On Finslerian Hypersurfaces Given by β-Changes

Masashi Kitayama

Abstract In 1984 C.Shibata has dealt with a change of Finsler metric which is called a β-change of metric [12]. For a β-change of Finsler metric, the differential oneform β play very important roles. In 1985 M.Matsumoto studied the theory of Finslerian hypersurfaces [6]. In there various types of Finslerian hypersurfaces are treated and they are called a hyperplane of the 1st kind, a hyperplane of the 2nd kind and a hyperplane of the 3rd kind.

The purpose of the present paper is to give some relations between the original Finslerian hypersurface and another Finslerian hypersurface given by the β-change of Finsler metrics under certain conditions.

The terminology and notations are referred to the Matsumoto’s monograph [8].

Mathematics Subject Classification: 53C60, 53B40 Key words: Finslerian hypersurface, hyperplane, curvature, Finsler metric

1 Preliminaries

Let M^{n} be an n-dimensional smooth manifold and F^{n} = (M^{n}, L) be an n-dimensional Finsler space equipped with a fundamental function L(x, y) on M^{n}. Then the metric tensor g_{ij}(x, y) and Cartan’s C-tenson C_{ijk}(x, y) are given by

(1.1)  g_{ij} = (∂^{2}L^{2}/∂y^{i}∂y^{j})/2, C_{ijk} = (∂g_{ij}/∂y^{k})/2,

and we can introduce in F^{n} the Cartan connection CΓ = (F_{j}^{i}_{k} , N^{i}_{j}, C_{j}^{i}_{k} ).

A hypersurface M^{n−1} of the underlying smooth manifold M^{n} may be parametrically represented by the equation x^{i} = x^{i}(u^{α}), where u^{α} are Gaussian coordinates on M^{n−1} and Greek indices run from 1 to n-1. Here, we shall assume that the matrix consisting of the pojection factors B_{α}^{i} = ∂x^{i}/∂u^{α} is of rank n-1. The following notations are also employed : B_{αβ}^{i} := ∂^{2}x^{i}/∂u^{α}∂u^{β}, B_{0β}^{i} := v^{α}B_{αβ}^{i}, B_{αβ...}^{ij...} := B_{α}^{i}B_{β}^{j}. . .. If the supporting element y^{i} at a point (u^{α}) of M^{n−1} is assumed to be tangential to M^{n−1}, we may then write y^{i} = B_{α}^{i}(u)v^{α}, so that v^{α} is thought of as the supporting element of M^{n−1} at the point (u^{α}). Since the function \underline{L}(u, v) := L(x(u), y(u, v)) gives rise to a Finsler metric of M^{n−1}, we get an (n-1)-dimensional Finsler space F^{n−1} = (M^{n−1}, \underline{L}(u, v)).

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 49-55.

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At each point (u^{α}) of F^{n−1}, the unit normal vector N^{i}(u, v) is defined by

(1.2)  g_{ij}B_{α}^{i}N^{j} = 0, g_{ij}N^{i}N^{j} = 1.

If (B_{i}^{α}, N_{i}) is the inverse matrix of (B_{α}^{i}, N^{i}), we have

(1.3)  B_{α}^{i}B_{i}^{β} = δ_{α}^{β}, B_{α}^{i}N_{i} = 0, N^{i}B_{i}^{α} = 0, N^{i}N_{i} = 1,

and further

(1.4)  B_{α}^{i}B_{j}^{α} + N^{i}N_{j} = δ_{j}^{i}.

Making use of the inverse matrix (g^{αβ}) of (g_{αβ}), we get B_{i}^{α} = g^{αβ}g_{ij}B_{β}^{j}, N_{i} = g_{ij}N^{j}.

For the induced Cartan connection ICΓ = (N_{β}^{α}, F_{β}^{α}_{γ} , C_{β}^{α}_{γ} ) on F^{n−1}, the second fundamental h-tensor H_{αβ} and the normal curvature vector H_{α} are given by

(1.5)  \begin{matrix}H_{αβ} := N_{i}(B_{αβ}^{i} + F_{j}^{i}_{k} B_{αβ}^{j k}) + M_{α}H_{β}, \\ H_{α} := N_{i}(B_{0α}^{i} + N^{i}_{j}B_{α}^{j}), \end{matrix}

where M_{α} := C_{ijk}B_{α}^{i}N^{j}N^{k} and B_{0α}^{i} = B_{βα}^{i} v^{β}.

