DERIVATIONAL FORMULAS OF A SUBMANIFOLD OF A GENERALIZED RIEMANNIAN SPACE

Vol. 36, No. 2, 2006, 91-100

DERIVATIONAL FORMULAS OF A SUBMANIFOLD OF A GENERALIZED RIEMANNIAN SPACE

Svetislav M. Minˇci´c¹, Ljubica S. Velimirovi´c²

Abstract. In the introduction is given basic information on a general- ized Riemannian space, as a differentiable manifold endowed with asym- metric basic tensor, and a subspace is defined (in local coordinates).

In§1., for a tensor whose certain indices are related to the space and the others to the subspace, four kinds of covariant derivative are intro- duced and, in this manner, also four connections.

Derivational formulas for tangents of the submanifold are expressed by means of the unit normals (Theorem 1.1 and Theorem 1.2). It is proved that by applying the third or the fourth kind of covariant derivative one concludes that induced connection is symmetric (Theorem 1.2).

§2. is related to the induced connection of the normal bundle (eq.

(2.9)). In this case also are possible four kinds of covariant derivatives on the obtained normal submanifoldX_N−M^N (eq. (2.10)). In Theorem 2.1.

is given the presentation of covariant derivative of the normals, using the first and the second kind of covariant derivatives. Theorem 2.2. is related to the properties of the coefficients of this connection.

In Theorem 2.3. is proved that, applying the third and the fourth kind of covariant derivative at X_N−M^N , we express the covariant derivative of normals by means of tangents, and in this case the induced connection at X_N−M^N is unique (¯Γ

1 = ¯Γ

2).

AMS Mathematics Subject Classification (2000): 53C25, 53A45, 53B05 Key words and phrases: Generalized Riemannian space, derivational for- mulas

0. Introduction

A generalized Riemannian space GR_N [2, 3, 9] is a differentiable manifold equipped with the asymmetric basic tensor Gij(x¹, ..., x^N) (the components) where xⁱ are the local coordinates. The symmetric, respectively antisymmetric part of Gij areHij andKij.

For the lowering and rasing of indices in GRN one uses Hij, respectively H^ij, where

(0.1) (H^ij) = (Hij)⁻¹, (det(Hij)6= 0).

1Faculty of Science and Mathematics, University of Niˇs, Viˇsegradska 33, 18000 Niˇs, Serbia

2Faculty of Science and Mathematics, University of Niˇs, Viˇsegradska 33, 18000 Niˇs, Serbia, e-mail: [email protected]

Cristoffel symbols atGRN are (0.2a, b) Γi.jk= 1

2(Gji,k−Gjk,i+Gik,j), Γⁱ_jk =H^ipΓp.jk,

where, for example, Gji,k = ∂Gji/∂x^k. Based on the asymmetry of Gij, it follows that the Cristoffel symbols are also asymmetric with respect to j, k in (2a, b).

By equations

(0.3) xⁱ=xⁱ(u¹, ..., u^M)≡xⁱ(u^α), i= 1, .., N,

a submanifoldXM is defined in local coordinates. Ifrank(B_αⁱ) = M (B_αⁱ =

∂xⁱ/∂u^α) and

(0.4) gαβ=Bⁱ_αB^j_βGij,

X_M becomes GR_M ⊂GR_N, withinduced basic tensor (0.4), which is gen- erally also asymmetric. Note that in the present work Latin indicesi, j, ...take values 1, . . . , N and refer to theGRN,while the Greek ones take values 1, . . . , M and refer to theGRM.

In theGRM are valid the relations similar to (0.1) and (0.2). The symmetric part ofgαβis denoted withhαβ,and antisymmetric one withkαβ, where e.g.

(0.4⁰a, b) hαβ=B_αⁱB^j_βHij,(h^αβ) = (hαβ)⁻¹.

Cristoffel symbols ˜Γα.βγ,Γ˜^α_βγ =h^απΓ˜π.βγ are expressed by gαβ analogously to (0.2).

