Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 22 (2006), 71–87 www.emis.de/journals ISSN 1786-0091 TANGENT BUNDLE OF THE HYPERSURFACES IN A EUCLIDEAN SPACE

Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 22 (2006), 71–87

www.emis.de/journals ISSN 1786-0091

TANGENT BUNDLE OF THE HYPERSURFACES IN A EUCLIDEAN SPACE

SHARIEF DESHMUKH, HAILA AL-ODAN AND TAHANY A. SHAMAN

Abstract. We consider an immersed orientable hypersurface f: M → Rⁿ⁺¹ of the Euclidean space (f an immersion), and observe that the tan- gent bundleT M of the hypersurfaceM is an immersed submanifold of the Euclidean spaceR²ⁿ⁺². Then we show that in general the induced metric on T M is not a natural metric and obtain expressions for the horizontal and vertical lifts of the vector fields onM. We also study the special case in which the induced metric on T M becomes a natural metric and show that in this case the tangent bundleT M is trivial.

1. Introduction

The geometry of the tangent bundle T M of a Riemannian manifold is an interesting field in differential geometry. The first attempt to define a Rie- mannian metric on T M was made by Sasaki [8], and since then the tangent bundle has become focus of study with this metric. Specially after the work of Dombrowoski [2], who has introduced a nice theory of linking the geometry of the tangent bundle with Sasaki metric to the geometry of the base man- ifold, many mathematicians have studied the geometry of the tangent bun- dle through various aspects (cf. the survey article [3] and references therein).

Since there is a naturally associated almost complex structureJ to the tangent bundle T M of a Riemannian manifold M, one naturally expects fairly good properties associated to this almost complex structure vis-a-vis the complex geometry. However, the Sasaki metric on T M offers a significant obstruction on the almost complex structure and does not even allow it to be a complex unless the base manifold is flat. This deficiency in the Sasaki metric lead math- ematicians to search for other metrics on the tangent bundle other than Sasaki metric, for instance Cheeger-Gromoll metric, Oproiu metric (cf. [1], [3], [7],

2000Mathematics Subject Classification. 53C25, 53C55.

Key words and phrases. Tangent bundle, hypersurfaces, submanifolds, trivial tangent bundle.

71

[9]). This lead to the class of metrics onT M which make the natural submer- sion π: T M → M into a Riemannian submersion and this class of metrics is known as natural metrics. In this paper we are interested in the tangent bundle T M of an immersed orientable hypersurface M in the Euclidean space Rⁿ⁺¹. Iff: M →Rⁿ⁺¹is the smooth immersion which makesM as an immersed hy- persurface ofRⁿ⁺¹, then we show that the smooth map F =df: T M →R²ⁿ⁺² is also an immersion, thereby makingT M a submanifold of R²ⁿ⁺² and conse- quently has an induced metricg. We study the Riemannian manifold (T M, g) as submanifold of the Euclidean space (R²ⁿ⁺²,h,i) and first show that in gen- eral the induced metricg is not a natural metric by calculating the horizontal and vertical lifts of vector fields on M to T M. Then we consider a special case, in which the metric g becomes a natural metric and observe that in this case the tangent bundle T M is trivial.

2. Preliminaries

Let (M, g) be a Riemannian manifold and T M be its tangent bundle with projection map π: T M →M. Then for each (p, u) ∈T M, the tangent space T_(p,u)T M =H_(p,u)⊕V_(p,u), where V_(p,u) is kernel of dπ_(p,u):T_(p,u)T M →T_pM and H_(p,u) is the kernel of the connection map K_(p,u): T_(p,u)T M → T_pM with respect to the Riemannian connection on (M, g). The subspaces H_(p,u), V_(p,u) are called the horizontal and vertical subspaces respectively. Consequently the Lie algebra of smooth vector fields X(T M) on the tangent bundle T M admits the decomposition X(T M) = H⊕V, where H is called the horizontal distribution and V is called the vertical distribution on the tangent bundle T M. For each X_p ∈T_pM, the horizontal lift ofX_p to a pointz = (p, u)∈T M is the unique vector X_z^h ∈ H_z such that dπ(X_z^h) = X_p ◦ π and the vertical lift of Xp to a point z = (p, u) ∈ T M is the unique vector X_z^v ∈ Vz such that X_z^v(df) = Xp(f) for all functions f ∈ C^∞(M), where df is the function defined by (df)(p, u) =u(f). Also for a vector fieldX ∈X(M), the horizontal lift of X is a vector field X^h ∈ X(T M) whose value at a point (p, u) is the horizontal lift of X(p) to (p, u), the vertical lift X^v of X is defined similarly.

