Loading [MathJax]/jax/output/SVG/jax.js
Research article Topical Sections

On eigenfunctions corresponding to first non-zero eigenvalue of the sphere Sn(c) on a Riemannian manifold

  • We recall classical themes such as "on hearing the shape of a drum" or "can one hear the shape of a drum?", and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under which an n-dimensional closed Riemannian manifold (Mn,g) having a non-zero eigenvalue nc for a positive constant c (that is, has same non-zero eigenvalue as first non-zero eigenvalue of the sphere Sn(c)), is isometric to Sn(c). We address this issue in two situations. First, we consider the compact (Mn,g) as the hypersurface of the Euclidean space (Rn+1,,) with isometric immersion f:(Mn,g) (Rn+1,,) and a constant unit vector a such that the function ρ=f,a satisfying Δρ=ncρ for a positive constant c is isometric to Sn(c) if and only if (Mn,g) is isometric to Sn(c) provided the integral of Ricci curvature Ric(ρ,ρ) has an appropriate lower bound. In the second situation, we consider that the compact (Mn,g) admits a non-trivial concircular vector field ξ with potential function σ satisfying Δσ=ncσ for a positive constant c and a specific function f related to ξ (called circular function) is constant along the integral curves of ξ if and only if (Mn,g) is isometric to Sn(c).

    Citation: Sharief Deshmukh, Amira Ishan, Olga Belova. On eigenfunctions corresponding to first non-zero eigenvalue of the sphere Sn(c) on a Riemannian manifold[J]. AIMS Mathematics, 2024, 9(12): 34272-34288. doi: 10.3934/math.20241633

    Related Papers:

    [1] Yanlin Li, Nasser Bin Turki, Sharief Deshmukh, Olga Belova . Euclidean hypersurfaces isometric to spheres. AIMS Mathematics, 2024, 9(10): 28306-28319. doi: 10.3934/math.20241373
    [2] Nasser Bin Turki, Sharief Deshmukh, Olga Belova . A note on closed vector fields. AIMS Mathematics, 2024, 9(1): 1509-1522. doi: 10.3934/math.2024074
    [3] Meraj Ali Khan, Ali H. Alkhaldi, Mohd. Aquib . Estimation of eigenvalues for the α-Laplace operator on pseudo-slant submanifolds of generalized Sasakian space forms. AIMS Mathematics, 2022, 7(9): 16054-16066. doi: 10.3934/math.2022879
    [4] Sharief Deshmukh, Mohammed Guediri . Some new characterizations of spheres and Euclidean spaces using conformal vector fields. AIMS Mathematics, 2024, 9(10): 28765-28777. doi: 10.3934/math.20241395
    [5] Songting Yin . Some rigidity theorems on Finsler manifolds. AIMS Mathematics, 2021, 6(3): 3025-3036. doi: 10.3934/math.2021184
    [6] Ibrahim Aldayel . Value of first eigenvalue of some minimal hypersurfaces embedded in the unit sphere. AIMS Mathematics, 2023, 8(11): 26532-26542. doi: 10.3934/math.20231355
    [7] Amira Ishan . On concurrent vector fields on Riemannian manifolds. AIMS Mathematics, 2023, 8(10): 25097-25103. doi: 10.3934/math.20231281
    [8] Ibrahim Al-Dayel, Meraj Ali Khan . Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting nearly Sasakian structure. AIMS Mathematics, 2021, 6(3): 2132-2151. doi: 10.3934/math.2021130
    [9] Hanan Alohali, Sharief Deshmukh . Some generic hypersurfaces in a Euclidean space. AIMS Mathematics, 2024, 9(6): 15008-15023. doi: 10.3934/math.2024727
    [10] Ibrahim Al-Dayel, Sharief Deshmukh, Olga Belova . Characterizing non-totally geodesic spheres in a unit sphere. AIMS Mathematics, 2023, 8(9): 21359-21370. doi: 10.3934/math.20231088
  • We recall classical themes such as "on hearing the shape of a drum" or "can one hear the shape of a drum?", and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under which an n-dimensional closed Riemannian manifold (Mn,g) having a non-zero eigenvalue nc for a positive constant c (that is, has same non-zero eigenvalue as first non-zero eigenvalue of the sphere Sn(c)), is isometric to Sn(c). We address this issue in two situations. First, we consider the compact (Mn,g) as the hypersurface of the Euclidean space (Rn+1,,) with isometric immersion f:(Mn,g) (Rn+1,,) and a constant unit vector a such that the function ρ=f,a satisfying Δρ=ncρ for a positive constant c is isometric to Sn(c) if and only if (Mn,g) is isometric to Sn(c) provided the integral of Ricci curvature Ric(ρ,ρ) has an appropriate lower bound. In the second situation, we consider that the compact (Mn,g) admits a non-trivial concircular vector field ξ with potential function σ satisfying Δσ=ncσ for a positive constant c and a specific function f related to ξ (called circular function) is constant along the integral curves of ξ if and only if (Mn,g) is isometric to Sn(c).