For the unit, mutually orthogonal vectorsN_Aⁱ, which are orthogonal to the GRM too, we have ([4]-[8], [10])

(0.5a) HijN_AⁱN_B^j =eAδ^A_B=hAB, eA∈ {−1,1},

(0.5b) HijN_AⁱB_α^j = 0,

whereA, B,· · · ∈ {M+ 1, . . . , N}.

1. Induced connection and derivational formulas on X

⊂ GR

1.1. As is known, the following relations between Cristoffel symbols of a gener- alized Riemannian space and its subspace are valid:

(1.1) Γ˜α.βγ = Γi.jkB_αⁱB_β^jB_γ^k+HijB_αⁱB_β,γ^j ,

(1.2) Γ˜^α_βγ=h^παΓ˜π.βγ =h^πα(Γi.jkB_πⁱB_β^jB^k_γ+HijB_πⁱB_β,γ^j ),

i.e.

(1.2⁰) Γ˜^α_βγ=h^παHpiB_π^p(Γⁱ_jkB_β^jB_γ^k+B_β,γⁱ ).

Supposing that the both connections, defined by the coefficients Γ and ˜Γ are asymmetric, one can define four kinds of covariant derivative for a tensor defined at points of the subspace [4]-[6]. For example, for a tensort^iα_jβ we have (1.3) ∆

1 23 4

µt^iα_jβ≡t^iα_jβ|

1 2 3 4

µ=t^iα_jβ,µ+ Γⁱ_pm

mp pm mp

t^pα_jβB^m_µ −Γ^p_jm

mj mj jm

t^iα_pβB_µ^m+ ˜Γ^α_πµ

µπ πµ µπ

t^iπ_jβ−Γ˜^π_βµ

µβ µβ βµ

t^iα_jπ

and in this manner are defined four connections∇

θ, θ∈ {1, . . . ,4} on the sub- manifold XM ⊂ GRN. The obtained structures we shall denote by (XM ⊂ GRN, gαβ,∇

θ, θ∈ {1, . . . ,4}).

1.2. We have to examine the presentation of covariant derivatives of tangent vectorsB_αⁱ =∂xⁱ/∂u^αand of the unit normals N_Aⁱ, with the help of the same magnitudes, so calledderivational formulas of the subspace for the tangents and normals.

Putting

(1.4) B_α|ⁱ

θ

µ= Φ

θ

παµB_πⁱ +X

P

ΩθP αµN_Pⁱ, θ∈ {1, . . . ,4}, we get

(1.5) Φ

θ

παµ=HijB_α|ⁱ

θ

µB^j_ρh^πρ. Let us investigate, firstly, Φ

1. Substituting in (1.5) with respect to (1.3), we obtain

Φ1

παµ=HijB_ρ^jh^πρ(B_α,µⁱ + Γⁱ_pmB_α^pB_µ^m−Γ˜^σ_αµBⁱ_σ)

=HijB_ρ^jh^πρ(Bⁱ_α,µ+ Γⁱ_pmB_α^pB_µ^m)−hρσh^πρΓ˜^σ_αµ. Further, we have

Φ1

παµ=h^πρ(HijB_α,µⁱ B_ρ^j+ Γj.pmB_ρ^jB_α^pB^m_µ −Γ˜ρ.αµ).

Taking into consideration (1.1), it follows that Φ

1

παµ = 0. In the same way one obtains Φ

2

παµ= 0.So,

(1.6) Φ

θ

παµ= 0, θ∈ {1,2}.

In order to determine Ω

θ at (1.4), we shall compose this equation withHijN_Q^j, and, by virtue of (0.5), we get

(1.7) HijBⁱ_α|

θ

µN_Q^j =X

P

ΩθP αµePδ_Q^P = Ω

θQαµeQ, eQ ∈ {−1,1}

(no summing wrp Q), i.e.

