A study of the relationship between pseudo-umbilical totally real submanifolds and minimal totally real submanifolds in complex space forms is presented in this paper. The paper studies totally real submanifolds in complex space forms. The moving-frame method and the DDVV inequality (a conjecture for the Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken) are used to obtain some rigidity theorems and an integral inequality, improving the associated results.
Citation: Fatimah Alghamdi, Fatemah Mofarreh, Akram Ali, Mohamed Lemine Bouleryah. Some rigidity theorems for totally real submanifolds in complex space forms[J]. AIMS Mathematics, 2025, 10(4): 8191-8202. doi: 10.3934/math.2025376
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A study of the relationship between pseudo-umbilical totally real submanifolds and minimal totally real submanifolds in complex space forms is presented in this paper. The paper studies totally real submanifolds in complex space forms. The moving-frame method and the DDVV inequality (a conjecture for the Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken) are used to obtain some rigidity theorems and an integral inequality, improving the associated results.
A key challenge in differential geometry is exploring the relationship between the geometry and topology of Riemannian manifolds. In submanifold theory, an exciting question is how the pinching conditions on intrinsic or extrinsic curvature invariants affect the geometry and topology of submanifolds in space forms. Simons first established a key result on minimal submanifolds of spheres with a sufficiently pinched second fundamental form in his seminal paper [18]. Later, Chern, et al. [6] proved a well-known rigidity theorem, which has since motivated numerous significant advances in the study of pinching conditions. The study of rigidity theorems is crucial in the theory of minimum submanifolds. Some pioneering work and substantial research on rigidity theorems for minimum submanifolds in spheres have been conducted by Lawson [8]. Let the square norm of the second fundamental form be represented by σ and a unit sphere be represented by Sn+m with the codimension m. If a compact minimal submanifold Nn in Sn+m with the following pinching condition:
0≤σ≤(n2−1m) |
then either
σ=0,orσ=(n2−1m) |
and N is the Clifford hypersurface or the Veronese surface in S4. Later, Li [9] and Chen [5] improved the pinching number n(2−1/m) to 2n3. They showed that if
0≤σ≤2n3, |
then either
σ=0orσ=2n3, |
and N is the Veronese surface in S4.
After initial motivation by Simons [18] and preliminary developments (for example, [5,8,12,16,17]), this topic has received much attention. These underlying works reveal, in particular, several similarities between the free boundary minimal surfaces in a Euclidean unit ball and closed minimal surfaces in the sphere. In this respect, the classical results and tactics for obtaining rigidity results, in conclusion, may indicate the direction of interest in exploring similar progress in the free boundary case. This study was motivated by the rigidity theorems for minimal submanifolds and submanifolds with parallel mean curvature in space forms see [9,11,19,20], etc.
On the other hand, the space forms are useful for understanding the geometric analysis. Several authors constructed the first eigenvalues for submanifolds in different space forms such as in C-totally real submanifolds in Sasakian space forms [1], Lagrangian submanifolds in complex space forms [2], slant submanifolds of Sasakian space form [13,15], semi-slant submanifolds of Sasakian space forms [14] and totally real submanifolds in generalized complex space forms [3] that contain a p-laplacian operator. It should be noted that little work has been done on the rigidity theorems for totally real submanifolds in space form geometry. Therefore, motivated by some previous work, we constructed the rigidity for a totally real submanifold in complex space form and discuss their consequences in the present paper.
Assume that ˜Nn is a complex space form of constant holomorphic sectional curvature 4κ, denoted ˜Nn(4κ). The curvature tensor ˜R of ˜Nn(4κ) can be expressed as:
˜R(V1,V2)V3=κ{g(V2,V3)V1−g(V1,V3)V2+g(V3,JV2)JV1−g(V3,JV1)JV2+2g(V1,JV2)JV3} | (2.1) |
for all V1,V2,V3∈Γ(T˜N). Based on the cases, κ<0,κ=0, and κ>0, ˜Nn(4κ) is the complex hyperbolic space CHn, the complex Euclidean space Cn and the complex projective space CPn. We call an m-dimensional Riemannian submanifold Nm of ˜Nn(4κ) as totally real if the standard complex structure J of ˜Nn(4κ) maps any tangent space of Nm into its corresponding normal space [4].