    The Laplace operator Δ acting on smooth functions of a Riemannian manifold (Mn,g), is defined by Δf=div(f), where f is the gradient of f. It is known that Δ is a self adjoint elliptic operator with respect to the inner product (,) defined on the algebra of smooth functions C(Mn) with compact support by

    (f,h)=MnfhdVg,

    where dVg is the volume form on Mn with respect to the metric g. If Δf=λf, for a constant λ, then λ is said to be an eigenvalue of the Laplace operator Δ, the negative sign in the definition chosen so that for non-zero eigenvalue λ, λ>0. The set of eigenvalues λi of the Laplace operator Δ on a Riemannian manifold (Mn,g) is called the spectrum of (Mn,g). Spectra of known Riemannian manifolds such as the sphere Sn(c) and the real projective space RPn are known, and a nice description could be found in [2] (Chapter 2, Section 5). Two Riemannian manifolds of same dimension (Mn,g) and (¯Mn,¯g) having the same spectra are said to be isospectral manifolds, whereas they are said to be isometric if there exists a diffeomorphism ϕ:Mn¯Mn that preserves the metric ϕ(¯g)=gϕ, that is, ϕ is an isometry. In the mid Nineteenth century, it was an open question whether isospectral Riemannian manifolds are isometric. We see that physicists followed this question under the topic "On hearing the shape of a drum" or "Can one hear the shape of a drum?" (cf. [5,11]), while Milnor constructed two flat tori in dimension 16, which are isospectral but not isometric (cf. [12]). In [12], the author constructed two 2-dimensional compact manifolds of constant negative curvature which are isospectral and not isometric. Further, Ejiri constructed two non-flat compact Riemannian manifolds which are isospectral but not isometric (cf. [6]). This initiated an interest in comparing spectra of two Riemannian manifolds, for further results in this direction, we refer to (cf. [1,7,8,9,10,15]). One of the natural questions is under what conditions two isospectral Riemannian manifolds are isometric? We know that the n-sphere Sn(c) has first non-zero eigenvalue λ1=nc, and, supposing an n-dimensional Riemannian manifold (Mn,g) also has non-zero eigenvalue λ1=nc, we reduce the above general question to the following specific question: Under what condition is an n-dimensional Riemannian manifold (Mn,g) that has non-zero eigenvalue λ1=nc for a positive constant c isometric to Sn(c)? In order to answer this question, naturally, we need a smooth function ρ on an n-dimensional Riemannian manifold (Mn,g) that satisfies Δρ=ncρ. In order to address this issue of finding a smooth function, we consider two situations:

    (ⅰ) Considering (Mn,g) as an isometrically immersed hypersurface in the Euclidean space (Rn+1,,) with isometric immersion f:(Mn,g)(Rn+1,,), though there are several known functions on the hypersurface (Mn,g), namely, the mean curvature function, the scalar curvature function, as well as the support function, but they cannot be eigenfunctions of the Laplace operator corresponding to a non-zero eigenvalue, while (Mn,g) is isometric to the sphere Sn(c). Then, we go for the alternative, namely, for a constant unit vector field a on the Euclidean space Rn+1, we get a smooth function ρ=f,a on the hypersurface (Mn,g), which we require to satisfy Δρ=ncρ, c>0.

    (ⅱ) We seek the Riemannian manifold (Mn,g), and there exists a non-trivial concircular vector field ξ with potential function ρ and we require that the potential function ρ satisfies Δρ=ncρ, c>0.

    In this article, we explore the above two situations that an n-dimensional Riemannian manifold (Mn,g) has an eigenvalue of the Laplace operator same as the first non-zero eigenvalue of the sphere Sn(c), and find an additional condition so that these two are isometric (See Theorems 1, 2, 3). In the language of physics, "yes, we could hear a node through an apparatus to predict the shape of a drum".