(1.7⁰) Ω

θP αµ=ePHijBⁱ_α|

θ

µN_P^j, from where, substituting B_α|ⁱ

θ

µ based on (1.3) and taking into consideration (0.5b), one obtains

(1.8a, b) Ω

1 2

P αµ= Ω

3 4

P αµ=ePHijN_Pⁱ(B_α,µ^j + Γ^j_pm

mp

B_α^pB_µ^m).

Based on (1.3),(1.4),(1.6), we have the following theorem.

Theorem 1.1. Derivational formulas for tangents of submanifoldX_M ⊂GR_N possessing the structure(XM, g_αβ,∇

θ,θ∈{1,2}), are

(1.9) B_α|ⁱ

θ

µ≡ ∇

θµBⁱ_α=X

P

ΩθP αµN_Pⁱ, θ∈ {1,2}, whereΩ

θ are given at (1.8) .

1.3. Consider now the same structure, but forθ∈ {3,4} and find Φ

θ. Based on (1.5), (1.3), (1.1) we get

Φ3

παµ=H_ijB_ρ^jh^πρ(B_α,µⁱ + Γⁱ_pmB^p_αB^m_µ −Γ˜^σ_µαB_σⁱ)

=HijB_ρ^jh^πρ(B_α,µⁱ + Γⁱ_pmB_α^pB_µ^m)−h^πρΓ˜ρ.µα=h^πρ(˜Γρ.αµ−Γ˜ρ.µα) = ˜T_αµ^π . In the same manner one finds Φ

4

παµ=−T˜_αµ^π , i.e.

(1.10) Φ

3

παµ=−Φ

4

παµ= ˜T_αµ^π .

Composing the equationHijB_αⁱB^j_ρ=hαρ withh^πρ, one gets Hijh^πρB_αⁱB_ρ^j=hαρh^πρ=δ^π_α, wherefrom, applying∇

3µ:

Hijh^πρ(Bⁱ_α|

3

µB_ρ^j+B_αⁱB_ρ|^j

3

µ) = 0, that is

(1.11) Φ

3 παµ+ ˆΦ

3 παµ= 0, where Φ

3 is given at (1.5), and Φˆ

3

παµ=Hijh^πρB_αⁱB_ρ|^j

3

µ.

Since

Hijh^πρB_ρ|^j

3

µ= (Hijh^πρB^j_ρ)|

3µ=B^π_i|

3

µ, by virtue of (1.3) the previous equation gives

Φˆ

3

παµ=B_αⁱB_i|^j

3

µ=B_αⁱ(B_i,µ^π −Γ^p_miB_p^πB^m_µ + ˜Γ^π_σµB_i^σ)

=B_αⁱ(B_i,µ^π −Γ^p_miB_p^πB^m_µ) +B_αⁱHijh^σρB^j_ρΓ˜^π_σµ

=B_αⁱ(B_i,µ^π −Γ^p_miB_p^πB^m_µ) + ˜Γ^π_αµ

(1.2=⁰)Bⁱ_αB_i,µ^π −Γ^p_miBⁱ_αB^π_pB_µ^m+h^ρπHpiB_ρ^pB_α^jB_µ^kΓⁱ_jk+h^ρπHpiB_ρ^pB_α,µⁱ

=B_αⁱ(Hpih^ρπB^p_ρ),µ−Γ^p_miB_αⁱB_p^πB_µ^m+B_i^πB_α^jB_µ^kΓⁱ_jk+Hpih^ρπB_ρ^pB_α,µⁱ where =

(1.2⁰)indicates ”= based on (1.2⁰)” . The first and the last addend give

(Bⁱ_αHpih^ρπB_ρ^p),µ= (h^ρπhρα),µ=δ^π_α,µ= 0,

and by corresponding changes of dummy indices at the rest ones, we finally obtain

(1.12) Φˆ

3

παµ=T_jkⁱ B_i^πB_α^jB_µ^k =

(1.2⁰)

T˜_αµ^π =

(1.10)Φ

3 παµ.