Contracting H_{βα} by v^{β}, we immediately get

(1.6)  H_{0α} := H_{βα}v^{β} = H_{α}.

Further we have put

(1.7)  M_{αβ} := C_{ijk}B_{αβ}^{i j}N^{k}, Q_{αβ} := C_{ijk |0}B_{αβ}^{i j}N^{k}, Q_{αβγ} := C_{ijk|0}B_{αβγ}^{i j k}.

The Gauss equation with respect to ICΓ is written as

(1.8)  \begin{aligned}R_{αβγδ} & = & R_{ijkh}B_{αβγδ}^{i j k h} + P_{ijkh}(B_{γ}^{k}H_{δ} − B_{δ}^{k}H_{γ})B_{αβ}^{i j}N^{h}+ \\ & + & (H_{αγ}H_{βδ} − H_{αδ}H_{βγ}). \end{aligned}

2 Hypersurfaces given by the β-change of a Finsler metric

Let F^{n} = (M^{n}, L) be an n-dimensional Finsler space with a fundamental function L(x, y). For a differential one-form β(x, dx) = b_{i}(x)dx^{i} on M^{n}, we shall consider a change of Finsler metric which is defined by L(x, y) ⟶ L̄(x, y) = f(L(x, y), β(x, y)), where f(L, β) is a positively homogeneous function of L and β of degree one. This is called a β-change of the metric. Then we can introduce in F̄^{n} = (M^{n}, L̄) the Cartan connection CΓ̄ = (F̄_{j}^{i}_{k} , N̄^{i}_{j}, C̄_{j}^{i}_{k}) from a β-change of the metric.

For the later use, we prepare here the following two lemmas. Lemma 1 (Shibata[12]). If the covariant vector b_{i}(x) is parallel with respect to the Cartan connection CΓ on F^{n}, the difference tensor D_{j}^{i}_{k} (:= F̄_{j}^{i}_{k} − F_{j}^{i}_{k}) vanishes.

This lemma leads us to N̄_{j}^{i} = N^{i}_{j} from D^{i}_{j} ≡ N̄_{j}^{i} − N^{i}_{j} = D_{j}^{i}_{k} y^{k} = D_{j}^{i}0. Lemma 2 (Shibata[12]). Assume that the covariant vector b_{i}(x) is parallel with respect to the Cartan connection CΓ on F^{n}. Then the h-curvature tensor R̄_{h}^{i}_{jk}(x, y) of F̄^{n}, obtained from F^{n} by the β-change, vanishes if and only if the h-curvature tensor R_{h}^{i}_{jk}(x, y) of F^{n} vanishes.

We now consider a Finslerian hypersurface F^{n−1} = (M^{n−1}, \underline{L}(u, v)) of F^{n} and another Finslerian hypersurface F̄^{n−1} = (M^{n}, \underline{L̄}(u, v)) of the F̄^{n} given by the βchange. Let N^{i} be a unit normal vector at each point of F^{n−1}, and (B_{i}^{α}, N_{i}) be the inverse matrix of (B_{α}^{i}, N^{i}). The functions B_{α}^{i}(u) may be considered as components of n-1 linearly independent vectors tangent to F^{n−1} and they are invariant under the β-change. And so a unit normal vector N̄^{i}(u, v) of F̄^{n−1} is uniquely determined by

(2.1)  ḡ_{ij}B_{α}^{i}N̄^{j} = 0,  ḡ_{ij}N̄^{i}N̄^{j} = 1.

