RIEMANNIAN METRICS ON THE TANGENT BUNDLE OF A FINSLER

˘ ¸ II “AL.I. CUZA” DIN IAS ANALELE S ¸ TIINT ¸ IFICE ALE UNIVERSITAT ¸ I (S.N.) ˘ Tomul LVI, 2010, f.2 MATEMATICA, DOI: 10.2478/v10157-010-0030-8

RIEMANNIAN METRICS ON THE TANGENT BUNDLE OF A FINSLER SUBMANIFOLD BY

AUREL BEJANCU and HANI REDA FARRAN

Abstract. Let IFn = (M, F ) be a Finsler submanifold of a Finsler manifold IFn+p = f, Fe). Then the induced non-linear connection HT M ◦ and the canonical non-linear (M connection GT M ◦ define two Riemannian metrics G and G∗ on T M ◦ = T M \ {0}, e on T M f◦ = both of Sasaki-Finsler type. On the other hand, the Sasaki-Finsler metric G ◦ n f T M \ {0} induces a Riemannian metric Gind on T M . We prove that IF is totally geodesic immersed in IFn+p if and only if G = G∗ = Gind on T M ◦ . Mathematics Subject Classification 2000: 53C60, 53C40. Key words: Finsler submanifold, Sasaki metric, totally geodesic immersion.

1. Preliminaries f, Fe) be an (n + p)-dimensional Finsler manifold, where Fe Let IFn+p = (M is the fundamental function of IFn+p that is of class C ∞ on the slit tangent f◦ = T M f\{0} (see Bao-Chern-Shen [1], Bejancu-Farran [3], bundle T M Matsumoto [4], Rund [5], for basic results on Finsler geometry). Denote f, where (xi ) are by (xi , y i ), i = {1, ..., n + p} the local coordinates on T M i f and (y ) are the coordinates of a the local coordinates of a point x ∈ M f f◦ on T M f◦ is endowed vector y ∈ Tx M . The vertical vector bundle V T M with a Riemannian metric whose local components are given by geij (x, y) =

1 ∂ 2 Fe2 . 2 ∂y i ∂y j

Denote by geij (x, y) the entries of the inverse of the (n + p) × (n + p)-matrix [e gij (x, y)]. Then the geodesics of IFn+p are given by the solutions of the

Unauthenticated Download Date | 3/22/16 8:42 PM

430


2

system of differential equations ( ) d2 xi dxj i j e + G x (s), = 0, ds2 ds e i are given by where G e i (x, y) = 1 geih G 4

{

∂ 2 Fe2 j ∂ Fe2 y − ∂y h ∂xj ∂xh

} .

f◦ is an integrable distribution on T M f◦ which is locally We note that V T M ∂ ◦ f there exists a distribution spanned by { ∂yi }, i ∈ {1, ..., n+p}. Also, on T M f◦ (in general, non-integrable) which is given by the non-holonomic GT M frame field: ∂ δ∗ e j ∂ , i ∈ {1, ..., n + p}, = −G i ∗ i i δ x ∂x ∂y j e j are given by where G i ej ej = ∂ G . G i ∂y i It is important to note that we have f◦ = GT M f◦ ⊕ V T M f◦ , TTM f◦ and V T M f◦ are complementary distributions on T M f◦ . We that is, GT M f◦ the canonical non-linear connection on T M f◦ . The above decall GT M ◦ f enables us to define the Sasaki-Finsler metric G e on composition of T T M ◦ f T M given locally as follows (cf. Bao-Chern-Shen [1], p. 48, BejancuFarran [3], p. 35, Matsumoto [4], p. 136) ( ∗ ) ( ) ( ∗ ) ∗ δ ∂ e δ , δ e ∂ , ∂ e (1) G = G = g e (x, y), G , = 0. ij δ ∗ xj δ ∗ xi ∂y j ∂y i δ ∗ xj ∂y i f that is locally given Next, let M be an n-dimensional submanifold of M by the equations xi = xi (u1 , ..., un ), i ∈ {1, ..., n + p}.