ForX ∈X(M) the horizontal and vertical liftsX^h, X^v of X are the uniquely determined vector fields on T M satisfying

dπ(X_z^h) =X_π(z), K(X_z^h) = 0_π(z), dπ(X_z^v) = 0_π(z), K(X_z^v) = X_π(z)

Also we have for a smooth function f ∈ C^∞(M) and vector fields X, Y ∈ X(M), that, (f X)^h = (f ◦π)X^h, (f X)^v = (f ◦π)X^v, (X +Y)^h = X^h +Y^h and (X +Y)^v = X^v +Y^v. If dimM = m and (U, φ) is a chart on M with local coordinates x¹, x², . . . , x^m, then (π⁻¹(U),Φ) is a chart on¯ T M with local coordinatesx¹, . . . , x^m, y¹, . . . , y^m, wherexⁱ =xⁱ◦πandyⁱ =dxⁱ,i= 1, . . . , m.

Throughout this paper we use Einstein summation, that is, the repeated indices are summed on their range. For horizontal and vertical lifts we have

Lemma 2.1 ([3]). Let (M, g) be a Riemannian manifold and X, Z ∈ X(M) which locally are represented by X = ξ^{i ∂}_∂xi and Z = η^{i ∂}_∂xi. Then the vertical and horizontal lifts X^v and X^h of X at the point Z ∈T M are given by

(X^v)_Z =ξⁱ ∂

∂yⁱ, (X^h)_Z =ξⁱ ∂

∂xⁱ −ξ^jη^kΓⁱ_jk ∂

∂yⁱ

where the coefficients Γⁱ_jk are the Christoffel symbols of the connection ∇ on (M, g).

A Riemannian metric ¯g on the tangent bundle T M is said to be natural metric with respect tog onM if ¯g(p,u)(X^h, Y^h) =gp(X, Y) and ¯g(p,u)(X^h, Y^v) = 0, for all vector fields X, Y ∈ X(M) and (p, u) ∈ T M, that is the projection map π: T M →M is the Riemannian submersion [6]

3. Tangent bundle of the hypersurface

Let M be an immersed hypersurface of the Euclidean space (Rⁿ⁺¹,h,i), where h,i is the Euclidean metric, with the immersion f: M → Rⁿ⁺¹. Then we have the smooth maps

F =df: T M →R²ⁿ⁺²,eπ: R²ⁿ⁺² →Rⁿ⁺¹

defined by F(p, X_p) = (f(p), df_p(X_p)) and πe(x, y) = x for x, y ∈ Rⁿ⁺¹, where df_p: T_pM →R is the differential of the mapf atp∈M. Clearlyf◦π =πe◦F holds, where π: T M → M is the projection of the tangent bundle. We have for the submersion eπ: (R²ⁿ⁺²,h,i) → (Rⁿ⁺¹,h,i), as πe is linear deπ_p = eπ, p ∈ R²ⁿ⁺², which implies that the vertical space ¯V_p = kerdeπ_p = (0, Rⁿ⁺¹) and since ¯Hp ⊥ V¯p we get ¯Hp = R²ⁿ⁺²/V¯p = (Rⁿ⁺¹,0). Also we see that deπ preserves lengths of horizontal vectors, that is, hX, Yi = hdeπ(X), deπ(Y)i for X, Y ∈ H¯ where deπ = £

I(n+1)×(n+1) 0(n+1)×(n+1)

¤ consequently it follows that e

π: (R²ⁿ⁺²,h,i)→(Rⁿ⁺¹,h,i) is a Riemannian submersion (cf. [6]).