Taking into account (1.10)−(1.12), we obtain

(1.13) Φ

3

παµ=−Φ

4

παµ= ˜T_αµ^π = 0.

So, we have proved the following theorem.

Theorem 1.2. Derivational formulas for tangents of a submanifold XM ⊂ GRN, possessing the structure (XM, gαβ,∇

θ, θ∈ {3,4}), are

(1.14) Bⁱ_α|

θ

µ≡ ∇

θ µBⁱ_α=X

P

Ωθ P αµN_Pⁱ, θ∈ {3,4}, whereΩ

θ are given at(1.8), and induced connectionΓ˜^α_βγin this case is symmetric ( ˜T = 0).

1.4. For the covariant derivative of the normals onXM, based on (1.3),we have

(1.15) N_A|ⁱ

1 2

µ=N_A|ⁱ

3 4

µ=N_A,µⁱ + Γⁱ_pm

mp

N_A^pB_µ^m,

provided that one supposes that the indices A, B,· · · ∈ {M + 1, . . . , N} have not a tensor character [7, 8, 4]. Starting from the presentation

(1.16) ∇

θµN_Aⁱ ≡N_A|ⁱ

θ

µ= Λ

θ

πAµBⁱ_π+X

P

ΨθP AµN_Pⁱ

one obtains, the known result [7, 8, 4] forderivational formulas of normals (1.17) N_A|ⁱ

θ

µ =−eAΩ

θAρµh^ρπB_πⁱ +X

P

ΨθP AµN_Pⁱ,Ψ

θAAµ = 0, θ∈ {1,2}

where Ω

θ is given at (1.8), and for Ψ

θ we have

(1.18) Ψ

θP Aµ =ePHijN_PⁱN_A|^j

θ

µ,

whereN_A|^j

θ

µ is given by virtue of (1.15).

So, the next theorem is valid:

Theorem 1.3. [7, 8, 4]Derivational formulas for normals of submanifoldXM ⊂ GRN with structure (XM, gαβ,∇

θ, θ∈ {1,2})are given at (1.17), whereΨ

θ have the values (1.18).

2. Induced connection on the normal bundle (normal subspace)

2.1. The set of normals of the submanifold XM ⊂ GRN make a normal bundleforXM, and we note itX_N−M^N . One can introduce a metric tensor on X_N^N_−M [10, 11, 1]

(2.1) gAB=GijN_AⁱN_B^j, which is asymmetric in a general case.

The symmetric part is (2.2) hAB =HijN_AⁱN_B^j =

(0.5a)eAδ_B^A=hBA=

(ea, A=B,

0, otherwise., eA∈ {−1,1}.

If

(2.3) (h^AB) = (hAB)⁻¹,

we have

(2.4) h^AB=eAδ_B^A=hAB =h^AB.

2.2. For a vector vⁱ one says that it belongs toX_N−M^N , if it is defined at the points ofXM and is a linear combination of the normals, i.e.

(2.5) vⁱ=v^PN_Pⁱ (i∈ {1, . . . , N}, P =M + 1. . . . , N, a summation on P)

One can define absolute differentialδvⁱ alongXM in two manners δ1

2

vⁱ=dvⁱ+ Γⁱ_jk

kj

v^jdx^k,

from where

(2.6) δ

12

vⁱ=N_Pⁱdv^P+ (N_P,µⁱ + Γⁱ_jk

kj

N_P^jB_µ^k)v^Pdu^µ. Composing the equation (2.6) with

(2.7) N_i^A=Hijh^ABN_B^j, we obtain the projection of δ

θvⁱ onX_N−M^N :

(2.8), δ

12

v^A=dv^A+ ¯Γ

12

AP µv^Pdu^µ,

where

(2.9) Γ¯

1 2

AP µ=N_i^A(N_P,µⁱ + Γⁱ_jk

kj

N_P^jB^k_µ)

are coefficients ofinduced connection of the normal bundle(submanifold, subspace) X_N^N_−M.