3

RIEMANNIAN METRICS

431

Throughout the paper we use the following ranges for indices: i, j, k, ... ∈ {1, ..., n + p}, α, β, γ, ... ∈ {1, ..., n}, a, b, c, ... ∈ {n + 1, ..., n + p}. Consider (uα , v α ) as local coordinates on T M ◦ = T M \ {0} and define the function F (uα , v α ) = Fe(xi (u), y i (u, v)), where we put y i (u, v) = Bαi v α , Bαi =

∂xi . ∂uα

Then IFn = (M, F ) is a Finsler submanifold of IFn+p . As above, we have the decomposition (2)

T T M ◦ = GT M ◦ ⊕ V T M ◦ ,

where V T M ◦ and GT M ◦ are the vertical vector bundle { ∂ }and the canonical ◦ non-linear connection on T M , locally spanned by ∂vα and δ∗ ∂ ∂ = − Gβα β , ∗ α α δ u ∂u ∂v respectively. Thus the Sasaki-Finsler metric G∗ on T M ◦ defined by the decomposition (1.2) is locally given by ( ∗ ) ( ) ( ∗ ) δ δ∗ ∂ ∂ δ ∂ ∗ ∗ ∗ (3) G , =G , = gαβ (u, v), G , = 0, δ ∗ uβ δ ∗ u α ∂v β ∂v α δ ∗ uβ ∂v α where we put gαβ (u, v) = (4)

1 ∂2F 2 2 ∂v α ∂v β .

By direct calculations it follows that

gαβ = geij Bαi Bβj .

e induces a Riemannian metric Gind on T M ◦ as follows On the other hand, G (5)

e Gind (X, Y ) = G(X, Y ), ∀X, Y ∈ Γ(T T M ◦ ).

Here, and in the sequel, we denote by Γ(T T M ◦ ) the F(T M ◦ )-module of vector fields on T M ◦ , where F(T M ◦ ) is the algebra of smooth functions on T M ◦ . The same notation will be used for the module of sections of any vector bundle. Now, we denote by V T M ◦⊥ the orthogonal complementary vector subf|T M ◦ with respect to G e and call it the Finsler bundle to V T M ◦ in V T M n n+p normal bundle of the immersion of IF in IF (cf. Bejancu [2], p. 47). Then we consider a local field of orthonormal frames {Ba = Bai ∂/∂y i },


432


4

a ∈ {n + 1, ..., n + p} in V T M ◦⊥ , and taking into account that {∂/∂v α = Bαi ∂/∂y i } is a local field of frames in V T M ◦ , we deduce that (a) geij Bαi Baj = 0, (b) geij Bai Bbj = δab .

(6)

eα B e a ] the inverse of the transition matrix [B i B i ] from the Denote by [B α a i i f◦ ◦ to the field of frames { ∂α , Ba }. natural field of frames { ∂y∂ i } on V T M |T M ∂v Next, we consider the induced non-linear connection HT M ◦ on T M ◦ , which is defined as the complementary orthogonal distribution to V T M ◦ in T T M ◦ with respect to Gind . Thus, apart from (2), T T M ◦ admits the decomposition T T M ◦ = HT M ◦ ⊕ V T M ◦ .