    On an n-dimensional Riemannian manifold (Mn,g), denote by the Riemannian connection. The curvature tensor R of (Mn,g) is given by

    R(X,Y)Z=[X,Y]Z[X,Y]ZX,Y,ZΓ(TMn), (2.1)

    where Γ(TMn) is the space of smooth sections of the tangent bundle TMn. The Ricci tensor Ric of (Mn,g) is a symmetric tensor given by

    Ric(X,Y)=nl=1g(R(el,X)Y,el)X,YΓ(TMn), (2.2)

    where {e1,..,en} is a local orthonormal frame (or lof) on (Mn,g). The Ricci operator Q of (Mn,g) is related to the Ricci tensor by

    Ric(X,Y)=g(QX,Y)X,YΓ(TMn)

    and therefore Q is a symmetric (1,1) tensor field. The scalar curvature τ of the Riemannian manifold (Mn,g) is given by

    τ=nl=1Ric(el,el).

    The following formula is well known (cf. [2]),

    12τ=nl=1(elQ)(el),

    where τ is the gradient of τ, and the covariant derivative is given by

    (XQ)(Y)=XQYQ(XY).

    Given a smooth function f:MnR on a Riemannian manifold (Mn,g), the Laplace operator Δ acts on f, given by

    Δf=div(f),

    where f is the gradient of f and

    divX=nl=1g(elX,el).

    If (Mn,g) is closed, then Stokes's theorem implies

    Mn(divX)dVg=0,

    where dVg is the volume element of (Mn,g).

    Given a symmetric (1,1) tensor field T on an n-dimensional Riemannian manifold (Mn,g) with trace t, that is,

    t=nl=1g(Tel,el),

    then the Cauchy–Schwartz inequality is

    T21nt2, (2.3)

    where

    T2=nl=1g(Tel,Tel).

    Moreover, the equality in (2.3) holds if and only if

    T=tnI,

    where I is the identity (1,1) tensor.

    Suppose on an n-dimensional closed Riemannian manifold (Mn,g) there is a smooth function f that satisfies Δf=ncf for a constant c. Then we have fΔf=ncf2. Then, integrating by parts, the last equation leads to

    Mnf2dVg=ncMnf2dVg. (2.4)

    In this section, we consider an n-dimensional Riemannian manifold (Mn,g) that admits an isometric immersion f:(Mn,g)(Rn+1,,) into the Euclidean space (Rn+1,,), where , is the Euclidean inner product. We denote by ς the unit normal to the hypersurface (Mn,g), and by B the shape operator with respect to the isometric immersion f. Then, denoting the Euclidean connection on (Rn+1,,) by ¯ and the Riemannian connection on the hypersurface (Mn,g) by , we have the following fundamental equations for the hypersurface (cf. [2])

    ¯XY=XY+gBX,YςX,YΓ(TMn) (3.1)

    and

    ¯Xς=BXXΓ(TMn). (3.2)

    The mean curvature β of the hypersurface (Mn,g) is given by

    β=1nnl=1g(Bel,el), (3.3)

    where {e1,..,en} is a local orthonormal frame on (Mn,g).

    The curvature tensor, the Ricci tensor, and the scalar curvature of the hypersurface (Mn,g) are given by (cf. [2])

    R(X,Y)Z=g(BY,Z)BXg(BX,Z)BXX,Y,ZΓ(TMn),
    Ric(X,Y)=nβg(BX,Y)g(BX.BY)X,YΓ(TMn)

    and

    τ=n2β2B2.

    Also, the shape operator B has the following wonderful property (Codazzi equation of the hypersurfaces in flat spaces)

    (XB)(Y)=(YB)(X)X,YΓ(TMn). (3.4)

    Using Eqs (3.3) and (3.4) as well as the symmetry of the shape operator, we have for any XΓ(TMn)

    nX(β)=nl=1g(XBel,el)+nl=1g(Bel,Xel)=nl=1g((XB)(el)+B(Xel),el)+nl=1g(Bel,Xel)=nl=1g((elB)(el),X)+2nl=1g(Bel,Xel).

    Note that Bel=jg(Bel,ej)ej and Xel=kωkl(X)ek, where g(Bel,ej) is symmetric while the connection forms ωkl are skew symmetric. Therefore,

    nl=1g(Bel,Xel)=jklg(Bel,ej)ωkl(X)g(ej,ek)=nl=1g(Bel,ej)ωjl(X)=0.

    Thus, above equation becomes

    nX(β)=nl=1g((elB)(el),X),

    that is, we have

    nl=1(elB)(el)=nβ. (3.5)

    Treating the isometric immersion f:(Mn,g)(Rn+1,,) as a position vector of points of Mn in Rn+1, and defining σ=f,ς, called the support function of the hypersurface (Mn,g), we express f as

    f=ξ+σς,

    where ξΓ(TMn) is tangential to (Mn,g). Differentiating equation (2.4), while using Eqs (3.1) and (3.2), we have upon equating the tangential and normal parts

    Xξ=X+σBX (3.6)

    and

    σ=Bξ.