For a tensor onXM,whose some indices are related toGRN and the others to X_N^N_−M, four kinds of covariant derivative are possible. For example,

(2.10)

∇¯

1 23 4

µt^iA_jB≡t^iA_jB⊥

1 2 3 4

µ

=t^iA_jB,µ+ Γⁱ_pm

mp pm mp

t^pA_jBB_µ^m−Γ^p_jm

mj mj jm

t^iA_pBB_µ^m+ ¯Γ

1 2 12

AP µt^iP_jB−Γ¯

1 2 21

PBµt^iA_jP.

In this way, 4 connections ¯∇

θ, θ∈ {1, . . . ,4}on the submanifoldX_N^N_−M ⊂GRN

are defined. We shall denote the obtained structures (X_N−M^N ⊂GRN, gAB,∇¯

θ, θ∈ {1, . . . ,4}).

Derivatives of the type (1.3) and (2.10) are van der Waeden-Bortoloti derivatives. Combining these two cases, we can observe also a derivative of a tensor containing simultaneously indices of all three types, e.g. t^iαA_jβB.

2.3. Consider now the explanation of ¯∇

θµN_Aⁱ. Analogously to (1.16) we have

(2.11) ∇¯

θµN_Aⁱ ≡N_A⊥ⁱ

θµ= ¯Λ

θ

πAµB_πⁱ +X

P

Ψ¯

θP AµN_Pⁱ,

from where, composing withHijB^j_ν, one gets (2.12) HijB^j_νN_A⊥ⁱ

θµ = ¯Λ

θ πAµhπν.

In order to determine ¯Λ

θ, consider the relation HijN_AⁱB_ν^j = 0 and apply the derivative ¯∇

θµ≡⊥

θ µ, which in the case ofB^j_ν is reduced to∇

θµ. So, Hij(N_A⊥ⁱ

θµB_ν^j+N_AⁱB^j_ν|

θ

µ) = 0, wherefrom, in relation to (2.12) and (1.7⁰): ¯Λ

θ

πAµhπν+eAΩ

θAνµ= 0,

(2.13) Λ¯

θ

πAµ=−eAΩ

θAρµh^πρ, θ∈ {1, . . . ,4}.

In order to determine ¯Ψ

θ in (2.11), we are composing withHijN_Q^j,and obtaining (2.14) HijN_A⊥ⁱ

θµN_Q^j =X

P

Ψ¯

θP AµePδ_Q^P =eQΨ¯

θQAµ. With respect to (2.10),(2.2),(2.9) the previous relation yields

eQΨ¯

1QAµ=HijN_Q^j(N_A,µⁱ + Γⁱ_pmN_A^pB^m_µ −Γ¯

1 PAµN_Pⁱ)

(2.2)= HijN_Q^j(N_A,µⁱ + Γⁱ_pmN_A^pB^m_µ)−hP QΓ¯

1 PAµ

(2.9)= HijN_Q^j(N_A,µⁱ + Γⁱ_pmN_A^pB^m_µ)−hP QN_i^P(N_A,µⁱ + Γⁱ_jkN_A^jB_µ^k)

= (N_A,µⁱ + Γⁱ_pmN_A^pB_µ^m)(HijN_Q^j −hP QN_i^P) = 0, because

(2.15) hP QN_i^P =HijN_Q^j. So, ¯Ψ

1QAµ= 0. In the same manner one proves that ¯Ψ

2QAµ= 0, and, based on (2.11) and (2.13), we have proved the following theorem.

Theorem 2.1.Derivational formulas for normals of a submanifoldXM ⊂GRN, considered in a structure(X_N−M^N , gAB,∇¯

θ, θ∈ {1,2})are (2.16) N_A⊥ⁱ

θµ≡∇¯

θµN_Aⁱ =−eAΩ

θAρµh^πρBⁱ_π, θ∈ {1,2}, whereΩ

θ are given at (1.8).