(7)

We should note that both (2) and (7) are orthogonal decompositions, but ◦ with respect to different metrics{G∗ and } Gind , respectively. Locally, HT M δ is given by the field of frames δuα , α ∈ {1, ..., n}, expressed as follows (cf. Bejancu-Farran [3], pp. 74, 75) (8)

( ) δ ∂ β β ∂ eβ Bi + Bj G ei , , N = B = − N α0 α j α i δuα ∂uα ∂v β α

or (9)

( ) ∗ δ i δ a a a i j ei e = B + H B , H = B B + B G α ∗ i α a α i α0 α j , δuα δ x

where we put i Bα0 =

∂ 2 xi vβ . ∂uα ∂uβ

Finally, by using the decomposition (7) we can define another SasakiFinsler metric G on T M ◦ given by ) ( ) ( ) ( δ ∂ ∂ δ ∂ δ =G = gαβ , G = 0. (10) G , , , δuβ δuα ∂v β ∂v α δuβ ∂v α Summing up, we constructed on T M ◦ three Riemannian metrics: G∗ , Gind and G given by (3), (5) and (10), respectively. Thus, it is natural to ask the following question. What class of Finsler submanifolds is characterized by the condition G∗ = Gind = G? In the second part of this paper we give the answer to this question.


5

433

RIEMANNIAN METRICS

2. The main result f, Fe). Then we Let IFn = (M, F ) be a Finsler submanifold of IFn+p = (M n n+p say that IF is totally geodesic immersed in IF if any geodesic of IFn is a geodesic of IFn+p . The main purpose of this section is to prove the following theorem. Theorem 2.1. Let IFn = (M, F ) be a Finsler submanifold of IFn+p = f, Fe). Then the following assertions are equivalent: (M (i) IFn is totally geodesic immersed in IFn+p . (ii) Gind = G∗ on T M ◦ . (iii) Gind = G on T M ◦ . As a direct consequence of this result we state the following. Corollary 2.1. IFn is a totally geodesic Finsler submanifold of IFn+p if and only if G∗ = Gind = G on T M ◦ . First, we prove the following lemma. Lemma 2.1. Let IFn = (M, F ) be a Finsler submanifold of IFn+p = f, Fe). Then we have: (M ( (a) Gind ( (11)

(b) Gind ( (c)

Gind

δ δ , α β δu δu δ ∂ , δuβ ∂v α ∂ ∂ , β ∂v ∂v α

) = gαβ +

n+p ∑

Hαa Hβa ,

a=n+1

) = 0, ) = gαβ .

Proof. By direct calculations using (5), (9), (1) and (4) we obtain ( ) ( ) ∗ ∗ δ δ j δ b i δ a e Gind , = G Bβ ∗ j + Hβ Bb , Bα ∗ i + Hα Ba δuβ δuα δ x δ x = geij Bαi Bβj + Hαa Hβb δab = gαβ +

n+p ∑ a=n+1


Hαa Hβa ,

434


6

which proves (11a). Then, (11b) follows from the definition of the non-linear connection HT M ◦ . Finally, by using (5), (1) and (4) we deduce that ( ) ( ) ∂ ∂ j ∂ i ∂ e Gind , = G Bβ j , Bα i = geij Bαi Bβj = gαβ , ∂v β ∂v α ∂y ∂y which completes the proof of the lemma.

Next, we recall the following characterization of totally geodesic Finsler submanifolds. Theorem 2.2 (Bejancu-Farran [3], p. 134). IFn =(M, F ) is a totally f, Fe) if and only if we have geodesic Finsler submanifold of IFn+p =(M (12)

Hαa = 0, ∀α ∈ {1, ..., n}, a ∈ {n + 1, ..., n + p}.

Now, we can state the following. Theorem 2.3. IFn is a totally geodesic Finsler submanifold of IFn+p if and only if Gind = G on T M ◦ . Proof. By comparing (11) with (10) we deduce that Gind = G on T M ◦ if and only if n+p ∑

Hαa Hβa = 0, ∀α, β ∈ {1, ..., n},

a=n+1

which is equivalent to (12). In order to find another characterization of totally geodesic Finsler submanifolds we present the relationship between the induced non-linear connection HT M ◦ and the canonical non-linear connection GT M ◦ on T M ◦ . First we consider the Cartan tensor field of type (0, 3) on IFn+p , whose local components are given by geijk =

1 ∂ 3 Fe2 . i 4 ∂y ∂y j ∂y k

Then we define on T M ◦ p Finsler tensor fields of type (1, 1) whose local components are given by ga β α = g βγ geijk Bai Bγj Bαk , a ∈ {n + 1, ..., n + p}, α, β ∈ {1, ..., n}.