We considered an orthonormal frame
{e1⋯em,em+1⋯em+h,e∗1=Je1⋯e∗m=Jem,e(m+1)∗=Jem+1⋯e(m+h)∗=Jem+h} |
in ˜Nm+h(4κ) restricted to Nm, e1⋯em is tangent to Nm. We provided the indices as follows:
A,B,C⋯=1,⋯,m+h,1∗⋯,m+h∗a,b,c⋯=1,⋯,m;a∗,b∗,c∗=m+1,⋯,m+h,1∗,⋯,m+h∗. |
Let Π denote the squared length of the second fundamental form ζ of Nm, which is defined by
Π=∑abk(ζkab)2. | (2.2) |
Similarly, the mean curvature of Nm is calculated as:
H=1m∑akζkaaek. | (2.3) |
If H=0 in (2.3), then Nm is minimal. From (2.1), we get the following equation for submanifold in complex space form:
˜KABCD=(δACδBD−δADδBC)κ+κ(JACJBD−JADJBC+2JABJCD) | (2.4) |
where ˜K is the sectional curvature of ˜Nn(4κ). The curvature tensor of indices for the submanifold is
Rabcl=˜Kabcl+∑α(ζαacζαbl−ζαalζαbc). | (2.5) |
We define the Ricci curvature for a totally real submanifold:
Rab=(m−1)κδab+∑a(ζkab∑cζkcc−∑cζkacζkcb). | (2.6) |
From the above, we can establish some notation
Π=‖ζ‖2,H=|ξ|,Hα=(ζαab)m×m. | (2.7) |
Let us assume that em+1 is parallel to H in which case we have
trHm+1=mH,Hα=0,α≠m+1 | (2.8) |
where tr stands for the trace of the matrix Hα=(ζαab). Taking account of (2.5) and (2.8), we have the scalar curvature as
R=m(m−1)κ+m2H2−Π | (2.9) |
where H stands for the mean curvature vector of Nm. Since H is constant, it can be concluded that the scalar curvature R is constant if and only if Π is constant by (2.9). Let ζkabc denote the second covariant derivative of ζkab in which case we have
∑cζαabcωc=dζαab−∑cζαcbωca−∑cζαacωcb+∑tζαabωtk | (2.10) |
where {ωa} is the dual frame of Nm. Taking the exterior derivative of the equation (2.10), we obtain
∑lζαabclωl=dζαabc−∑lζαabcωla+∑tζtabcωtk−∑lζαalcωab−∑lζαablωlc. | (2.11) |
Moreover, the Laplacian of ζαab is
Δζαab=∑cζαabcc=∑cζαccab+∑cd(ζαcdRdabc+ζαdaRdcbc)−∑βcζβcaRαβbc. | (2.12) |
Lemma 2.1. [10] Let T1,⋯,Tn be symmetric (m×m)-matrices, in which case
n∑r,s=1‖[Tr,Ts]‖2≤(n∑r=1‖Tr‖2)2 |
such that equality holds if and only if the following matrices are satisfied:
Tr=P(0μ0⋯0μ00⋯0000⋯0⋮⋮⋮⋱⋮000⋯0)Pt,Ts==P(μ00⋯00−μ0⋯0000⋯0⋮⋮⋮⋱⋮000⋯0)Pt |
where P is an orthogonal (m×m)-matrix and [Tr,Ts]=TrTs−TsTr is the commutator of the matrices Tr,Ts.
Lemma 2.2. Let us T1,T2,⋯Tm(n≥2) be symmetric (m×m)-matrices, in this case
−2m∑α,β=1(tr(T2αT2β−tr(TαTβ)2)≥m∑α,β=1[tr(TαTβ)]2−32(m∑α=1tr(T2α))2. | (2.13) |
We can now estimate our first main result, which is as follows.
Theorem 2.1. If the mean curvature vector of an m-dimensional compact totally real submanifold Nm in complex space form ˜Nm+h(4κ) is parallel and satisfies the following inequality
RN≥(m+2h−12(m+2h))(κ+H2), | (2.14) |
then Nm is a totally umbilical sphere Sm(1√κ+H2), where H denotes the mean curvature of Nm.