The fundamental tensor ḡ_{ij} = (∂^{2}L̄^{2}/∂y^{i}∂y^{j})/2 of the Finsler space F̄^{n} given by a β-change is as follows [12]:

(2.2)  ḡ_{ij}(x, y) = pg_{ij}(x, y) + p_{0}b_{i}b_{j} + p_{−1}(b_{i}y_{j} + b_{j}y_{i}) + p_{−2}y_{i}y_{j},

where we put p = ff_{L}/L

(2.3)  \begin{matrix}p_{0} = ff_{ββ} + f_{β}^{2},  p_{−1} = (ff_{Lβ} + f_{L}f_{β})/L, \\ p_{−2} = (ff_{LL} + f_{L}^{2} − ff_{L}/L)L^{2}, \end{matrix}

and subscripts L, β denote partial differetiations by L, β respectively. Now contracting (1.2) by v^{α}, we immediately get

(2.4)  y_{i}N^{i} = 0.

Further contracting (2.2) by N^{i}N^{j} and paying attention to (1.2) and (2.4), we have

(2.5)  ḡ_{ij}N^{i}N^{j} = p + p_{0}(b_{i}N^{i})^{2}.

Then we obtain

(2.6)  ḡ_{ij}(±N^{i}/\sqrt{p + p_{0}(b_{i}N^{i})^{2}})(±N^{j}/\sqrt{p + p_{0}(b_{i}N^{i})^{2}}) = 1,

provided p + p_{0}(b_{i}N^{i})^{2} > 0. Therefore we can put

(2.7)  N̄^{i} = N^{i}/\sqrt{p + p_{0}(b_{i}N^{i})^{2}},

where we have chosen the sign ”+” in order to fix an orientation.

Using (1.2) and (2.4), the first condition of (2.1) gives us

(2.8)  (b_{i}N^{i})(p_{0}b_{j}B_{α}^{j} + p_{−1}y_{j}B_{α}^{j}) = 0.

Now, assuming that p_{0}b_{j}B_{α}^{j} + p_{−1}y_{j}B_{α}^{j} = 0 and contracting this by v^{α}, we find p_{0}β + p_{−1}L^{2} = 0. By (2.3) this equation lead us to ff_{β} = 0, where we have used Lf_{Lβ} + βf_{ββ} = 0 and Lf_{L} + βf_{β} = f owing to the homogeneity of f. Thus we have f_{β} = 0 because of f ≠ 0. This fact means L̄ = f(L) and contradicts the definition of a β-change of metric. Consequently (2.8) gives us

(2.9)  b_{i}N^{i} = 0.

Therefore (2.7) is rewritten as

(2.10)  N̄^{i} = N^{i}/\sqrt{p}  (p > 0),

and then it is clear N̄^{i} satistifies (2.1). Summarizing the above, we obtain Theorem 2.1. For a field of linear frame (B_{1}^{i}, …, B_{n−1}^{i}, N^{i}) of F^{n}, there exists a field of linear frame (B_{1}^{i}, …, B_{n−1}^{i}, N̄^{i} = N^{i}/\sqrt{p}) of the F̄^{n} given by the β-change such that (2.1) is satisfied along F̄^{n−1}, and then we get (2.9).

The quantities B̄_{i}^{α} are uniquely defined along F̄^{n−1} by

(2.11)  B̄^{α}_{i} = ḡ^{αβ}ḡ_{ij}B_{β}^{j},

where (ḡ^{αβ}) is the inverse matrix of (ḡ_{αβ}).

Let (B̄_{i}^{α}, N̄_{i}) be the inverse matrix of (B_{α}^{i}, N̄^{i}), and then we have

(2.12)  B_{α}^{i}B̄_{i}^{β} = δ_{α}^{β}, B_{α}^{i}N̄_{i} = 0, N̄^{i}B̄_{i}^{α} = 0, N̄^{i}N̄_{i} = 1,

and further

(2.13)  B_{α}^{i}B̄_{j}^{α} + N̄^{i}N̄_{j} = δ^{i}_{j}.

We also get N̄_{i} = ḡ_{ij}N̄^{j}, that is,

(2.14)  N̄_{i} = \sqrt{p}N_{i}.

If each path of a hypersurface F^{n−1} with respect to the induced connection is also a path of the ambient space F^{n}, then F^{n−1} is called a hyperplane of the first kind. A hyperplane of the 1st kind is characterized by H_{α} = 0.

