on the occasion of his 85thbirthday
ON THE ROTER TYPE
OF WINTGEN IDEAL SUBMANIFOLDS
SIMONA DECU, MIROSLAVA PETROVI ´C-TORGAˇSEV, ALEKSANDAR ˇSEBEKOVI ´C and LEOPOLD VERSTRAELEN
For all submanifoldsMnof all real space formsMfn+m(c) with constant sectional curvaturec, for all dimensionsn≥2 and for all co-dimensionsm≥1, the so-called Wintgen inequalityasserts thatρ≤ H2−ρ⊥+c, wherebyρis thenormalised scalar curvature of the Riemannian manifold M (which is an intrinsic invariant), and wherebyH2 andρ⊥, are thesquared mean curvatureand thenormalised normal scalar curvatureof M in the ambient spaceMf, respectively (which are extrinsic invariants), and submanifoldsM which satisfy the equality in this optimal general
“soft” inequality are called Wintgen ideal submanifolds of Mf; cf. [1]–[11]. For Wintgen ideal submanifoldsM of real space forms Mf, in the present article (in Sections 3 and 4), we will show some properties concerning theirDeszcz symmetry and theirRoter type, herewith finalising (and in the instance ofn≥4 andm≥3 amending) some results on which was reported before in [12]–[15]. The geometrical notions involved being, at least in our opinion, rather basic and also rather young, an attempt is made to present these notions with some care (in Sections 1 and 2).
AMS 2010 Subject Classification: 53B20, 53B25.
Key words: Wintgen ideal submanifolds, Deszcz spaces, Roter spaces.
1. ON RIEMANNIAN SPACES [16]–[19]
Let Mn be an n-dimensional Riemannian manifold with (positive defi- nite)metric(0,2)tensorg. LetRdenote the (0,4)Riemann–Christoffel curva- ture tensor and S the (0,2) Ricci tensor of M. Since S is symmetric, (i) all its eigenvalues are real: the Ricci curvatures of M, and (ii) S determines on M an orthogonal set of eigendirections: the Ricci principal directions, or, still, the intrinsic principal directions of M. A Riemannian manifold with constant sectional curvature K = c is called a real space form of curvature c; for n ≥ 3, these spaces are characterised by the fact that R = 2cg∧g, whereby ∧ denotes the Kulkarni–Nomizu product (for symmetric (0,2) ten- sors t1 andt2: (t1∧t2)(X, Y, Z, W) =t1(X, W)t2(Y, Z) +t1(Y, Z)t2(X, W)−
REV. ROUMAINE MATH. PURES APPL.,57(2012),1, 75-90
t1(X, Z)t2(Y, W)−t1(Y, W)t2(X, Z), wherebyX, Y, Z, W are arbitrary vector fields on M). A Riemannian manifold for which the Ricci tensor S is propor- tional to the metric tensorg,S=γ g(for some functionγ), or, still, for which all Ricci curvatures are equal, or, still, for which S has an eigenvalue with multiplicity n, is called an Einstein space; for n≥3, thenγ is automatically constant. Trivially every real space form is an Einstein space and, by a Theo- rem of Schouten and Struik, every 3D Einstein space has constant sectional curvature. A Riemannian manifold Mn for which the Ricci tensor S has an eigenvalue of multiplicity ≥n−1, is said to bequasi-Einstein; in case of mul- tiplicity =n−1, one uses the term “properly quasi-Einstein”. By a Theorem of Weyl, forn= 3, theconformal curvature tensorCvanishes identically, and, for n ≥4, C = 0 if and only if the Riemannian manifold Mn is conformally flat (we recall that the (0,4) conformal curvature tensor of Weyl is defined by C = R−[1/(n−2)]g∧S+ [τ /(n−1)(n−2)]g∧g, whereby τ denotes the scalar curvature of M).