7

RIEMANNIAN METRICS

435

Finally, we consider the deformation Finsler tensor field D with respect to the pair (HT M ◦ , GT M ◦ ), whose local components are given by Dαβ = ga β α Hγa v γ .

(13)

Then it{ is proved in [3], p. 112, that the local fields of { Bejancu-Farran } ∗ } frames δ∗δuα and δuδα in GT M ◦ and HT M ◦ are related by δ δ∗ ∂ = + Dαβ β · α ∗ α δu δ u ∂v

(14)

Lemma 2.2. Let IFn = (M, F ) be a Finsler submanifold of IFn+p = f, Fe). Then we have: (M (

) n+p ∑ δ∗ δ∗ , = g + Hαa Hβa + gγµ Dαγ Dβµ , αβ δ ∗ uβ δ ∗ u α a=n+1 ) ( ∗ δ ∂ = −gαγ Dβγ . (b) Gind ∗ β , δ u ∂v α (a) Gind

(15)

Proof. First by using (14) and (11) we obtain ( ∗ ( ) ) δ δ δ∗ δ µ ∂ γ ∂ Gind ∗ β , ∗ α = Gind − D , − D α β ∂v µ δuα δ u δ u δuβ ∂v γ = gαβ +

n+p ∑

Hαa Hβa + gγµ Dαγ Dβµ ,

a=n+1

and (15a) is proved. In a similar way we obtain (15b).

Theorem 2.4. IFn is a totally geodesic Finsler submanifold of IFn+p if and only if Gind = G∗ . Proof. Suppose IFn is totally geodesic in IFn+p . Then by (12) and (13) we have Hαa = 0 and Dαβ = 0, ∀α, β ∈ {1, ..., n}, a ∈ {n + 1, ..., n + p}. Using these in (15) and comparing with (3) we deduce that Gind = G∗ . Conversely, if Gind = G∗ , from (15) and (3) we deduce that gαγ Dβγ = 0 and

n+p ∑

Hαa Hβa + gγµ Dαγ Dβµ = 0, ∀α, β ∈ {1, ..., n},

a=n+1


436


which imply (12). Hence IFn is totally geodesic.

8

By Theorems 2.3 and 2.4 we have a proof of Theorem 2.1. f, Fe) a Riemannian manifold, In particular, let us consider IFn+p =(M that is geijk = 0, ∀i, j, k ∈ {1, ..., n+p}. Then by (13) we deduce that Dαβ = 0 for all α, β ∈ {1, ..., n}, and therefore by (14) we obtain HT M ◦ = GT M ◦ . As a conclusion we have G = G∗ . Therefore, in the particular case of Riemannian submanifolds we obtain the following result. Theorem 2.5. Let (M, g) be a submanifold of a Riemannian manifold f (M , ge). Then (M, g) is totally geodesic if and only if the Riemannian metric f coincides with the Sasaki metric induced on T M by the Sasaki metric of T M induced on T M by g.

REFERENCES

1. Bao, D.; Chern, S.S.; Shen, Z. – An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer-Verlag (2000). 2. Bejancu, A. – Finsler Geometry and Applications, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, New York, 1990. 3. Bejancu, A.; Farran, H.R. – Geometry of Pseudo-Finsler Submanifolds, Kluwer Academic Publishers, Dordrecht, 2000. 4. Matsumoto, M. – Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Shigaken, 1986. 5. Rund, H. – The Differential Geometry of Finsler Spaces, Springer-Verlag, BerlinG¨ ottingen-Heidelberg, 1959. Received: 17.III.2009

Kuwait University, Department of Mathematics and Computer Science, P.O. Box 5969, Safat 13060, KUWAIT [email protected] [email protected]