2.4. In order to investigate N_A⊥ⁱ

θµ for θ ∈ {3,4}, we shall firstly consider properties of the coefficients ¯Γ

1,Γ¯

2. For the Kronecker symbols, being constants, we have

(2.17) δ^A_B⊥

θµ =δAB⊥

θµ=δ^AB_⊥

θµ = 0,∀θ∈ {1, . . . ,4}.

From here and because of (2.2),(2.4),we obtain

(2.18) hAB⊥

θµ=h^AB_⊥

θµ = 0,∀θ∈ {1, . . . ,4}.

On the other hand, from (2.10) one gets δ_⊥^AB

1µ = 0 + ¯Γ

1

AP µδ^{P B}+ ¯Γ

1

BP µδ^AP = ¯Γ

1 ABµ+ ¯Γ

1 BAµ =

(2.17)0.

The analogous is valid for ¯Γ

2, and we have

(2.19) Γ¯

ω

ABµ=−Γ¯

ω

BAµ,∀ω∈ {1,2},

i.e. an antisymmetry is in force with respect toA, B. Further, we have δ_B⊥^A

3µ =

(2.10)

Γ¯

1 ABµ−Γ¯

2 ABµ =

(2.17)0, and the result is analogous by applying ¯∇

4, so

(2.20) Γ¯

1 ABµ= ¯Γ

2

ABµ for ¯∇

θ, θ∈ {3,4}.

Based on (2.19) and (2.20), we conclude that applying ¯∇

1 and ¯∇

3 or ¯∇

1 and ¯∇

4 or

∇¯

2 and ¯∇

3 or ¯∇

2 and ¯∇

4 one obtains

(2.21) Γ¯

1

ABµ=−Γ¯

2 BAµ.

From the above, we state the following theorem Theorem 2.2. The coefficientsΓ¯

1,Γ¯

2 (2.9) of induced connection in the normal submanifoldX_N−M^N ⊂GR_N have the properties:

a)the property(2.19)in the structures(X_N^N_−M, gAB,∇¯

θ, θ∈ {1,2}), b)the property (2.20)in the structures(X_N−M^N , gAB,∇¯

θ, θ∈ {3,4}), c) the property(2.21) in the structures (X_N^N_−M, gAB,∇¯

θ,∇¯

ω,(θ, ω)∈ {(1,3), (1,4),(2,3),(2,4)}).

2.5. Let us investigate now ¯Ψ

θ forθ∈ {3,4} at (2.11). In relation to (2.9) is (2.22) hP Q(¯Γ

1 p Aµ−Γ¯

2 p

Aµ) =hP QN_i^PT_jkⁱ N_A^jB_µ^k=HijT_pmⁱ N_Q^jN_A^pB^m_µ, and based on (2.14),(2.9) and (2.15):

eQΨ¯

3QAµ=HijN_Q^j(N_A,µⁱ + Γⁱ_pmN_A^pB^m_µ)−hP QN_i^P(N_A,µⁱ + Γⁱ_mpN_A^pB_µ^m)

(2.15)= HijT_pmⁱ N_Q^jN_A^pB_µ^m =

(2.22)hP Q(¯Γ

1 p Aµ−Γ¯

2 p Aµ).

An analogous equation is valid for ¯Ψ

4 too.

Taking into account (2.20), we conclude that, from the previous equation

(2.23) Ψ

θQAµ= 0,∀θ∈ {3,4},

and, by virtue of (2.11) and (2.13) we have the following theorem.

Theorem 2.3. In the structure (X_N−M^N , gAB,∇¯

θ, θ ∈ {3,4}) derivational for- mulas for normals of submanifoldXM ⊂GRN are

(2.24) N_A⊥ⁱ

θµ≡∇¯

θµN_Aⁱ =−eAΩ

θAρµh^πρB_πⁱ, θ{3,4},

and then in X_N^N_−M there exists a unique connection (2.9) with the coefficients Γ¯

1= ¯Γ

2 = ¯Γ.

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Received by the editors September 28, 2006