Ifx¹, . . . , xⁿ are the local coordinates onM then the corresponding coordi- nates on T M are x¹, x², . . . , xⁿ, y¹, y², . . . , yⁿ where xⁱ = xⁱ◦π, yⁱ = dxⁱ, i = 1, . . . , n. Similarly if u¹, . . . , uⁿ⁺¹ are the local coordinates on Rⁿ⁺¹ then we get a corresponding coordinates u¹, . . . , uⁿ⁺¹, v¹, . . . , vⁿ⁺¹ on R²ⁿ⁺² where we know that

µ ∂

∂uⁱ

¶_v

= ∂

∂vⁱ µ ∂

∂uⁱ

¶_h

= ∂

∂uⁱ, i= 1, . . . , n+ 1.

Let us denote byD,D¯ the Euclidean connections onRⁿ⁺¹, R²ⁿ⁺² respectively, then recall that the connection coefficients (Christoffel symbols) Γ^k_ij of the Euclidean connections are zero.

For the Riemannian submersion eπ: R²ⁿ⁺² →Rⁿ⁺¹ we have the following:

Theorem 3.1. eπ: R²ⁿ⁺² → Rⁿ⁺¹ is the Riemannian submersion with totally geodesic fibersRⁿ⁺¹, that is, T = 0. The tensor fieldA onR²ⁿ⁺² also vanishes.

Proof. Recall that for E, F ∈X(R²ⁿ⁺²) we have [6]

T_EF =H( ¯D_VEVF) +V( ¯D_VEHF) A_EF =V( ¯D_HEHF) +H( ¯D_HEVF).

LetE =X+U, F =Y +V where X, Y ∈H,U, V¯ ∈V,, that is¯ X =a^{i ∂}_∂ui, Y =b^{i ∂}_∂ui, U =c^{i ∂}_∂vi and V =d^{i ∂}_∂vi. Then we have

T_EF =H( ¯D_UV) +V( ¯D_UY)

=H(U(dⁱ) ∂

∂vⁱ) +V(U(bⁱ) ∂

∂uⁱ) = 0, A_EF =V( ¯D_XY) +H( ¯D_XV)

=V(X(bⁱ) ∂

∂uⁱ) +H(X(dⁱ) ∂

∂vⁱ) = 0.

¤ The following theorem is a consequence of the fact that an immersion of M inN induces an immersion of T M in T N, yet we sketch the proof for the sake of our need for an explicit expression for the differential of the induced immersion of T M in T N.

Theorem 3.2. The map F: T M →R²ⁿ⁺² is an immersion.

Proof. Let p∈ M and P = (p, X_p)∈T M, then we have for local coordinates x¹, . . . , xⁿaroundp,X_p =yⁱ(P)(_∂x^∂i)_pandF(P) = df(p, X_p) = (f(p), df_p(X_p)).

The matrix for df_p: T_pM →T_f(p)Rⁿ⁺¹ is the (n+ 1)×n matrix.

df_p =





∂f¹

∂x¹(p) · · · · _∂xn^∂f1(p) ... ... ... ...

∂fⁿ⁺¹

∂x¹ (p) · · · · ^∂f_∂xnⁿ⁺¹(p)





where f^α =u^α◦f, α = 1, . . . , n+ 1. This gives df_p(X_p) =





∂f¹

∂xⁱ(p)yⁱ(P) ...

∂fⁿ⁺¹

∂xⁱ (p)yⁱ(P)





consequently

F(P) = (f¹(p), f²(p), . . . , fⁿ⁺¹(p),∂f¹

∂xⁱ(p)yⁱ(P), . . . ,∂fⁿ⁺¹

∂xⁱ (p)yⁱ(P)) that is

F = (f¹◦π, f²◦π, . . . , fⁿ⁺¹◦π,(∂f¹

∂xⁱ ◦π)yⁱ, . . . ,(∂fⁿ⁺¹

∂xⁱ ◦π)yⁱ).