Proof. Assume that Nm is a totally real submanifold of complex space form ˜Nm+h(4κ) with the parallel mean curvature vector H. Consider an em+1 that it is parallel to H and
trHm+1=mH,trHα=0,α=m+1. | (2.15) |
We assume that the mean curvature vector H is parallel, so we have
D⊥H=dHem+1+HD⊥em+1=dHem+1+H∑βωm+1βeβ=0 | (2.16) |
where D is a Levi-Civita connection. From the structure equation and (2.16), we derive
dωm+1β=−∑γωm+1γ∧ωγβ+12∑clRm+1βclωc∧ωl=12∑clRm+1βclωc∧ωl=0. | (2.17) |
If we consider, from (2.12), that Nm has as parallel mean curvature vector and ∑cHαccab=0, we can derive
12ΔΠH=∑abc(ζm+1abc)2+∑ijζm+1abΔζm+1ab=∑abc(ζm+1abc)2+∑abclζm+1ab(ζm+1clRlabc+ζm+1laRlcbc). | (2.18) |
Let RN(p,π) the represent the sectional curvature of Nm for the 2-plane π⊂TpN at the point p∈Nm. Then set
Rmin(p)=minπ⊂TpNRN(p,π). |
Therefore, the orthonormal fields are {ei} such that ζm+1ab=λiδab, where λi represents the eigenvalues; hence, we get
∑abclζm+1ab(ζm+1clRlabc+ζm+1laRmlcbc)=12∑ab(λa−λb)2Rabab≥12∑ab(λa−λb)2Rmin. | (2.19) |
Taking (2.18) and (2.19), we have
12ΔΠH≥∑abc(ζn+1abc)2+12∑ab(λa−λb)2Rmin. | (2.20) |
It follows from RN≥m+2h−12(m+2h)(κ+H2) and the lemma of Hopf that ΠH is a constant [21], and we derive
12∑ab(λa−λb)2Rmin=0. | (2.21) |
It is implied that λa=λb. In this case, Nm is pseudo-umbilical. Again, from (2.12), ∑cHαccab=0 and mean curvature of Nm is parallel; we can constructs
12Δτ=∑α≠m+1∑abc(ζαabc)2+∑α≠m+1∑abclζαab(ζαclRlabc+ζαlaRlcbc)−∑α≠m+1∑βabcζαabζβcaRαβbc | (2.22) |
where τ is the scalar curvature of Nm. From (2.5) and (2.15), we get
∑α≠m+1∑abclζαab(ζαclRlabc+ζαlaRlcbc)=m(κ+H2)τ+∑αβ≠m+1(tr(HαHβ)2−tr(H2αH2β))−∑αβ≠m+1(tr(HαHβ))2. | (2.23) |
Again (2.5), we derive
∑α≠m+1∑βabcζαabζβcaRαβbc=∑itrH2i∗−∑α,β≠m+1(tr(HαHβ)2−tr(H2αH2β)). | (2.24) |
Inserting (2.24) and (2.23) into (2.22), we obtain
12Δτ=∑α≠m+1∑abc(ζαabc)2+∑itrH2i∗−a′m(1+H2)τ+(1+a′)∑α≠m+1∑abclζαab(ζαclRlabc+ζαlaRlcbc)+a′∑αβ≠m+1(tr(HαHβ))2+(1−a′)∑αβ≠m+1(tr(HαHβ)2−tr(H2αH2β)). | (2.25) |
For a fixed α, we choose the orthonormal frame field {ea} such that ζαab=λαaδab. From (2.15), we get
∑abclζαab(ζαclRlabc+ζαlaRlcbc)=12∑ab(λαa−λαb)2Rabab≥12∑ab(λαi−λαj)2Rmin=mtrH2αRmin |
which implies that
∑α≠m+1∑abclζαab(ζαclRlabc+ζαlaRlcbc)≥mτRmin. | (2.26) |
In the implementation of DDVV (a conjecture for the Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken), demonstrated by the article [7], inequality of Lemma 2.2, we construct the following:
∑αβ≠m+1{tr(H2αH2β)−tr(HαHβ)2}=12∑αβ≠m+1tr(HαHβ−HβHα)2≤12(∑α≠m+1trH2α)2=12τ2. | (2.27) |
On the other hand, we have
∑αβ≠m+1(tr(HαHβ))2≥τ2m+2h−1. | (2.28) |
