×= j,24 ) =  2 2 3  2  x 0000 0 00 0 x 00e 0 0e 00 000 0 0 0 0 0 4 = x x4 3  e x 0 001 e 00e 0 000 0 0 00 0 0 00 0 00 1  1 2x 4 4 2x 1 e 0 00e 00 0 00 0 2 2

Page 5 of 6 Manelo A, et al. J Generalized Lie Theory Appl, Volume 14:3, 2020

Conclusion References 1. Joseph, Grifone. “Structure presque-tangente et connexions I.” Ann Inst Fourier To recognize a connection without torsion, that is to say, Grenoble 22 (1972): 287-334. G=[,]J S and [,] C S= S 2. Joseph, Grifone. ‘’Structure presque-tangente et connexions II.’’ Ann Inst comes from an energy function. Fourier Grenoble 22 (1972): 291-338. Case 1 3. Frölicher, Alfred, and Albert Nijenhuis. “Theory of vector-valued differential form.” Proc Kond Ned Akad A 59 (1996): 338-359. The connection is of regular curvature, that is to say, the module generated on (TM) by ImR is of dimension n-1. The necessary and 4. Anona, Manelo. “Semi-Simplicity of a Lie Algebra of Isometries.” J Generalized sufficient condition is written: Lie Theory Appl 14 (2020): 1-10.

i. H Å ImR is involutive 5. Manelo, Anona. “Sur la dL-cohomologies.” C R Acad Sc Paris T. 290 (1980): 649-651. ii. There is an energy function E such that dER  = 0. 6. Joseph, Klein and Voutier André. “Formes extérieures génératrices de sprays.” Indeed, according to proposition 1, we have d E=0, this implies d E=0. h R Ann Inst Fourier 18 (1968): 241-260. So H Å ImR is the kernel of dE. Then, H Å ImR is involutive, and dEv  is a completely integrable 1-form. 7. Randriambololondrantomalala, Princy, HSG Ravelonirina, and MF Anona. “Sur les algèbres de Lie associées à une connexion.” Can Math Bull 58 (2015): 692- This result is valid for a Finslerian manifold. 703. Case 2 8. Lehmann-Lejeune, Josiane. ”Cohomologies sur le fibré transverse à un dim(ImR)

H Å ImR is involutive, the scalar 1-form dEv  is then completely integrable. 40-45.

How to cite this article: Anona, Manelo and Ratovoarimanana Hasina. “On the Existence of a Riemannian Manifolds at a Given Connection.” J Generalized Lie Theory Appl 14 (2020): 306. doi: 10.37421/GLTA.2020.14.306.

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