From (1.5), (2.14) and Lemma 2, we have H̄_{α} = \sqrt{p}H_{α}. Thus we obtain Theorem 2.2. Let b_{i}(x) be parallel with respect to CΓ on F^{n}. Then a hypersurface F^{n−1} is a hyperplane of the 1st kind, if and only if the hypersurface F̄^{n−1} is a hyperplane of the 1st kind.

If each h-path of a hypersurface F^{n−1} with respect to the induced connection is also an h-path of the ambient space F^{n}, then F^{n−1} is called a hyperplane of the second kind. A hyperplane of the 2nd kind is characterlized by H_{αβ} = 0.

From (1.5), (1.6), (2.14) and Lemma 2, we obtain Theorem 2.3 Let b_{i}(x) be parallel with respect to CΓ on F^{n}.Then a hypersurface F^{n−1} is a hyperplane of the 2nd kind, if and only if the hypersurface F̄^{n−1} is a hyperplane of the 2nd kind.

As to the torsion tensor C̄_{ijk} of F̄^{n}, Shibata [12] gave:

(2.15)  C̄_{ijk} = pC_{ijk} + p_{−1}(h_{ij}m_{k} + h_{jk}m_{i} + h_{ki}m_{j})/2 + p_{0β}m_{i}m_{j}m_{k}/2,

where we put

(2.16)  m_{i} = b_{i} − βy_{i}/L^{2}.

Using (2.4) and (2.9), we easily get

(2.17)  m_{i}N^{i} = 0.

As for the angular metric tensor h_{ij} = g_{ij} − l_{i}l_{j}, (1.2) and (2.4) yield

(2.18)  h_{ij}B_{α}^{i}N^{j} = 0.

Contracting (2.15) by B_{αβ}^{ij}N^{k}, (2.17) and (2.18) lead to

(2.19)  C̄_{ijk}B_{αβ}^{ij}N^{k} = pC_{ijk}B_{αβ}^{ij}N^{k}.

On using (1.7) and (2.10), (2.19) is rewritten as

(2.19)  M̄_{αβ} = \sqrt{p}M_{αβ}.

If the unit normal vector of F^{n−1} is parallel along each curve of F^{n−1}, then F^{n−1} is called a hyperplane of the third kind. A hyperplane of the 3rd kind is characterized by H_{αβ} = M_{αβ} = 0.

Thus, from Theorem 2.3 and (2.20), we obtain Theorem 2.4. Let b_{i}(x) be parallel with respect to CΓ on F^{n}. Then a hypersurface F^{n−1} is a hyperplane of the 3rd kind, if and only if the hypersurface F̄^{n−1} is a hyperplane of the 3rd kind.

Taking account of Lemma 1, as to BΓ we have [6]

(2.21)  G_{β}^{α}_{γ} = B_{i}^{α}A_{βγ}^{i}

where A_{βγ}^{i} := B_{βγ}^{i} + G_{j}^{i}_{k}B_{βγ}^{jk}. Now using (1.4), then (2.21) becomes

(2.22)  A_{βγ}^{i} = B_{δ}^{i}G_{β}^{δ}_{γ} + N^{i}N_{h}A_{βγ}^{h}

Since Lemma 1 leads to Ā_{βγ}^{i} = A_{βγ}^{i}, we immediately get

(2.23)  Ḡ_{βγ}^{α} = B̄_{i}^{α}A_{βγ}^{i}.

On substituting (2.22) in (2.23) and paying attention to (2.10) and (2.12), we find Ḡ_{β}^{α}_{γ} = G_{β}^{α}_{γ}. Thus we obtain Theorem 2.5. Let b_{i}(x) be parallel with respect to CΓ on F^{n}. Then a hyperplane F^{n−1} of the 1st kind is a Berwald space, if and only if the hyperplane F̄^{n−1} of the 1st kind is a Berwald space.

Paying attention to Lemma 1, as to CΓ the (v)hv-torsion tensor is written as

(2.24)  P^{α}_{βγ} = B_{i}^{α}K_{βγ}^{i},

where K_{βγ}^{i} := P_{jk}^{i}B_{βγ}^{jk}. On using (1.4), then (2.24) becomes

(2.25)  K_{βγ}^{i} = B_{δ}^{i}P^{δ}_{βγ} + N^{i}N_{h}K_{βγ}^{h}.