Setting a′=m+2h−1m+2h+1 in (2.25), and combining (2.26)–(2.28), we derive
12Δτ≥{−(m+2h−1m+2h+1)(κ+H2)+(2m+4hm+2h+1)Rmin}mτ. | (2.29) |
If our assumption (2.14) is satisfied, then we have
12Δτ≥0 |
by Hopf's lemma. This concludes that Δτ=0. Hence, we get the following:
τ=0orRN=(m+2h−12(m+2h))(κ+H2). |
For the first case, τ=0, and thus Nm is totally umbilical. For the second case, on the basis of (2.5), we derive
Rabab=κ+H2 |
and conclude that Nm is a totally umbilical sphere
Sm(1√κ+H2). |
Moreover, all inequalities (2.26)–(2.29) changed to equalities if
RN=(m+2h−12(m+2h))(κ+H2). |
Now, we will show that the second case can not occur. For this, we consider the equality (2.27) implies that either all H′αs are zero or two of the H′αs are nonzero α≠m+1. We estimate the following if the inequality in (2.28) and (2.29) converts into equalities:
trH2α=trH2β(α,β≠m+1),and∑ttrH2t∗=0. |
Thus Nm is totally umbilical with Rabab=κ+H2, as the H′sα are zero (α≠m+1). This leads to a contradiction. This completes the proof of the theorem.
In the results below, we have the following:
Theorem 2.2. Let Jξ be normal to an (m≥2)-dimensional totally real submanifold Nm in the complex space form ˜Nm+h(4κ). Then either Nm is totally umbilical or it satisfies the inequality
infρ≤m(κ+H2)(m−53) | (2.30) |
where the scalar curvature, the mean curvature, and the mean curvature vector are represented by ρ,H, and ξ, respectively.
Proof. Let Jξ be normal to Nm. We can consider em+1 is parallel to ξ, and we have
trHm+1=mH,trHα=0,α≠m+1. | (2.31) |
From (2.12), we have
12ΔΠ=∑αijk(ζαabc)2+∑αabcζαabζαccab+∑αabclζαab(ζαclRlabc+ζαlaRlcbc)−∑αβabcζαabζβcaRαβbc. | (2.32) |
From (2.5), (2.31), and if Nm is totally umbilical, we obtain
∑αabclζαab(ζαclRlabc+ζαlaRlcbc)=m(κ+H2)Π−m2H2+∑αβ{tr(HαHβ)2−tr(H2αH2β)}−∑αβ(tr(HαHβ))2, | (2.33) |
∑αβabcζαabζβcaRαβbc=−∑αβ{tr(HαHβ)2−tr(H2αH2β)}−∑atrH2a∗. | (2.34) |
In view of (2.31), and the pseudo-umbilical condition such that ζm+1ab=Hδab, we derive
∑αabcζαabζαccab=mHΔH, | (2.35) |
∑αabc(ζαabc)2≥∑ac(ζm+1aac)2=m∑a(∇aH)2, | (2.36) |
12ΔH2=HΔH+∑a(∇aH)2. | (2.37) |
By Lemma 2.2 and pseudo-umbilical condition ζm+1ab=Hδab, we have
2∑αβ{tr(HαHβ)2−tr(H2αH2β)}−∑αβtr(HαHβ)2=2∑αβ≠m+1{tr(HαHβ)2−tr(H2αH2β)}−∑αβ≠m+1tr(HαHβ)2−(trH2m+1)2≥−32τ2−m2H4=−32(Π−mH2)2−m2H4. | (2.38) |
Substituting (2.33)–(2.38) into (2.32), we have
12ΔΠ≥12mΔH2+m(κ+H2)Π−32(Π−mH2)2−m2H4−m2H2=12mΔH2+(Π−mH2){m(κ+H2)−32(Π−mH2)}=12mΔH2+τ{m(κ+H2)−32τ}. | (2.39) |
By the same argument as in [16], we conclude that either Mn is totally umbilical or
infρ≤m(κ+H2)(m−53). |
This completes the proof of the theorem.