Thus the matrix fordF_P: T_P(T M)→T_F(P)(R²ⁿ⁺²) is the (2n+2)×2nmatrix.

dF_P =













∂F¹

∂x¹(P) · · · ^∂F_∂xn¹(P) ^∂F_∂y1¹(P) · · · ^∂F_∂yn¹(P)

... ... ... ... ... ...

∂Fⁿ⁺¹

∂x¹ (P) · · · ^∂F_∂xⁿ⁺¹n (P) ^∂F_∂yⁿ⁺¹1 (P) · · · ^∂F_∂yⁿ⁺¹n (P)

∂Fⁿ⁺²

∂x¹ (P) · · · · · · · · · · · · ^∂F_∂yⁿ⁺²n (P)

... ... ... ... ... ...

∂F²ⁿ⁺²

∂x¹ (P) · · · · · · · · · · · · ^∂F_∂y²ⁿ⁺²n (P)













Note that for α= 1, . . . , n+ 1 andj = 1, . . . , nwe have:

∂F^α

∂x^j (P) = ∂(f^α◦π)

∂x^j (p) = ∂f^α

∂x^j(p),

∂F^n+1+α

∂y^j (P) = ∂((^∂f_∂x^αi ◦π)yⁱ)

∂y^j (P) = ∂f^α

∂x^j(p),

∂F^α

∂y^j (P) = ∂f^α

∂y^j(p) = 0,

∂F^n+1+α

∂x^j (P) = ∂((^∂f_∂x^αi ◦π)yⁱ)

∂x^j (P) = ∂²f^α

∂x^j∂xⁱ(p)yⁱ(P) thus we arrive at

dF_P =

· df_p(n+1)×n 0_(n+1)×n

(_∂x^∂j²∂x^fik(p)y^k(P))_(n+1)×n df_p(n+1)×n

¸ .

Hence dF_P has rank 2n that is F: T M →R²ⁿ⁺² is an immersion. ¤ Thus the tangent bundleT M of the hypersurfaceM of the Euclidean space Rⁿ⁺¹is a submanifold ofR²ⁿ⁺². We denote the induced Riemannian metrics on M and T M respectively by g and g respectively. Also we denote by ∇,∇¯ the Riemannian connections on M, T M respectively. We denote by N the unit normal vector field of the orientable hypersurface M. For the hypersurface M of the Euclidean space Rⁿ⁺¹ we have the following Gauss and Weingarten formulae

D_XY = ∇_XY +hS(X), YiN (1)

D_XN = −S(X) (2)

where X, Y ∈ X(M) and S denotes the Weingarten map S: X(M) → X(M).

Similarly for the submanifold T M of the Euclidean space R²ⁿ⁺² we have the Gauss and Weingarten formulae:

D_XY = ¯∇_XY +h(X, Y) (3)

D_XNˆ = −S_Nˆ(X) +∇^⊥_XNˆ (4)

whereX, Y ∈X(T M) and S_Nˆ denotes the Weingarten map in the direction of the normal ˆN which is SNˆ: X(T M) → X(T M), and is related to the second

fundamental form h by D

h(X, Y),Nˆ E

=­S¯Nˆ(X), Y® .

Also we observe that for X ∈ X(M) the vertical lift X^v of X to T M, as X^v ∈kerdπ we havedπ(X^v) = 0 that isdf(dπ(X^v)) = 0 or equivalently we get d(f◦π)(X^v) = 0, that is d(eπ◦F)(X^v) = 0 which givesdF(X^v)∈kerdeπ =V.

Moreover we have the following lemmas:

Lemma 3.1. For P = (p, X_p)∈T M

dFP(X_P^v) = (dfp(Xp))^v.

Proof. For X =ξ^{i ∂}_∂xi we know that X_P^v =ξ^{i ∂}_∂yi. Thus we have

dF_P(X_P^v) =











 0...

0

∂f¹

∂xⁱ(p)ξⁱ ...