From an algebraic point of view,Roter spacesmay well be considered to bethe simplest Riemannian manifolds next to the real space forms. According to [20], for n ≥ 3, a Riemannian manifold Mn is called a Roter space or a Roter manifold when its (0,4) curvature tensor R can be expressed as a linear combination of Kulkarni–Nomizu products of its main symmetric (0,2) tensors, i.e., in the first place the Riemannian metric tensorgand in the second place the Ricci tensor S, thusM being a Roter space when
(1) R=λ(g∧g) +µ(g∧S) +ν(S∧S)
for some functions λ, µ, ν : M → R. Of course, in particular, in a trivial way: (i) the real space forms of curvature c are the Roter spaces for which λ = c/2 and µ = ν = 0, (ii) Einstein Roter spaces are real space forms, and, (iii) all 3D Riemannian manifolds and all conformally flat Riemannian manifolds of dimensionsn≥4 are Roter spaces for whichλ=τ /2(n−1)(n−2), µ= 1/(n−2) andν = 0. Moreover, from [21], it is known that every properly quasi-Einstein Roter space of dimension n ≥ 4 is conformally flat. A.o. in view of some previous considerations, in [21] was introduced the more subtle notion ofRiemannian manifoldsMnof Roter-type, hereby defining, forn≥4, the Riemannian manifolds Mn for which (1) holds on the open setU of their points whereC 6= 0 and whereS6=φ g+ψ ω⊗ω, (for some functionsφ, ψand some 1-form ω on M). OnU, for manifolds of Roter type the decomposition of R in terms of g∧g, g∧S and S∧S is unique and the Ricci tensorS has exactly two distinct eigenvalues; for these and more properties of manifolds of Roter type, see [21]–[27].
From a geometric point of view, the Deszcz symmetric spaces may well be considered to be the simplest Riemannian manifolds next to the real space forms. Whereas the real space forms are the most perfect symmetric spaces in
the sense that they are homogeneousand isotropic, i.e., that they “behave” in the same way at all of their points and that at all of their points they “behave”
in the same way in all directions, or, still, the real space forms are the Rie- mannian manifolds which satisfy the axiom of free mobility, the Deszcz sym- metric, or, still, thepseudo-symmetric spacesare homogeneous but, though no longer being isotropic, might be considered to be only “mildly” anisotropic;
for some surveys on these spaces, see [28], [29]. Now, we recall their defini- tion in the present context, while referring to [29], [30] for more details. The natural metrical endomorphism X∧gY on a Riemannian manifold associated with vector fields X and Y, given by (X ∧g Y)Z = g(Y, Z)X−g(X, Z)Y, in first approximation, measures the change of directions under infinitesimal rotations in planes. The curvature operator R(X, Y) associated with vector fieldsXandY, given byR(X, Y)Z =∇X∇YZ−∇Y∇XZ−∇[X,Y]Z, whereby
∇ and [ ,] respectively denote the Levi–Civita connection on the Riemann- ian manifold M and the Lie-bracketon the differential manifoldM, as shown by Schouten [31], in first approximation, measures the change of directions under parallel transport fully around infinitesimal co-ordinate parallelograms.
In some sense, the Riemannian sectional curvature K(p, π) at a pointp of a Riemannian manifold for a 2-planeπ =−→x ∧ −→y spanned by two (linearly inde- pendent) tangent vectors −→x and −→y toM at p, concerns a sort of calibration of the latter change of directions on M with respect to the former, in that (2) K(p, π) =g(R(−→x ,−→y)−→y ,−→x)/g((−→x ∧g−→y)−→y ,−→x)
(the linear independence of −→x and −→y asserting that the denominator in (2) is different from zero). And, since the (0,4) curvature tensorR is defined by R(X, Y, Z, W) =g(R(X, Y)Z, W), it follows that, for plane fieldsπ =X∧Y on a Riemmanin manifold M, the sectional curvature K(π) is given by (3) K(π) =R(X, Y, Y, X)/ 12g∧g
(X, Y, Y, X).
A Riemannian manifold is locally Euclidean or flat (R = 0 or K = 0), if and only if at all of its points all its directions remain invariant under par- allel transport around all infinitesimal co-ordinate parallelograms (Schouten [31]). And, by a Theorem of Beltrami (cf. also [32]), the projective class of the locally Euclidean spaces is nothing but the class of the real space forms.