Theorem 2.3. Let Jξ be normal to an (m≥2)-dimensional compact totally real submanifold Nm in the complex space form ˜Nm+h(4κ). Then we have the inequality
∫{2(κ+H2)Π−3Π2−5m2H4−4m2H2+2mH2}dV≤0 | (2.40) |
where H and Π denote the mean curvature of Nm and the squared norm of the length of the second fundamental form of Nm, respectively.
Proof. Without loss of generality, we consider e1∗ such that it is parallel to ξ and trH1∗=mH. In this case trHα=0, for α≠1∗, and Jξ is normal to Nm. Taking this together with (2.5), we have
∑αβabcζαabζβcaRαβbc=m2H2−∑atrH2a∗−∑αβ{tr(HαHβ)2−tr(H2αH2β)}≤m2H2−trH21∗−∑αβ{tr(HαHβ)2−tr(H2αH2β)}=m2H2−mH2−∑αβ{tr(HαHβ)2−tr(H2αH2β)}. | (2.41) |
By the same argument as in Theorem 2.2, we conclude that
12ΔΠ≥12mΔH2+m(κ+H2)Π−32(Π−mH2)2−m2H4−2m2H2+mH2. |
The boundary of Nm is compact, by Stokes's theorem, we obtain
∫{m(κ+H2)Π−32(Π−mH2)2−m2H4−2m2H2+mH2}≤0 |
which implies (2.40). This completes the proof of the theorem.
Remark 2.1. Note that if κ=1 in (2.36), then the complex space form ˜Nm+h(4κ) turns to a complex projective space with constant curvature of 1.
From the hypothesis, Theorem 2.1, we rewrite that
Theorem 2.4. Let Nm be an m-dimensional compact totally real submanifold in complex projective space CPm+h with parallel mean curvature. In this case, we have
RN≥(m+2h−12(m+2h))((1+H2), |
and thus Nm is a totally umbilical sphere Sm(1√1+H2), where H and Π denote the mean curvature of Nm and the squared norm of the length of the second fundamental form of Nm, respectively. Moreover, CPm+h has the constant sectional curvature 1.
Theorem 2.5. Let Nm be an (m≥2)-dimensional totally real submanifold in complex projective spaces CPm+h. If Jξ is normal to Nm, then either Nm is totally umbilical or satisfies the following inequality:
infρ≤m{1+H2}(m−53). |
Similarly, Theorem 2.3 can be written as if the mean curvature is minimal from Theorem 2.4
Corollary 2.1. Let Nm be an (m≥2)-dimensional compact totally real submanifold in complex projective spaces CPm+h. If Jξ is normal to Nm, then we have the following inequality:
∫{2(1+H2)Π−3Π2−5m2H4−4m2H2+2mH2}dV≤0. |
Remark 2.2. Using Remark 2.1 in Theorems 2.4 and 2.5, then Theorems 2.4 and 2.5 coincided with Theorems 1 and 2 in [21].
The study of totally real submanifolds in complex space forms is a rich area in differential geometry, with deep connections to complex geometry, curvature theory, and minimal submanifold theory. Their extrinsic curvature properties (like second fundamental form, mean curvature) are deeply influenced by the complex structure of the ambient space. This leads to classification results that help understand the geometric landscape of complex manifolds. In theoretical physics (e.g., string theory), totally real submanifolds relate to real slices of complexified spaces. In all these, totally real submanifolds offer tools to probe the nature of curvature, symmetry, and submanifold geometry.
Fatimah Alghamdi: Conceptualization, Investigation; Fatemah Mofarreh: Conceptualization, Methodology, Writing-review and editing Writing-original draft preparation, Investigation, Funding acquisition; Akram Ali: Conceptualization, Methodology, Investigation, Writing-review and editing; Mohamed Lemine Bouleryah: Methodology, Writing-original draft preparation, Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through a Large Research Project under grant number R.G.P.2/22/45. The author, Fatemah Mofarreh, expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
The authors declare no competing interests.
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