∂fⁿ⁺¹

∂xⁱ (p)ξⁱ













and on the other hand

df_p(X_p) =





∂f¹

∂xⁱ(p)ξⁱ ...

∂fⁿ⁺¹

∂xⁱ (p)ξⁱ



.

Thus we get (df_p(X_p))^v =dF_P(X_P^v). ¤

Remark. On a Riemannian manifold (M, g) for a smooth functionf ∈C^∞(M), the Hessian of the function f is defined by Hf(X, Y) = X(Y(f))− ∇XY(f), X, Y ∈X(M), where∇is the Riemannian connection onM. IfX =ξ^{i ∂}_∂xi and Y =η^{j ∂}_∂xj then we have

H_f(X, Y) =X(η^j ∂f

∂x^j)−ξⁱ(∇ ^∂

∂xiη^j ∂

∂x^j)(f)

=X(η^j)∂f

∂x^j +η^jX(∂f

∂x^j)−ξⁱη^jΓ^k_ij ∂f

∂x^k −ξⁱ∂η^j

∂xⁱ

∂f

∂x^j

=ξⁱη^j ∂²f

∂xⁱ∂x^j −ξⁱη^jΓ^k_ij ∂f

∂x^k

where Γ^k_ij are the Christoffel symbols for the Riemannian connection. Thus at a pointp if X_p =λⁱ(_∂x^∂i)_p and Y_p =µ^j(_∂x^∂j)_p we have

Hf(Xp, Yp) = λⁱµ^j ∂²f

∂xⁱ∂x^j(p)−λⁱµ^jΓ^k_ij(p)∂f

∂x^k(p).

Lemma 3.2. Let N be the unit normal vector field to the hypersurface M and P = (p, Xp)∈T M. Then the horizontal lift Y_P^h of Yp ∈TpM satisfies

dFP(Y_P^h) = (dfp(Yp))^h+VP

where V_P ∈V_P is given by V_P =hS_p(X_p), Y_piN_P^v. Proof. Since

dF_P =







df_p 0

∂²f¹

∂x¹∂x^k(p)y^k(P) · · · _∂x^∂n^2f∂x^1k(p)y^k(P)

... ... ...

∂²fⁿ⁺¹

∂x¹∂x^k(p)y^k(P) · · · _∂x^∂2fnⁿ⁺¹∂x^k(p)y^k(P) df_p







for X_p = ξⁱ¡ _∂

∂xⁱ

¢

p and Y_p = η^j¡ _∂

∂x^j

¢

p as Y_P^h = ηⁱ¡ _∂

∂xi

¢

P −ξ^kη^jΓⁱ_jk(p)

³ ∂

∂yⁱ

´ we have P

dFP(Y_P^h) =







df_p(Y_p)

∂²f¹

∂x^α∂x^k(p)y^k(P)η^α−ξ^kη^jΓ^α_jk(p)_∂x^∂fα¹(p) ...

∂²fⁿ⁺¹

∂x^α∂x^k(p)y^k(P)η^α−ξ^kη^jΓ^α_jk(p)^∂f_∂xⁿ⁺¹α (p)







=

· df_p(Y_p) 0

¸ +







0 H_f¹(Y_p, X_p)

...

H_fⁿ⁺¹(Y_p, X_p)





.

(Note thatX_p =ξⁱ(_∂x^∂i)_p =yⁱ(P)(_∂x^∂i)_p, that is,ξⁱ =yⁱ(P)). Consequently we get

dF_P(Y_P^h) = (df_p(Y_p))^h+V_P

where V_P ∈ V_P and V_P = H_f^α(Y_p, X_p)_∂v^∂α. We know that to compute the horizontal liftY_P^h atP = (p, Xp) we need to assume that∇YX = 0 (that is,X is parallel along integral curves ofY) (cf. [3], p.8 ). Thus we have from Gauss equation

D_df(Y)df(X) =∇_YX+hS(X), Y)iN =hS(X), Y)iN.