Also, by the Lemma of Schur, for n ≥ 3, the real space forms are charac- terised as the Riemannian manifolds Mn for which at all of their points p, the Riemannian sectional curvature function K(p, π) is isotropic, i.e., as the spaces for which at all of their points p the sectional curvature K(p, π) has the same value K(p) for all possible tangent 2D planes π to M at p (this latter fact thus implying that these sectional curvatures K(p) moreover then actually have the same value K(p) = c at all of their points p). As shown in Eisenhart [33], a Riemannian manifold Mn of dimension n ≥ 3 is a real
space form if and only if its Tachibana tensor Q(g, R) vanishes identically, whereby Q(g, R) essentially is the (0,6) tensor which results from the action as a derivation on the (0,4) curvature tensor R, by the metrical endomor- phism, i.e., Q(g, R) =− ∧g·R, or, in extenso,Q(g, R)(X1, X2, X3, X4;X, Y) =
−(∧g·R)(X1, X2, X3, X4;X, Y) =−((X∧gY)·R)(X1, X2, X3, X4) =R((X∧g Y)X1, X2, X3, X4)+R(X1,(X∧gY)X2, X3, X4)+R(X1, X2,(X∧gY)X3, X4)+
R(X1, X2, X3,(X∧gY)X4), wherebyX1, X2, X3, X4 are further arbitrary tan- gent vector fields on M. The geometrical meaning of the Tachibana tensor lies in that it measures, in first approximation, the change of the sectional curvatures K(p, π) at each point p of Mn for any tangent 2D plane π at p under the infinitesimal rotations of π at p with respect to any tangent 2D plane ¯π atp [30]. By the action of the curvature operator R as a derivation on the (0,4) curvature tensor R similarly results the (0,6) curvature tensor R·R, i.e., (R·R)(X1, X2, X3, X4;X, Y) = (R(X, Y)·R)(X1, X2, X3, X4) =
−R(R(X, Y)X1, X2, X3, X4)−R(X1, R(X, Y)X2, X3, X4)−R(X1, X2, R(X, Y) X3, X4)−R(X1, X2, X3, R(X, Y)X4). And, the geometrical meaning of the (0,6) tensor R·R lies in that it measures, in first approximation, the change of the sectional curvatures K(p, π) at any point p of Mn for any tangent 2D plane π atp under the parallel transport ofπ all around any infinitesimal co- ordinate parallelogram in M cornered at p and tangent there to any tangent 2Dplane ¯π atp. The Riemannian manifoldsMnfor whichR·R= 0, i.e., the manifolds Mn for which all Riemannian sectional curvatures K(p, π), for all pointsp and 2D planesπ, in first approximation, remain unchanged after the parallel transport of π all around all infinitesimal co-ordinate parallelograms ofM cornered atpand tangent there to arbitrary 2Dplanes ¯π, were classified by Szab´o [34], [35], and are called Szab´o symmetric spaces orsemi-symmetric spaces. This conditionR·R= 0 first appeared as the integrability condition of
∇R = 0 during the study of theCartan symmetric spaces or thelocally sym- metric spaces, i.e., of the Riemannian manifolds M for which ∇R= 0 holds, and which spaces were classified by E. Cartan. In analogy with the step from the locally Euclidean spaces to the real space forms, one can go from the Szab´o symmetric spaces to the Deszcz symmetric spaces or, in other words, to the pseudo-symmetric spaces, as follows. As a sort of calibration of the changes of Riemannian sectional curvaturesK(p, π) for planesπ =−→u ∧ −→v under parallel transport around infinitesimal co-ordinate parallelograms cornered at p and tangent there to planes ¯π=−→x ∧ −→y by the changes ofK(p, π) under infinites- imal rotations of π with respect to ¯π (for curvature dependent planes π and
¯
π atp, i.e., for planes for which (∧g·R)(−→u ,−→v ,−→v ,−→u;−→x ,−→y) 6= 0), arises the Deszcz sectional curvature or, still, the double sectional curvature of M at p for curvature dependent planes π and ¯π
(4) L(p, π,¯π) = [(R·R)/(− ∧g·R)](−→u ,−→v ,−→v ,−→u;−→x ,−→y).