    Taking a constant unit vector field a on the Euclidean space Rn+1 (for instance a coordinate vector field), we define a smooth function h on the hypersurface (Mn,g), by h=a,ς. Denoting the tangential part of a to the hypersurface (Mn,g) by ζ, we have

    a=ζ+hς. (3.7)

    Differentiating the above equation with respect to XΓ(TMn), while using Eqs (3.1) and (3.2), we have upon equating the tangential and normal parts

    Xζ=hBXXΓ(TMn) (3.8)

    and

    h=Bζ. (3.9)

    Now, we prove the main result of this section.

    Theorem 1. An n-dimensional compact and connected isometrically immersed hypersurface f:(Mn,g)(Rn+1,,) in the Euclidean space (Rn+1,,) with mean curvature β and a constant unit vector a=ζ+hς on Rn+1, where the function ρ=f,a satisfies Δρ=ncρ for a positive constant c, is isometric to the sphere Sn(c) if and only if the Ricci curvature Ric(ζ,ζ) satisfies

    MnRic(ζ,ζ)dVgn(n1)Mnh2β2dVg.

    Proof. Consider an n-dimensional compact and connected Riemannian manifold (Mn,g) that admits an isometric immersion f:(Mn,g)(Rn+1,,) in the Euclidean space (Rn+1,,) with shape operator B, mean curvature β, and a constant unit vector a=ζ+hς on Rn+1, where the function ρ=f,a satisfies

    Δρ=ncρ (3.10)

    for a positive constant c. Also, the Ricci curvature Ric(ζ,ζ) satisfies

    MnRic(ζ,ζ)dVgn(n1)Mnh2β2dVg. (3.11)

    Now, differentiating ρ=f,ain the direction of XΓ(TMn), we get X(ρ)=X,a=f,ζ. This gives us the gradient of ρ as

    ρ=ζ. (3.12)

    The Hessian operator Hρ of the function ρ is given by HρX=Xρ, XΓ(TMn), and using Eqs (3.8) and (3.12), we arrive at

    HρX=hBXXΓ(TMn). (3.13)

    Taking the trace in the above equation and taking account of Eq (3.10), we get

    cρ=hβ (3.14)

    and therefore, through Eq (3.13), we conclude

    HρX+cρX=hBXhβXXΓ(TMn).

    From the above equation, we reach

    Hρ+cρI2=h2BβI2. (3.15)

    Next, using Eq (3.8), we have

    h(BXβX)=XζhβX,

    which yields

    h2BβI2=ζ2+nh2β22hβdivζ.

    Inserting divζ=nhβ (an outcome of Eq (3.8)), in the above equation, we arrive at

    h2BβI2=ζ2nh2β2. (3.16)

    Recalling the following well known integral formula (cf. [16])

    Mn(Ric(ζ,ζ)+12|£ζg|2ζ2(divζ)2)dVg=0,

    and integrating Eq (3.16) while using the above integral formula, we conclude

    Mnh2BβI2dVg=Mn(Ric(ζ,ζ)+12|£ζg|2(divζ)2nh2β2)dVg. (3.17)

    Using Eq (3.8), we compute

    (£ζg)(X,Y)=2hg(BX,Y)X,YΓ(TMn)

    and consequently, we have

    12|£ζg|2=2h2B2.

    Thus, inserting above equation and divζ=nhβ in Eq (3.17) confirms

    Mnh2BβI2dVg=Mn(Ric(ζ,ζ)+2h2B2n2h2β2nh2β2)dVg,

    that is,

    Mnh2BβI2dVg=Mn(Ric(ζ,ζ)+2h2(B2nβ2)n(n1)h2β2)dVg. (3.18)

    For a local orthonormal frame {e1,..,en}, we have

    BβI2=kg(Bekβek,Bekβek)=B2+nβ22βkg(Bek,ek)=B2nβ2.

    Utilizing the above equation in (3.18), we arrive at

    Mnh2BβI2dVg=Mn(n(n1)h2β2Ric(ζ,ζ))dVg.

    Inserting from Eq (3.15) in the above equation, we have

    MnHρ+cρI2dVg=Mn(n(n1)h2β2Ric(ζ,ζ))dVg

    and treating it with inequality (3.11) allows us to reach the conclusion

    Hρ=cρI.