Now for df(X) =λ^{α ∂}_∂uα, λ^α = df(X)(u^α) = X(u^α◦f) = X(f^α) and that D being Euclidean connection:

D_df(Y)df(X) =Y X(f^α) ∂

∂u^α

= (Y X(f^α)−(∇_YX)(f^α)) ∂

∂u^α

=Hf^α(Y, X) ∂

∂u^α.

Thus

H_f^α(Y, X) ∂

∂u^α =hS(X), Y)iN =hS(X), Y)i(h^α ∂

∂u^α) impliesH_f^α(Y, X) =hS(X)Y)ih^α that is

V_P = µ

hS(X), Y)ih^α ∂

∂v^α

¶

(P) = hS_p(X_p), Y_piN_P^v

¤ Lemma 3.3. Let N = (N,0)∈X(R²ⁿ⁺²), where N is the unit normal vector field of the hypersurface M in Rⁿ⁺¹. Then

(1) N =N^h.

(2) N is a normal vector field to T M as a submanifold of R²ⁿ⁺².

Proof. 1. We denote by H and V the horizontal and vertical distributions of the tangent bundle T Rⁿ⁺¹. Then clearly N ∈ H, which implies K(N) = 0, where K is the connection map of the connection D, and since the matrix of deπ is deπ = [I 0], we get deπ(N) = N ◦eπ. This proves N^h = N. Note that we can prove this part from the known formula for the horizontal lift given in Lemma 2.1 as follows:

SinceN =h^{α ∂}_∂uα and Γ^k_jifor the connectionDvanish,N^h = (h^α◦eπ)_∂u^∂α =N. 2. It is enough to prove that for anyX, Y ∈X(M)

­dF(X^h), N®

= 0 and ­

dF(Y^v), N®

= 0.

Now since eπ is a Riemannian submersion we have

­dF(X^h), N®

=­

(df(X))^h, N^h®

=­

deπ(df(X))^h), deπ(N^h)®

=hdf(X), Ni ◦πe= 0

asdf(X)∈X(Rⁿ⁺¹) andN be the normal vector field to M inRⁿ⁺¹. Also by Lemma 3.1 since dF(Y^v) = (df(Y))^v we have ­

dF(Y^v), N®

= 0. This proves

that N is normal vector field to T M. ¤

Remark. The Euclidean space R²ⁿ⁺² has natural complex structure J, and if we put ˜N =JN then from the definition of J we have ˜N =JN^h =N^v. Now forX, Y ∈X(M) we have

D

dF(X^h),N˜ E

=­

(df(X))^h+V, N^v®

=hV, N^vi and D

dF(Y^v),N˜ E

=h(df(Y))^v, N^vi. Let Y =η^{j ∂}_∂xj, then we have df(Y) = (∂f^α

∂xⁱηⁱ) ∂

∂u^α, (df(Y))^v = ((∂f^α

∂xⁱηⁱ)◦eπ) ∂

∂v^α and N^v = (h^α◦eπ)_∂v^∂α which implies

D

dF(Y^v),N˜ E

= ((∂f^α

∂xⁱηⁱ)h^α)◦eπ =hdf(Y), Ni ◦πe= 0.

But since hV, N^vi 6= 0 in general, so ˜N can not be a normal vector field to T M.

We choose N^∗ as a unit normal vector field to T M in R²ⁿ⁺² which is or- thogonal toN so that for X, Y ∈X(T M) we have

h(X, Y) =­

h(X, Y), N®

N+hh(X, Y), N^∗iN^∗ =­S¯_NX, Y®

N+­S¯_N^∗X, Y® N^∗. Lemma 3.4. The unit normal N^∗ to T M is a vertical vector field on the tangent bundle T Rⁿ⁺¹.