And a Riemannian manifoldMnof dimensionn≥3 is calledDeszcz symmet- ricorpseudo-symmetricwhen itsdouble sectional curvature functionL(p, π,π)¯ is isotropic, i.e., at all of its pointsphas the same valueL(p) for all curvature dependent tangent planesπ and ¯π atp; (it should be observed that every Rie- mannian surface M2 is automatically semi-symmetric and that there do exist ample pseudo-symmetric spaces with non-constant Deszcz sectional curvature L:Mn→R; the Deszcz symmetric spaces for which this function Lalso does not depend on the points p, i.e., the spaces of constant Deszcz sectional cur- vature L, were called the pseudo-symmetric spaces of constant type [36]). In analogy with the situation of the real space forms, atensorial characterisation of the Deszcz symmetric Riemannian spaces Mn of dimension n≥ 3 can be given by the proportionality of the (0,6) tensors R·R and Q(g, R), i.e., by the fact that
(5) R·R=L Q(g, R)
for some functionL:M →R(on the open subset ofM on whichQ(g, R)6= 0), the Deszcz sectional curvature functionof Mn. Moreover, cf. the above Theo- rem of Beltrami, now the class of Deszcz symmetric spaces also is closed un- der geodesic mappings and if a semi-symmetric space M (L = 0), admits a geodesic mapping onto a Riemannian manifold Mf, then in general, Mf is a
“proper” pseudo-symmetric space (Le6= 0). As special case of the fact that if C = 0 a Riemannian manifold Mn, n ≥ 3, is pseudo-symmetric if and only if its Ricci tensor has at most two distinct eigenvalues, it is clear that a 3D Riemannian space is Deszcz symmetric if and only if it is quasi-Einstein(and in the above notation hasL= (φ+ψ)/2, [37]). From the latter situation, e.g., it readily follows that all 3D Thurston model Riemannian spaces are pseudo- symmetric spaces of constant type L = 0,+1 or −1, [38]. With the usual care, all previous definitions go through to the indefinite, semi-Riemannian case, and, in this respect, a classification of the pseudo-symmetric spacetimes was given in [39]. When e.g. replacing the (0,4) curvature tensor R by the (0,2) Ricci tensorS, in a similar fashion, results the class of theRicci pseudo- symmetric spaces. While every Deszcz symmetric space is automatically also Ricci pseudo-symmetric, the converse does not hold in general; see, e.g., [28].
The property to be a pseudo-symmetric space is not invariant under con- formal transformations of Riemannian spaces. The formally related curvature condition for a Riemannian manifoldMnof dimensionn≥4 to have apseudo- symmetric Weyl tensor C however is conformally invariant indeed; we recall that the latter property means that C·C = LCQ(g, C) for some function LC :Mn →R (on the open part of M where Q(g, C) 6= 0). For the extrinsic immediate relevance of this intrinsic conformal curvature condition, see [40]:
e.g., for hypersurfaces Mn in Euclidean spaces En+1, n ≥ 4, basically this
condition amounts to 2-quasi-umbilicity (the hypersurface having a principal curvature of multiplicity ≥n−2), whereas quasi-umbilicity itself (the hyper- surface having a principal curvature of multiplicity ≥n−1), since Cartan, is known to characterise its conformal flatness.
Amongst the connections between Roter spaces and pseudo-symmetry properties of Riemannian manifolds, from [22]–[26], we recall the following:
(i)The open submanifoldU of a Riemannian manifoldMn of Roter type (whereby U is used in the same sense as above), is Deszcz symmetric, has a pseudo-symmetric Weyl conformal curvature tensor C and its Ricci tensor S has exactly two distinct principal curvatures;
(ii) the open submanifold U of a Deszcz symmetric space with pseudo- symmetric Weyl tensor C is a space of Roter type.
2. ON RIEMANNIAN SUBMANIFOLDS [16]–[19]
LetMnbe annD submanifoldin an (n+m)DRiemannian spaceMfn+m. The Riemannian metric and correspondingLevi–Civita connection on Mfwill be denoted by ge and by ∇, and thee induced Riemannian metric and corre- sponding Levi–Civita connection on M will be denoted byg and ∇. Tangent vector fields on M will be written as X, Y, . . . and normal vector fields on M in Mfwill be written as ξ, η, . . .. The formulae of Gauss and Weingartenare given by
(6) ∇eXY =∇XY +h(X, Y) and
(7) ∇eXξ =−Aξ(X) +∇⊥Xξ,
whereby ∇⊥ is the connection induced in the normal bundle of M in Mf, h is the second fundamental form of the submanifold and Aξ the shape ope- rator or Weingarten map on M with respect to the normal vector field ξ, such that g(Aξ(X), Y) =eg(ξ, h(X, Y)), or, still, thus having that h(X, Y) = P
αg(Aα(X), Y)ξα, whereby {ξα}, α, β, . . . ∈ {1,2, . . . , m}, is any local or- thonormal normal frame on M in Mf and whereby we have put Aα = Aξα. The mean curvature vector fieldof M inMfis defined by
(8) −→
H = 1
ntrh= 1 n
X
i
h(Ei, Ei) = 1 n
X
α
(trAα)ξα,
whereby {Ei}, i, j, . . .∈ {1,2, . . . , n}, is anylocal orthonormal tangent frame on M.
A submanifoldM inMfis totally geodesic when h= 0, totally umbilical whenh=g−→
H,minimalwhen−→
H = 0, or, equivalently, when its (squared)mean
curvature function H2 = eg(−→ H ,−→
H) vanishes identically, and pseudo-umbilical when the mean curvature vector field −→
H determines an umbilical normal di- rection on M inMf, i.e., when A−→H =χId, whereby Id stands for the identity operator on T M and χ is some real function onM.