Lemma 2 gives us K̄_{βγ}^{i} = K_{βγ}^{i}, and then we immediately obtain

(2.26)  P̄_{βγ}^{α} = B̄_{i}^{α}K_{βγ}^{i}.

On substituting (2.25) in (2.26) and taking account of (2.10) and (2.12), we find P̄_{βγ}^{α} = P^{α}_{βγ}. Thus we obtain Theorem 2.6. Let b_{i}(x) be parallel with respect to CΓ on F^{n}. Then a hyperplane F^{n−1} of the 1st kind is Landsberg, if and only if the hyperplane F̄^{n−1} of the 1st kind is Landsberg.

From (1.8) the Gauss equation of hyperplane of the 1st kind is rewritten as

(2.27)  R_{αβγδ} = R_{ijkh}B_{αβγδ}^{ijkh} + (H_{αγ}H_{βδ} − H_{αδ}H_{βγ}).

Then Lemma 2 and H̄_{αβ} = \sqrt{p}H_{αβ} give us the following. Theorem 2.7. Let b_{i}(x) be parallel with respect to CΓ on F^{n}. Then the curvature tensor R_{αβγδ} of a hyperplane F^{n−1} of the 1st kind of F^{n} with R_{ijkh} = 0 vanishes, if and only if the curvature tensor R̄_{αβγδ} of the hyperplane F̄^{n−1} of the 1st kind of F̄^{n} with R̄_{ijkh} = 0 vanishes.

Further Theorem 2.5 and Theorem 2.7 immediately give Theorem 2.8. Let b_{i}(x) be parallel with respect to CΓ on F^{n}. Then a hyperplane F^{n−1} of the 1st kind of F^{n} with R_{ijkh} = 0 is a locally Minkowski space, if and only if the hyperplane F̄^{n−1} of the 1st kind of F̄^{n} with R̄_{ijkh} = 0 is locally Minkowskian.

References

[1] P.L.Antonelli, R.S.Ingarden and M.Matsumoto, The theory of sprays and Finsler spaces with applications in physics and biology, Kluwer Academic Publishers, Dordrecht, 1993.

[2] G.M.Brown, A study of tensors which characterize a hypersurface of a Finsler space, Canad. J.Math. 20(1968), 1025-1036.

[3] S.Kikuchi, On the theory of subspace, Tensor, N.S.,2(1952), 67-79.

[4] M.Kitayama, A.Sato and C.Shibata, The second fundamental H-tensor of a Hypersurface of Finsler spaces, J.Hokkaido Univ. of Education, 37-1(1986), 17-21.

[5] M.Kitayama, On a hypersurface of Finsler spaces with R_{ohk}^{i} = 0, J. Hokkaido Univ. of Education, 40-1(1989), 23-26.

[6] M.Matsumoto, The induced and intrinsic connections of a hypersurface and Finslerian projective geometry, J.Math. Kyoto Univ., 25(1985), 107-144.

[7] M.Matsumoto, Foundation of Finsler geometry and special Finsler spaces, Kaiseisha Press, Otsu, Japan, 1986.

[8] M.Matsumoto, Theory of Finsler spaces with (α, β)-metric, Reports on Math. Phys., 31(1992), 43-83.

[9] R.Miron and M.Anastasiei, The geometry of Lagrange spaces. Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1994.

[10] H.Rund, The differential geometry of Finsler spaces, Springer Verlag, Berlin, 1959.

[11] C.Shibata, U.P.Singh and A.K.Singh, On induced and intrinsic theories of hypersurface of Kropina spaces, J.Hokkaido Univ. of Education, 34-1(1983).

[12] C.Shibata, On invariant tensors of β-change of Finsler metrics, J.Math. Kyoto Univ., 24(1984), 163-188.

Department of Mathematics, Kushiro Campus Hokkaido University of Education, Kushiro, Hokkaido 085-8580, JAPAN