    Note that ρ satisfies Eq (3.10), that is, Δρ=ncρ for a non-zero constant c. We claim that ρ can not be a constant, for if it were, Equation (3.10) will imply ρ=0, and then Eq (3.14) will imply hβ=0. Note that by Eq (3.6) we have divξ=n(1+σβ), and therefore on the compact hypersurface (Mn,g), we have (cf. [4])

    Mn(1+σβ)=0,

    which does not allow β=0. Hence, in the situation where ρ is a constant, we have h=0, and, also, by Eq (3.12), ζ=0, and by virtue of Eq (3.7), we will reach the conclusion a=0, contrary to the assumption that a is a unit vector. Thus, ρ is a non-constant function which satisfies Obata's equation (3.8) proving that (Mn,g) is isometric to the sphere Sn(c) (cf. [13,14]).

    Conversely, consider the isometric immersion f:Sn(c)(Rn+1,,) of the sphere Sn(c) in the Euclidean space (Rn+1,,) given by f(x)=x. Then, the unit normal ς=cf, the shape operator B=cI, and the mean curvature β=c. Consider the unit constant vector a given by the first Euclidean coordinate vector field, that is,

    a=x1=ζ+hς,

    where ζ is tangent to the sphere Sn(c) and h=a,ς=a,cf=cf,a. Thus, defining ρ=f,a, we have

    h=cρ. (3.19)

    Now, differentiating (3.19) in the direction of XΓ(TSn(c)) and equating the tangential and normal parts, we confirm

    Xζ=chXh=cζ (3.20)

    Using Eqs (3.19) and (3.20), we see cζ=cρ, that is,

    ζ=ρ, (3.21)

    which, in view of the first equation in (3.20), provides

    Δρ=divζ=nch=ncρ. (3.22)

    Finally, using Eq (3.21), the Ricci curvature Ric(ζ,ζ) of the sphere Sn(c) is given by

    Ric(ζ,ζ)=(n1)cζ2=(n1)cρ2. (3.23)

    However, Equations (3.19) and (3.22) confirm

    Sn(c)ρ2dVg=ncSn(c)ρ2dVg=nSn(c)h2dVg=ncSn(c)h2β2dVg.

    Now, integrating Eq (3.23) while using above equation yields

    Sn(c)Ric(ζ,ζ)dVg=n(n1)Sn(c)h2β2dVg.

    Hence, the converse also holds.

    Next, we prove the following result for the complete hypersurface (Mn,g) of the Euclidean space (Rn+1,,).

    Theorem 2. An n-dimensional complete and simply connected isometrically immersed hypersurface f:(Mn,g)(Rn+1,,), n>1, in the Euclidean space (Rn+1,,) with mean curvature β and a constant unit vector a=ζ+hς on Rn+1, where the function h=a,ς0 satisfies Δh=nch for a positive constant c, is isometric to the sphere Sn(c) if and only if the mean curvature β is a constant along the integral curves of ζ and β2c holds.

    Proof. Consider an n-dimensional complete and simply connected Riemannian manifold (Mn,g) that admits an isometric immersion f:(Mn,g)(Rn+1,,) in the Euclidean space (Rn+1,,) such that the function h=a,ς0 satisfies

    Δh=nch, (3.24)

    with mean curvature β satisfying

    ζ(β)=0

    and

    β2c. (3.25)

    We use Eqs (3.5) and (3.8), the symmetry of the shape operator B, and a local orthonormal frame {e1,..,en} in order to find div(Bζ),

    div(Bζ)=nl=1g(elBζ,el)=nl=1g((elB)(ζ)+B(hBel),el)=nl=1g(ζ,(elB)(el))+hB2=nζ(β)+hB2.

    Using Eq (3.15), we get

    div(Bζ)=hB2.

    Now, taking the divergence in Eq (3.9) and using the above equation with Eq (3.24) yields

    hB2=nch,

    that is,

    h2(B2nβ2)=nh2(cβ2). (3.26)

    Combining above equation with inequality (3.25), while keeping in view Cauchy–Schwartz's inequality B2nβ2, we get

    h2(B2nβ2)=0.

    Since h0, we get

    B2=nβ2,

    which, being an inequality in Cauchy–Schwartz's inequality B2nβ2, we must have

    B=βI. (3.27)

    The above equation implies

    (XB)(Y)=X(β)YX,YΓ(TMn),

    which gives

    nl=1(elB)(el)=β.

    Combining the above equation with Eq (3.5) yields

    (n1)β=0

    and, as n>1, we get that β is a constant, and by virtue of Eqs (3.26) and (3.27), we have

    β2=c.