The normalised scalar curvature of an nD Riemannian manifold M is defined to be
(9) ρ= [2/(n(n−1))]X
i<j
R(Ei, Ej, Ej, Ei) = 2τ /(n(n−1)),
and, similarly, the normalised scalar normal curvature of an nD Riemannian submanifold M inMfis defined to be
(10) ρ⊥: = 2
n(n−1)
X
i<j
X
α<β
[R⊥(Ei, Ej;ξα, ξβ)]2 12
,
wherebyR⊥denotes thenormal curvature tensorofMinMf, i.e., the curvature tensor of the normal connection ∇⊥ of M in Mf, or, still, R⊥(X, Y, ξ, η) = g(Re ⊥(X, Y)ξ, η) (R⊥(X, Y) =∇⊥X∇⊥Y − ∇⊥Y∇⊥X − ∇⊥[X,Y]).
From here on, we will consider submanifolds Mn in ambient real space forms Mfn+m(c) of constant sectional curvature Ke = c. Then the equations of Gaussand of Ricciexplicitly relate the Riemann–Christoffel curvature ten- sor R of M and the normal curvature tensor R⊥ of M in Mf to the second fundamental form and the shape operators of M inMf, as follows:
R(X, Y, Z, W) =eg(h(Y, Z), h(X, W))−eg(h(X, Z), h(Y, W))+
(11)
+c{g(Y, Z)g(X, W)−g(X, Z)g(Y, W)}, and
(12) R⊥(X, Y, ξ, η) =g([Aξ, Aη]X, Y), whereby [Aξ, Aη] =AξAη −AηAξ.
And, in the present article, we further mainly will be concerned with the so-called Wintgen ideal submanifolds, for whose general literature, for brevity here, we refer to [1]–[15] and the references therein for various classification re- sults and many examples, old and new. The following theorem of Ge and Tang [4] and of Lu [2] states the Wintgen inequalityfor submanifoldsMn of any di- mensionnwith any codimensionmin a real space formMfn+m(c),and charac- terises its equality case, after previous intermediate developments mainly made by Rouxel, Rodrigues–Guadalupe, De Smet–Dillen–Vrancken and one of the authors while working on the original inequality obtained by Wintgen in the late 19seventies for surfaces M2 in E4. It concerns a basic general optimal inequality between likely the most primitive scalar valued geometric quanti- ties that can be defined on submanifolds: as intrinsic invariant it involves
the scalar curvature and as extrinsic invariants it involves the scalar normal curvature and the squared mean curvature.
Theorem A. Let Mn be a submanifold in a real space form Mfn+m(c).
Then
(∗) ρ≤H2−ρ⊥+c,
and in (∗) actually the equality holds if and only if, with respect to a suitable adapted orthonormal frame {Ei, ξα} on M in M, the shape operators of thef submanifold take the following forms
A1 =
eλ1 µe 0 . . . 0 eµ λe1 0 . . . 0 0 0 eλ1 . . . 0 ... ... ... ... 0 0 0 . . . λe1
,
A2 =
eλ2+eµ 0 0 . . . 0 0 λe2−µe 0 . . . 0 0 0 eλ2 . . . 0 ... ... ... ... 0 0 0 . . . eλ2
, (13)
A3 =
eλ3 0 0 . . . 0 0 λe3 0 . . . 0 0 0 eλ3 . . . 0 ... ... ... ... 0 0 0 . . . λe3
,
A4 =· · ·=Am = 0,
whereby eλ1,eλ2,eλ3 and µe are real functions onM.
The submanifoldsM inMf which satisfy the equality
(¯∗) ρ=H2−ρ⊥+c
in Wintgen’s general inequality (∗) are calledWintgen ideal submanifolds, and frames {Ei, ξα} in which the shape operators assume the forms of (13) will further be called Choi–Lu frames and moreover the corresponding tangent E1E2-planes will be called the Choi–Lu planesof M inMf. The term “ideal”
might be justified as follows. The right hand side of (∗) can be considered as a kind of external stress which the submanifold M experiences due to the shape that it takes in Mf and equality (¯∗) then reflects the property that
M would succeed to take on such convenient shape or form in Mf that this external stress would be as small as possible, given the intrinsic curvature situation ofMas this is expressed by the magnitude of its scalar curvature. For Wintgen ideal submanifolds M in M, the shape operators (13) clearly reflectf the presence of two noteworthy orthogonally complementary distributions on such submanifolds, a totally umbilical one of dimension n−2 in general and the particular 2 dimensional one spanned by{E1, E2}, which later on therefore was given the above proper name.