Proof. Take U ∈ X(Rⁿ⁺¹) |M. Then we can express it as U = df(X) +ϕN, ϕ∈C^∞(M), X ∈X(M), consequently we have

(5) U^h = (df(X))^h+ (ϕ◦π)N =dF(X^h)−Vp+ (ϕ◦π)N

Now sincedF(X^h) = (df(X))^h+V,if (df(X))^h =Y^h+bN andV_p =γN^vwhere Y^h, bN are the tangential and normal components of (df(X))^h respectively and γ =g(S(X), Y). We havedF(X^h) =Y^h+bN +γN^v, wherebN +γN^v must be tangential toT M (asdF(X^h) is tangent to T M). Thus g(bN +γN^v, N^∗) = 0 which implies γg(N^v, N^∗) = 0. Also g(bN +γN^v, N) = 0 proves b = 0, that is γN^v =V_p must be tangential. Taking inner product in equation (3.5) with N^∗, we get ­

U^h, N^∗®

= 0 for eachU ∈ X(Rⁿ⁺¹) |_Mwhich implies N^∗ must be

vertical. ¤

Lemma 3.5. For X ∈X(M) and N = (N,0)∈X(R²ⁿ⁺²) we have D¯_X^hN = (D_XN)^h and D¯_X^vN = 0.

Proof. Expressing locally N = (h^α◦eπ)_∂u^∂α, h^α ∈C^∞(Rⁿ⁺¹) we compute D¯_X^hN = (dF(X^h))(h^α◦π)e ∂

∂u^α = (df(X))^h(h^α◦eπ) ∂

∂u^α

= ((df(X))^h(h^α)◦eπ) ∂

∂u^α

On the other hand we haveD_XN = (df(X))(h^α)_∂u^∂α and (D_XN)^h = ((df(X))^h(h^α)◦eπ) ∂

∂u^α = ¯D_X^hN For the second relation we have

D¯_X^vN = (dF(X^v))(h^α◦eπ) ∂

∂u^α = ((df(X))^v(h^α◦eπ) ∂

∂u^α = 0.

¤ Corollary 3.1. For X ∈X(M) we have (S(X))^h =−D¯_X^hN.

Proof. From equation (3.2) and Lemma 3.5 we have (S(X))^h =−(D_XN)^h =

−D¯_X^hN. ¤

Example. TakeM =S²andf:S² →R³the inclusionf(z¹, z², z³) = (z¹, z², z³) for (z¹, z², z³)∈S². Letp= (z¹, z², z³) ∈S² be a point with z³ >0 and take a chart (U, φ) around p whereU ={(z¹, z², z³)∈S² :z³ >0} and

φ:U →B1(0)⊂R²,φ(z¹, z², z³) = (z¹, z²), φ⁻¹(u¹, u²) = (u¹, u²,p

1−(u¹)²−(u²)²).

Letx¹, x² be the local coordinates on U and u¹, u², u³ be the Euclidean coor- dinates on R³. Then

f^α =u^α◦f =z^α, α= 1,2,3

∂fⁱ

∂x^j(p) =δ_jⁱ, i, j = 1,2.

∂f³

∂xⁱ(p) = ∂(f³◦φ⁻¹)

∂uⁱ (φ(p)) = ∂(p

1−(u¹)²−(u²)²)

∂uⁱ (φ(p))

= −uⁱ

p1−(u¹)²−(u²)²(φ(p))

= −zⁱ

z³ , i= 1,2 and

df_p =



 1 0

0 1

−z¹ z³ −z²

z³



.

Now let P = (p, X_p) whereX =ξ^{i ∂}_∂xi ∈X(S²), then

F =df = (f¹◦π, f²◦π, f³◦π, y¹, y²,−uⁱ u³yⁱ)

where x¹, x², y¹, y² are the local coordinates with respect to the chart onT S² corresponding to (U, φ) on S². We get

∂F^3+α

∂xⁱ (P) = ∂(y^α)

∂xⁱ (P) = 0, α, i= 1,2

∂F⁶

∂x^j (P) =−∂(^u_u^iy3ⁱ)

∂x^j (P) = −(u³yⁱδ_jⁱ −uⁱyⁱ(^−u_u3^j) (u³)² )(P)

= −(u³)²y^j−P

iuⁱyⁱu^j

(u³)³ (P) = −(z³)²ξ^j −zⁱξⁱz^j

(z³)³ j = 1,2.

Note thaty^α(P) =ξ^α, α= 1,2, so we get

dFP =











1 0 0 0

0 1 0 0

−z¹

z³ −z²

z³ 0 0

0 0 1 0

0 0 0 1

−(z³)²ξ¹ −zⁱξⁱz¹ (z³)³

−(z³)²ξ² −zⁱξⁱz² (z³)³

−z¹ z³

−z² z³









 .

Now for Y =η^{i ∂}_∂xi ∈X(S²) we have Y^h =η^{i ∂}_∂xi −η^jξ^kΓⁱ_jk∂y^∂i consequently

dFP(Y_P^h) =











η¹ η²

−z¹η¹−z²η² z³

−η^jξ^kΓ¹_jk

−η^jξ^kΓ²_jk

{(^−(z3)2_(z^ξα3)^{−z3 iξizα})η^α+^z_z^α3(η^jξ^kΓ^α_jk)}











that is

dF_P(Y_P^h) =

· (df(Y_p))^h 0

¸ +







0

−η^jξ^kΓ¹_jk

−η^jξ^kΓ²_jk

{(^−(z3)2_(z^ξα3)^{−z3 iξizα})η^α+ ^z_z^α3(η^jξ^kΓ^α_jk)}





.

Now we need to compute the connection coefficients of Γ^k_ji of the connection

∇ with respect to this chart onS². Since

∂

∂xⁱ = ∂

∂uⁱ − uⁱ u³

∂

∂u³ we get for i, j = 1,2

gij = g µ ∂

∂xⁱ, ∂

∂x^j

¶

=

¿ ∂

∂xⁱ, ∂

∂x^j À

=

¿ ∂

∂uⁱ − uⁱ u³

∂

∂u³, ∂

∂u^j − u^j u³

∂

∂u³ À

=δ_jⁱ + uⁱu^j (u³)² and consequently

(g_ij) =



 1 +

³u¹ u³

´₂

u¹u² (u³)² u¹u²

(u³)² 1 +³

u² u³

´₂





and ¡

g^ij¢

=

· (u³)² + (u²)² −u¹u²

−u¹u² (u³)²+ (u¹)²

¸ .

Using

Γ^k_ij = 1 2g^αk

½∂g_iα

∂u^j − ∂g_ij

∂u^α + ∂g_αj

∂uⁱ

¾

we arrive at

Γ¹₁₁ = u¹((u³)²+ (u¹)²) (u³)² Γ²₁₁ = u²((u³)²+ (u¹)²)

(u³)² Γ¹₁₂ = Γ¹₂₁= (u¹)²u²

(u³)² Γ²₁₂ = Γ²₂₁= (u²)²u¹

(u³)² Γ¹₂₂ = u¹((u³)²+ (u²)²)

(u³)² Γ²₂₂ = u²((u³)²+ (u²)²)

(u³)² which gives

dF_P(Y_P^h) =

· (df(Y_p))^h 0

¸ +









0 0 0

−u¹hX, Yi

−u²hX, Yi

−u³hX, Yi









where N =u^{α ∂}_∂uα ∈X(R³) is the unit normal vector field to S² and hX, Yi=η¹ξ¹+η²ξ²+ 1

(u³)²(η¹u¹+η²u²)(ξ¹u¹+ξ²u²).

Remark. We observe that the metrics defined on T M using the Riemannian metric of M (such as Sasaki metric, Cheeger-Gromoll metric, Oproiu metric) are natural metrics in the sense that the submersion π: T M →M becomes a Riemannian submersion with respect to these metrics. However, the induced metric on the tangent bundleT M of a hypersurfaceM of the Euclidean space Rⁿ⁺¹, as a submanifold ofR²ⁿ⁺² is not a natural metric because of the presence of the term VP (see Lemma 3.2).

4. A special case

In this section we study the hypersurfacesf: M →Rⁿ⁺¹ satisfying dF_P(X_P^h) = (df_p(X_p))^h,