    Now, using Eq (3.27) in the expression of the curvature tensor of the hypersurface with the above equation gives

    R(X,Y)Z=c{g(Y,Z)Xg(X,Z)Y}X,,Y,ZΓ(TMn),

    that is, (Mn,g) is a complete and simply connected space of constant positive curvature c. Hence, (Mn,g) is isometric to Sn(c). The converse is trivial.

    Consider an n-dimensional Riemannian manifold (Mn,g) that possesses a concircular vector field ξ (cf. [3]), that is, the vector field satisfies

    Xξ=σXXΓ(TMn), (4.1)

    where σ is a smooth function, called the potential function of the concircular vector field. A concircular vector field is said to be non-trivial if the potential function σ0. Using Eq (4.1), we immediately have

    divξ=nσ. (4.2)

    In this section, we are interested in an n-dimensional compact Riemannian manifold (Mn,g) that possesses a non-trivial concircular vector field ξ with potential function σ satisfying

    Δσ=ncσ,

    where c>0 is a constant, that is, σ is an eigenfunction of the Laplace operator with eigenvalue the same as the first non-zero eigenvalue of the sphere Sn(c), and we find a condition under which (Mn,g) is isometric to the sphere Sn(c).

    Before we approach this issue, we first prepare an auxiliary result to prove the main result. First, for Eq (4.1), using Eq (2.1) immediately gives the following expression of the curvature tensor, namely

    R(X,Y)ξ=X(σ)YY(σ)XX,YΓ(TMn).

    Taking the trace in the above equation and using Eq (2.2), we reach

    Ric(X,ξ)=(n1)X(σ) (4.3)

    and this equation gives the following expression for the Ricci operator Q operating on ξ, namely

    Q(ξ)=(n1)σ,

    where σ is the gradient of the potential function σ.

    In the following paragraph, we show that for each concircular vector field ξ on a connected Riemannian manifold (Mn,g) there corresponds a smooth function f, which we call a concircular function of the concircular vector field ξ. Note that Eq (4.2) implies

    R(X,ξ)ξ=X(σ)ξξ(σ)XXΓ(TMn)

    and the operator R(X,ξ)ξ, XΓ(TMn) is symmetric in X, and, therefore, the above equation implies

    X(σ)g(ξ,Y)=Y(σ)g(ξ,X)XΓ(TMn).

    The above equation implies

    X(σ)ξ=g(ξ,X)σ

    and taking the inner product in the above equation with σ and replacing X by ξ, we conclude

    (ξ(σ))2=ξ2σ2,

    that is

    ξ2σ2=g(ξ,σ)2.

    This proves that vector fields σ and ξ are parallel, and, consequently, there exists a smooth function f such that

    σ=fξ.

    We call this function f the concircular function of the concircular vector field ξ.

    First, we prove the following proposition.

    Proposition 1. Let ξ be a non-trivial concircular vector field on an n-dimensional compact Riemannian manifold (Mn,g) with potential function σ and concircular function f. If the potential function σ satisfies

    Δσ=ncσ

    for a positive constant c, then

    Mn(Hσ21n(Δσ)2)dVg=n1nMn(ξ(f))2dVg.

    Proof. Let ξ be a non-trivial concircular vector field on an n-dimensional compact Riemannian manifold (Mn,g) with potential function σ and concircular function f, and the potential function σ satisfies

    Δσ=ncσ (4.4)

    for a positive constant c. Using Eqs (4.2) and (4.3), we have

    Ric(ξ,ξ)=(n1)ξ(σ)=(n1)[div(σξ)nσ2]

    and integrating the above equation, we confirm

    MnRic(ξ,ξ)dVg=n(n1)Mnσ2dVg.

    Using the integral formula in [16], we have for a vector field ζ on (Mn,g)

    Mn(Ric(ζ,ζ)+12|£ζg|2ζ2(divζ)2)dVg=0.

    Replacing ζ in the above equation by σ and noting that

    12|£ζg|2=2Hσ2ζ2=Hσ2,

    we conclude

    Mn(Ric(σ,σ)+Hσ2(Δσ)2)dVg=0.

    Thus, we have

    Mn(Hσ21n(Δσ)2)dVg=Mn(n1n(Δσ)2Ric(σ,σ))dVg

    and, inserting Eq (4.4), we reach

    Mn(Hσ21n(Δσ)2)dVg=Mn(n(n1)c2σ2Ric(σ,σ))dVg.