3. PSEUDO-SYMMETRY PROPERTIES OF WINTGEN IDEAL SUBMANIFOLDS
Clearly, the intrinsic properties of a Riemannian manifold (Mn, g) of dimension n to be pseudo-symmetric, everywhere in this paper, are relevant only for manifolds of dimension n > 2, and so, at least in this section, we will only consider Wintgen ideal submanifolds Mn of dimensionn≥3 and of arbitrary co-dimension m in ambient spaces Mfn+m. From [13] and [12], we recall the following.
Theorem B. A Wintgen ideal submanifold M3 in a real space form Mf3+m(c) is Deszcz symmetric if and only if (I) M3 is a totally umbilical submanifold with Deszcz sectional curvature L = 0 (M3 then being a space of constant sectional curvature K), or, (II) M3 is a minimal submanifold or a pseudo-umbilical submanifold with Deszcz sectional curvature L= supK = c+H2, or, (III) M3 is characterised by the property infRic = 2KChoi−Lu= 2 infK and has Deszcz sectional curvatureL= infK.
Theorem C.A Wintgen ideal submanifold Mn of dimensionn >3and of co-dimension m = 2 in a real space form Mfn+2(c) is Deszcz symmetric if and only if (I) Mn is a totally umbilical submanifold with Deszcz sectional curvature L= 0 (Mn then being itself a space of constant sectional curvature K), or, (II) Mn is a minimal submanifold having Deszcz sectional curvature L=c.
Theorem D. A Wintgen ideal submanifold of dimension > 3 and of co-dimension = 2 in a real space form is Deszcz symmetric if and only if it is Deszcz Ricci symmetric.
Theorem E.Every Wintgen ideal submanifold Mn of dimensionn >3 and of co-dimension m = 2 in a real space form Mfn+2(c) has a pseudo- symmetrical conformal curvature tensorC of Weyl. A Wintgen ideal submani- fold Mn of dimension n >3 in Mfn+2(c) is minimal if and only if
LC = n−3
(n−1)(n−2)(c−infK).
Next, we consider the Deszcz symmetry of the Wintgen ideal submani- folds Mn in Mfn+m(c) for dimensions n >3 and co-dimensions m >2. Their Riemann–Christoffel curvature tensors are obtained by inserting the shape operators (13) in the equation of Gauss (whereby we may assume thateλ36= 0, since otherwise all algebraic considerations for the submanifolds studied at present actually reduce to the considerations already made before in the case of co-dimensionm= 2). Up to the algebraic symmetries of the (0,4) curvature tensor R of such Wintgen ideal submanifolds, all components of R are zero except possibly the following ones
R1221 =c+eλ21+eλ22+eλ23−2µe2,
R1kk1 =c+λe21+eλ22+eλ23+µeeλ2, k≥3, R1kk2 =µeeλ1, k≥3,
R2kk2 =c+λe21+eλ22+eλ23−µeeλ2, k≥3, Rkllk =c+eλ21+eλ22+eλ23, k6=l, k, l≥3.
(14)
Then expressing the condition R·R=L(− ∧g·R) for Deszcz symmetry to be satisfied by the (0,6) tensorsR·Rand∧g·Rfor some functionL:M →R, by evaluating these tensors on the tangent vectorsE1, . . . , Enof a Choi–Lu frame one finds that this pseudo-symmetry is characterised by the following system of algebraic equations (obtained by the evaluations of these tensors on the six combinations (E1, E3, E3, E1;E1, E2),(E1, E3, E3, E2;E1, E2),(E1, E2, E1, E3;E2, E3),(E1, E4, E3, E4;E1, E3),(E1, E4, E3, E4;E2, E3) and (E2, E4, E3, E4;E2, E3), all other choices of combinations of basic vectorsE1, . . . , En lead- ing either to one or other equation of the following system or to a triviality):
eλ1µ(2e µe2+ Λ) = 0, eλ1µΛ = 0,e eλ21µe2+µ(2e µe+eλ2)(eλ2µe+ Λ) = 0, eλ21µe2+µeλe2(µeeλ2−Λ) = 0, eλ2µ(2e µe2+ Λ) = 0,
eλ21µe2+µeλe2(µeeλ2+ Λ) = 0, (15)
whereby we have put Λ =L−c−eλ21−λe22−eλ23. And this system can readily be seen to be satisfied if and only if (I) µe = 0, in which case L = 0, or, (II) µe6= 0 and eλ1 =eλ2= 0, in which case L=c+eλ23. From (13), and also taking into account Theorem C, we thus obtained the following.
Theorem 1. A Wintgen ideal submanifoldMn of dimension n≥4 and of co-dimension m≥2 in a real space formMfn+m(c)is a Deszcz symmetrical Riemannian manifold if and only if it is a totally umbilical (withL= 0) or a minimal or pseudo-umbilical submanifold (withL=c+H2)of this space form Mfn+m(c).
From (14), up to algebraic symmetries, the eventually non-trivial compo- nents of the (0,2)Ricci tensorof Wintgen ideal submanifoldsMninMfn+m(c) in a Choi–Lu frame are found to be
S11= (n−1)eλ+ (n−2)µeeλ2−2µe2, S22= (n−1)eλ−(n−2)µeeλ2−2µe2, S12= (n−2)µeeλ1,
Skk= (n−1)eλ, k≥3,
S1k =S2k=Slk= 0, k, l≥3, k 6=l, (16)
where we have putλe=c+eλ21+eλ22+λe23. And, in a straightforward way, basing on (14) and (16), it follows that Theorem D actually holds for all dimensions and co-dimensions.
Theorem 2. Any Wintgen ideal submanifold Mn in a real space form Mfn+m(c), of dimension n ≥3 and of co-dimensions m ≥ 2, is Deszcz sym- metric if and only if it is Deszcz Ricci symmetric.
From (16), the scalar curvatureτ of a Wintgen ideal submanifoldMn in Mfn+m(c), in a Choi–Lu frame, is found to be given by
(17) τ =n(n−1)eλ−4eµ2.
Then, combining (14), (16) and (17), up to algebraic symmetries, the compo- nents of the Weyl conformal curvature tensorCofMnare trivially zero or are determined by the following expressions
C1221=−2(n−3) n−1 µe2, C1kk1=C2kk2= 2(n−3)
(n−1)(n−2)µe2, k≥3, Clkkl=− 4
(n−1)(n−2)µe2, k6=l, k, l≥3.
(18)
Accordingly, the components of the (0,6) tensors C·C and Q(g, C) =− ∧g
·C which are not automatically trivial, by the algebraic symmetries of these
tensors, are all determined by the following (C·C)(E2, E3, E1, E2;E1, E3) =
= (n−3)(C·C)(E1, E4, E3, E4;E1, E3) = 4(n−3)2 (n−1)(n−2)2µe2, (Q(g, C))(E2, E3, E1, E2;E1, E3) =
= (n−3)(Q(g, C))(E1, E4, E3, E4;E1, E3) =−2(n−3) n−2 µe2, (19)
such that always C·C =LC(Q(g, C)) for LC =−(n−1)(n−2)2(n−3) eµ2. Since, from (14), we can observe that, for Wintgen ideal submanifolds Mn,
(20) infK=KChoi−Lu=K12=eλ−2µe2, from (17) and (20) we obtain that, for such submanifolds,
(21) µe2 = 1
2(n+ 1)(n−2)[τ−n(n−1) infK].
Thus, we can state the following.
Theorem 3.LetMnbe a Wintgen ideal submanifold of dimensionn >3 in a real space form Mfn+m(c).
(i) Then M is conformally flat if and only if M is a totally umbilical submanifold in Mf (and, hence, M is a real space form itself).
(ii) M always has a pseudo-symmetrical conformal Weyl tensor C and the corresponding function of pseudo-symmetry is given by
LC =− n−3
(n+ 1)(n−1)[τ −n(n−1) infK].
For some geometrical comments in this respect concerning the extrema of K in the context of some general remarks of Berger on this matter and for reference to concrete examples of Wintgen ideal submanifolds, see [6], [7], [11], [13]. Finally, it should be observed that for submanifoldsMninMfn+m(c) with flat normal connection, and thus in particular forhypersurfaces (m = 1), the Wintgen inequality actually reduces to a Chen inequalityand the correspond- ing ideal submanifoldsM then aretotally umbilical inMfand hencespaces of constant curvature (and so are special Deszcz symmetric spaces, withL= 0);
and, as general basic reference for optimal inequalities relating various extrin- sic and intrinsic characteristics of submanifolds, we refer to B.Y. Chen’s new book [19].