    Now, inserting from Eq (4.4) in the above equation takes us to

    Mn(Hσ21n(Δσ)2)dVg=Mn(n(n1)c2σ2f2Ric(ξ,ξ))dVg. (4.5)

    Using Eqs (4.3) and (4.4), we have

    Ric(ξ,ξ)=(n1)ξ(σ)=(n1)g(ξ,σ)=(n1)fξ2

    and using this in Eq (4.5) leads to

    Mn(Hσ21n(Δσ)2)dVg=(n1)Mn(nc2σ2+f3ξ2)dVg. (4.6)

    Note that taking the divergence on both sides of Eq (4.4) and using Eq (4.2) gives

    Δσ=ξ(f)+nfσ

    and combining this with Eq (4.4) allows us to conclude

    ξ(f)=nσ(c+f). (4.7)

    Using Eqs (4.2), (4.4), and (4.7), we compute

    div(f2σξ)=ξ(f2σ)+nf2σ2=f2g(ξ,σ)+2σfξ(f)+nf2σ2=f3ξ22nσ2f(c+f)+nf2σ2

    and, integrating the above equation, gives

    Mnf3ξ2dVg=Mn(nf2σ2+2ncfσ2)dVg.

    Inserting the above equation into Eq (4.6) leads to

    Mn(Hσ21n(Δσ)2)dVg=n(n1)Mnσ2(c+f)2dVg

    and combining it with Eq (4.7) yields

    Mn(Hσ21n(Δσ)2)dVg=(n1)nMn(ξ(f))2dVg.

    As a straightforward application of the above result, we have the following theorem.

    Theorem 3. An n-dimensional compact and connected Riemannian manifold (Mn,g) admits a non-trivial concircular vector field ξ with potential function σ and concircular function f such that the potential function σ satisfies

    Δσ=ncσ

    for a positive constant c, and the concircular function f is a constant along the integral curves of ξ if and only if (Mn,g) is isometric to Sn(c).

    Proof. Suppose an n-dimensional compact and connected Riemannian manifold (Mn,g) admits a non-trivial concircular vector field ξ with potential function σ and concircular function f such that the potential function σ satisfies

    Δσ=ncσ (4.8)

    for a positive constant c, and the concircular function f is a constant along the integral curves of ξ. Then, by Proposition 1 we have

    Mn(Hσ21n(Δσ)2)dVg=0. (4.9)

    The Cauchy–Schwartz inequality implies

    Hσ21n(Δσ)2 (4.10)

    and equality holds if and only if

    Hσ=ΔσnI. (4.11)

    In view of inequality (4.10) and Eq (4.9), we are ready to conclude the equality

    Hσ2=1n(Δσ)2

    and, therefore, Eq (4.11) holds. Combining Eqs (4.8) and (4.11), we arrive at

    Hσ=cσI. (4.12)

    Note that the potential function σ can not be a constant, for if it were a constant, the above equation would imply σ=0, which is contrary to the assumption that ξ is a non-trivial concircular vector field. Thus, Equation (4.12) is Obata's equation, and therefore (Mn,g) is isometric to the sphere Sn(c) (cf. [13,14]).

    Conversely, take a constant unit vector a on the Euclidean space (Rn+1,,) while treating Sn(c) as a hypersurface of (Rn+1,,) with unit normal ς, shape operator B=cI, and expressing a as

    a=ξ+hς, (4.13)

    where ξ is tangent to Sn(c) and h=a,ς. Differentiating equation (4.13) with respect to XΓ(TSn(c)) and equating the tangential and normal parts, we arrive at

    Xξ=chXh=cξ. (4.14)

    This confirms that ξ is a concircular vector field on Sn(c) with potential function σ=ch, and the second equation gives σ=cξ. This proves that the concircular function f=c. Moreover, if the potential function σ=0, we get h=0, and by the second equation in (4.14), we get ξ=0. In this case, Eq (4.13) confirms a=0, a contradiction to the fact that a is a unit vector. Thus, the potential function σ0, that is, the concircular vector field ξ on Sn(c) is non-trivial. Note that divξ=nch=nσ, and, combining it with the equation σ=cξ, we get Δσ=ncσ with c a positive constant. Hence, the converse holds.