4. WINTGEN IDEAL SUBMANIFOLDS WHICH ARE ROTER SPACES
In view of the definition itself of Roter spaces, further on we only need to consider manifolds Mn of dimensions n ≥ 4, thus excluding the trivial situations which do occur for dimensions n = 2 and n = 3 in this context.
So, in the following we only consider Wintgen ideal submanifolds Mn in real space forms Mfn+m(c) of dimensions n ≥ 4. First, we note that, as follows e.g. from (13), (16) and (18), such Wintgen ideal submanifolds M which are either assumed to be conformally flat (C = 0) or to be quasi-Einstein (S=φ g+ψ ω⊗ω), clearly are totally umbilical in real space forms M, and,f so, are themselves real space forms too, and thus in particular are evidently pseudo-symmetric, with L= 0. Recalling next that, as was shown by Deszcz and Hotlo´s [22], the open submanifolds U of Riemannian spaces Mn of Roter type are always pseudo-symmetric, we thus see, in combination with the above, that for all the Wintgen ideal submanifoldsMn in real space formsMfn+m(c) under consideration the property to be a Roter space immediately implies to be a Deszcz symmetric space as well.
We now will prove (explicitly) that also the converse statement holds good. From Theorem 1, and an above made comment about the case of hyper- surfaces, it follows that for Deszcz symmetric Wintgen ideal submanifoldsMn with n≥ 4 in real space forms Mfn+m(c), the shape operators in a Choi–Lu frame are given by (13) whereby (I) µe = 0 or (II) µe 6= 0 and eλ1 = λe2 = 0.
In case (I), Mnis itself a real space form and hence a Roter space in a trivial way. In case (II), from (14) and (16), it follows that the only possibly non zero components of the (0,4) Riemann tensorR and of the (0,2) Ricci tensor S of Mn are then given by
R1221 =c+λe23−2µe2,
R1kk1 =R2kk2 =c+eλ23, k≥3, Rlkkl=c+eλ23, k6=l, k, l≥3, (22)
and by
(23) S11=S22= (n−1)(c+λe23)−2eµ2, Skk= (n−1)(c+eλ23), k≥3,
respectively. Consequently, the condition (1) for Mn to be a Roter space is equivalent to the following system of linear equations admitting a solution for
the coefficientsλ, µ andν of (1):
2ν[(n−1)(c+λe23)−2µe2]2+2µ(n−1)(c+λe23−2µe2)+λ=c+eλ23−2µe2, 2ν(n−1)(c+λe23)[(n−1)(c+eλ23)−2µe2]+µ[n(c+eλ23)−2µe2]+λ=c+eλ23, (24)
2ν(n−1)2(c+eλ23)2+2µ(n−1)(c+eλ23)+λ=c+eλ23,
(for instance obtained by taking the choices 1221,1331 and 3443 as indices of the (0,4) tensors occurring in (1) in a Choi–Lu frame). And, indeed, (24) ac- tually is found to have λ= (c+eλ23)[2µe2−(n−1)(c+e2 λ23)]
µe2 , µ= (n−1)(c+e2 λ23)
µe2 ,ν =−41
eµ2
as unique solution (the above small calculation gives the precise expressions of the coefficients λ, µ, ν concerned, whereas from the closing statement (ii) of Section 1 and the observations made in the first paragraph of the present Section, the validity of the converse statement as such does follow at once).
All in all, we thus obtained the following result, which essentially asserts that for the Wintgen ideal submanifolds of real space forms the algebraic and the geometric most natural generalisations of the property to be a most basic Rie- mannian manifold, namely a Riemannian space of constant curvature, actually do agree, whereas this is not at all so in the class of Riemannian manifolds as such.
Theorem 4. Let M be a Wintgen ideal submanifold of dimension ≥ 4 and of arbitrary codimension in a real space form. Then M is Deszcz symme- tric if and only if M is a Roter space.
Acknowledgements.The second named author is supported by the Ministry of Edu- cation and Science of the Republic of Serbia, grant IO-174012. The fourth named au- thor is supported by the Research Foundation–Flanders project G.0432.07(Belgium).
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Received 15 January 2012 SIAT
Bucharest, Romania simona.decu@gmail.com University of Kragujevac
Faculty of Sciences, Department of Mathematics Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia
mirapt@kg.ac.rs Miodraga Jovanovi´ca 40 36300 Novi Pazar, Serbia Katholieke Universiteit Leuven
Fakulteit Wetenschappen, Departement Wiskunde Celestijnenlaan 200B, 3001 Heverlee, Belgium
leopold.verstraelen@wis.kuleuven.be