    We have initiated the study of an n-dimensional compact Riemannian manifold (Mn,g) that has an eigenvalue nc for a positive constant c of the Laplace operator the same as the first non-zero eigenvalue of the n-sphere Sn(c) of constant curvature c, and searched for an additional condition under which (Mn,g) is isometric to the sphere Sn(c). The main aim was to find an appropriate smooth function on (Mn,g) that will become the eigenfunction of the Laplace operator with eigenvalue nc as seen in Theorems 1 and 3. Naturally, the scope of this study is quite modest, for instance one can consider an n-dimensional compact Riemannian manifold (Mn,g) that admits a torse forming vector field ξ (cf. [17]). Recall that a torse forming vector field ξ on (Mn,g) satisfies

    Xξ=σX+ω(X)ξXΓ(TMn),

    where σ is a smooth function defined on Mn called the conformal scalar and ω is a smooth 1-form on Mn called the generating form of the a torse forming vector field ξ. It will be an interesting question to consider torse forming vector field ξ on an n-dimensional compact Riemannian manifold (Mn,g) such that its conformal scalar σ satisfies Δσ=ncσ for a positive constant c, and find conditions under which (Mn,g) is isometric to Sn(c).

    We know that the second non-zero eigenvalue of the sphere Sn(c) is given by λ2=2(n+1)c, and another aspect of our work could be, if there is a smooth function f on an n-dimensional compact Riemannian manifold (Mn,g) such that Δf=2(n+1)cf, that is, (Mn,g) has an eigenvalue same as second non-zero eigenvalue of the sphere Sn(c), to find additional condition on (Mn,g) so that (Mn,g) is isometric to Sn(c).

    Sharief Deshmukh: Conceptualization, Methodology, Writing-original draft, Writing-review and editing, Supervision; Amira Ishan: Conceptualization, Methodology, Writing-review and editing; Olga Belova: Formal analysis, Writing-original draft, Writing-review and editing. All authors have read and agreed to the published version of the manuscript.

    The authors would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

    The authors declare no conflicts of interest in this paper.



    [1] R. L. Bishop, B. O'Neill, Manifolds of Negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1–49. https://doi.org/10.1090/S0002-9947-1969-0251664-4 doi: 10.1090/S0002-9947-1969-0251664-4
    [2] B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific: Singapore, 1983. https: //doi.org/10.1142/0065
    [3] B.-Y. Chen, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc., 52 (2015), 1535–1547. https://doi.org/10.4134/BKMS.2015.52.5.1535 doi: 10.4134/BKMS.2015.52.5.1535
    [4] S. Deshmukh, An integral formula for compact hypersurfaces in a Euclidean space and its applications, Glasgow Math. J., 34 (1992), 309–311. https://doi.org/10.1017/S0017089500008867 doi: 10.1017/S0017089500008867
    [5] M. E. Fisher, On hearing the shape of a drum, J. Combinatorial Theory, 1 (1966), 105–125. https://doi.org/10.1016/S0021-9800(66)80008-X doi: 10.1016/S0021-9800(66)80008-X
    [6] N. Ejiri, A construction of Non-flat compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z., 168 (1979), 207–212. https://doi.org/10.1007/BF01214512 doi: 10.1007/BF01214512
    [7] P. B. Gilkey, On spherical space forms with meta-cyclic fundamental group which are isospectral but not equivariant cobordant, Compositio Mathematica, 56 (1985), 171–200.
    [8] R. Gornet, J. McGowan, Lens spaces, isospectral on forms but not on functions, LMS J. Comput. Math., 9 (2006), 270–286. https://doi.org/10.1112/S1461157000001273 doi: 10.1112/S1461157000001273
    [9] A. Ikeda, Y. Yamamoto, On the spectra of a 3-dimensional lens space, Osaka J. Math., 16 (1979), 447–469.
    [10] A. Ikeda, On the spectrum of a riemannian manifold of positive constant curvature, Osaka J. Math., 17 (1980), 75–93.
    [11] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1–23. https://doi.org/10.1080/00029890.1966.11970915 doi: 10.1080/00029890.1966.11970915
    [12] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci., 51 (1964), 542. https://doi.org/10.1073/pnas.51.4.542 doi: 10.1073/pnas.51.4.542
    [13] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14 (1962), 333–340. https://doi.org/10.2969/jmsj/01430333 doi: 10.2969/jmsj/01430333
    [14] M. Obata, The conjectures about conformal transformations, J. Diff. Geom., 6 (1971), 247–258. https://doi.org/10.4310/jdg/1214430407 doi: 10.4310/jdg/1214430407
    [15] M. F. Vigneras, Theorie des nombres, C. R. Acad. Sci. Paris. Ser. A, 287 (1978), 47–49.
    [16] K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker Inc.: New York, NY, USA, 1970.
    [17] K. Yano, On the torse-forming directions in a Riemannian spaces, Proc. Imp. Acad., 20 (1944), 340–345. https://doi.org/10.3792/pia/1195572958 doi: 10.3792/pia/1195572958
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(445) PDF downloads(62) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog