Contact CR-submanifolds in odd-dimensional spheres:

Contact CR-submanifolds in odd-dimensional spheres:

new examples

ECM 2021, Portoro˘z, Slovenia

Marian Ioan Munteanu

University Alexandru Ioan Cuza of Iaşi Romania

June 22, 2021

Table of Contents

1.CR-submanifolds

2. Semi-invariant submanifolds in almost contact metric manifolds

3. Minimal contactCRsubmanifolds inS²ⁿ⁺¹satisfying theδ(2)-Chen’s equality

4. ContactCRsubmanifolds inS⁷

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CR-submanifolds

History

(M, g),→

iso(M ,f g, Je )– Kähler manifold

T(M)its tangent bundle;T(M)^⊥its normal bundle Two important situations occur:

• _Tx(M)is invariant under the action ofJ:

J(T_x(M)) =T_x(M) for allx∈M

Mis calledcomplexsubmanifold orholomorphicsubmanifold

• _Tx(M)is anti-invariant under the action ofJ:

J(Tx(M))⊂T(M)^⊥_x for allx∈M Mis know as atotally realsubmanifold

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History

(M, g),→

iso(M ,f g, Je )– Kähler manifold

T(M)its tangent bundle;T(M)^⊥its normal bundle Two important situations occur:

• _Tx(M)is invariant under the action ofJ:

J(T_x(M)) =T_x(M) for allx∈M

M is calledcomplexsubmanifold or holomorphic submanifold

• _Tx(M)is anti-invariant under the action ofJ:

J(Tx(M))⊂T(M)^⊥_x for allx∈M Mis know as atotally realsubmanifold

History

(M, g),→

iso(M ,f g, Je )– Kähler manifold

T(M)its tangent bundle;T(M)^⊥its normal bundle Two important situations occur:

• _Tx(M)is invariant under the action ofJ:

J(T_x(M)) =T_x(M) for allx∈M

Mis calledcomplexsubmanifold orholomorphicsubmanifold

• T_x(M)is anti-invariant under the action ofJ:

J(Tx(M))⊂T(M)^⊥_x for allx∈M

M is know as a totally real submanifold 3/28

History

In1978A. Bejancu

•CR-submanifolds of a Kähler manifold. I, Proc. Amer. Math. Soc., 69 (1978), 135-142

•CR- submanifolds of a Kähler manifold. II, Trans. Amer. Math. Soc., 250 (1979), 333-345

started a study of the geometry of a class of submanifolds situated between the two classes mentioned above.

Such submanifolds were namedCR–submanifolds:

Mis aCR-submanifoldof a Kähler manifold(M ,f eg, J)if there exists a holomorphic distributionDonM, i.e.JD_x=D_x,∀x∈M and such that its orthogonal complementD^⊥is anti-invariant, namelyJD_x^⊥⊂T(M)^⊥_x,∀x∈M.

History

In1978A. Bejancu

•CR-submanifolds of a Kähler manifold. I, Proc. Amer. Math. Soc., 69 (1978), 135-142

•CR- submanifolds of a Kähler manifold. II, Trans. Amer. Math. Soc., 250 (1979), 333-345

started a study of the geometry of a class of submanifolds situated between the two classes mentioned above.

Such submanifolds were namedCR–submanifolds:

Mis aCR-submanifoldof a Kähler manifold(M ,f eg, J)if there exists a holomorphic distributionDonM, i.e.JD_x=D_x,∀x∈M and such that its orthogonal complementD^⊥is anti-invariant, namelyJD_x^⊥⊂T(M)^⊥_x,∀x∈M.

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Sasakian manifolds

(Mf^2m+1, ϕ, ξ, η,eg)Sasakian manifold:

ϕ∈ T₁¹(M),f ξ∈χ(Mf),η∈Λ¹(Mf):

ϕ² =−I+η⊗ξ, ϕξ = 0, η◦ϕ= 0, η(ξ) = 1

dη(X, Y) =eg(ϕX, Y) (the contact condition) eg(ϕX, ϕY) =eg(X, Y)−η(X)η(Y) (the compatibility condition) N =N_ϕ+ 2dη⊗ξ = 0 (the normality condition)

(∇e_Uϕ)V =−eg(U, V)ξ+η(V)U, U, V ∈χ(Mf)

Semi-invariant

submanifolds in almost contact metric

manifolds

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Contact CR-submanifolds

Sasakian geometry: odd dimensional version of Kählerian geometry, A new concept:contactCR-submanifold:

a submanifoldMof an almost contact Riemannian manifold(M ,f (ϕ, ξ,η,e eg)) carrying an invariant distributionD, i.e.ϕxD_x⊆ D_x, for anyx∈M, such that the orthogonal complementD^⊥ofDinT(M)is anti-invariant, i.e.

ϕ_xD_x^⊥⊆T(M)^⊥_x, for anyx∈M.

This notion was introduced byA.Bejancu & N.Papaghiucin Semi-invariant submanifolds of a Sasakian manifold,

An. Şt. Univ. "Al.I.Cuza" Iaşi, Matem., 1(1981), 163-170. by using the terminology ofsemi-invariant submanifold.

It is customary to require thatξbe tangent toMrather than normal which is too restrictive(K. Yano & M. Kon):Mmust be anti-invariant.

Contact CR-submanifolds

Sasakian geometry: odd dimensional version of Kählerian geometry, A new concept:contactCR-submanifold:

a submanifoldMof an almost contact Riemannian manifold(M ,f (ϕ, ξ,η,e eg)) carrying an invariant distributionD, i.e.ϕxD_x⊆ D_x, for anyx∈M, such that the orthogonal complementD^⊥ofDinT(M)is anti-invariant, i.e.

ϕ_xD^⊥_x ⊆T(M)^⊥_x, for anyx∈M.

This notion was introduced byA.Bejancu & N.Papaghiucin Semi-invariant submanifolds of a Sasakian manifold,

An. Şt. Univ. "Al.I.Cuza" Iaşi, Matem., 1(1981), 163-170. by using the terminology ofsemi-invariant submanifold.

It is customary to require thatξbe tangent toMrather than normal which is too restrictive(K. Yano & M. Kon):Mmust be anti-invariant.

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Contact CR-submanifolds

Sasakian geometry: odd dimensional version of Kählerian geometry, A new concept:contactCR-submanifold:

a submanifoldMof an almost contact Riemannian manifold(M ,f (ϕ, ξ,η,e eg)) carrying an invariant distributionD, i.e.ϕxD_x⊆ D_x, for anyx∈M, such that the orthogonal complementD^⊥ofDinT(M)is anti-invariant, i.e.

ϕ_xD^⊥_x ⊆T(M)^⊥_x, for anyx∈M.

This notion was introduced byA.Bejancu & N.Papaghiucin Semi-invariant submanifolds of a Sasakian manifold,

An. Şt. Univ. "Al.I.Cuza" Iaşi, Matem., 1(1981), 163-170.

by using the terminology ofsemi-invariant submanifold.

It is customary to require thatξbe tangent toMrather than normal which is too restrictive(K. Yano & M. Kon):Mmust be anti-invariant.

Contact CR-submanifolds

Sasakian geometry: odd dimensional version of Kählerian geometry, A new concept:contactCR-submanifold:

a submanifoldMof an almost contact Riemannian manifold(M ,f (ϕ, ξ,η,e eg)) carrying an invariant distributionD, i.e.ϕxD_x⊆ D_x, for anyx∈M, such that the orthogonal complementD^⊥ofDinT(M)is anti-invariant, i.e.

ϕ_xD^⊥_x ⊆T(M)^⊥_x, for anyx∈M.

This notion was introduced byA.Bejancu & N.Papaghiucin Semi-invariant submanifolds of a Sasakian manifold,

An. Şt. Univ. "Al.I.Cuza" Iaşi, Matem., 1(1981), 163-170.

by using the terminology ofsemi-invariant submanifold.

It is customary to require thatξbe tangent toMrather than normal which is too restrictive(K. Yano & M. Kon):Mmust be anti-invariant.

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Contact CR-submanifolds

Given a contactCRsubmanifoldM of a Sasakian manifoldMf eitherξ∈ D, orξ ∈ D^⊥. Therefore

T(M) =H(M)⊕Rξ⊕E(M)

H(M)is the maximally complex, distribution ofM;ϕE(M)⊆T(M)^⊥.

BothD:=H(M), D^⊥:=E(M)⊕Rξ D:=H(M)⊕Rξ, D^⊥:=E(M) organizeM as a contactCRsubmanifold

H(M)is never integrable (e.g.Capursi & Dragomir - 1990)

Contact CR-submanifolds

Given a contactCRsubmanifoldM of a Sasakian manifoldMf eitherξ∈ D, orξ ∈ D^⊥. Therefore

T(M) =H(M)⊕Rξ⊕E(M)

H(M)is the maximally complex, distribution ofM;ϕE(M)⊆T(M)^⊥. BothD:=H(M), D^⊥:=E(M)⊕Rξ

D:=H(M)⊕Rξ, D^⊥:=E(M) organizeM as a contactCRsubmanifold

H(M)is never integrable (e.g.Capursi & Dragomir - 1990)

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Contact CR-submanifolds

Given a contactCRsubmanifoldM of a Sasakian manifoldMf eitherξ∈ D, orξ ∈ D^⊥. Therefore

T(M) =H(M)⊕Rξ⊕E(M)

H(M)is the maximally complex, distribution ofM;ϕE(M)⊆T(M)^⊥. BothD:=H(M), D^⊥:=E(M)⊕Rξ

D:=H(M)⊕Rξ, D^⊥:=E(M) organizeM as a contactCRsubmanifold

H(M)is never integrable (e.g.Capursi & Dragomir - 1990)

Some inequalities

Theorem (Papaghiuc - 1984, M. - 2005)

LetMf^2m+1(c)be a Sasakian space form and letM =N^⊤×N^⊥be a contact CRproduct inM .f Then the norm of the second fundamental form ofM satisfies the inequality

||h||² ≥q ((c+ 3)s+ 2).

"=" holds if and only if bothN^⊤andN^⊥are totally geodesic inM .f

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Some inequalities

r:S^2s+1×S^2q+1−→S^2m+1 m=sq+s+q

(x0, y0, . . . , xs, ys; u0, v0, . . . , uq, vq)7−→(. . . , xjuα−yjvα, xjvα+yjuα, . . .) M =S^2s+1×S^p −→S^2s+1×S^2q+1−→^r S^2m+1

contactCRproduct inS^2m+1for which the equality holds.

Some inequalities

Theorem (Papaghiuc - 1984, M. - 2005)

LetMbe a strictly proper contactCRproduct in a Sasakian space form Mf^2m+1(c),withc̸=−3.Then

m≥sq+s+q.

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Some inequalities

Theorem (M. - 2005, Theorem 3.3)

LetM =N₁×_fN₂be a contactCRwarped product of a Sasakian space formMf^2m+1(c), that is a contactCR-submanifold inMf, such thatN₁is φ-invariant and tangent toξ, whileN₂isφ-anti-invariant. Then the second fundamental form ofMsatisfies the following inequality

||h||²≥2p

||∇lnf||²−∆ lnf+ c+ 3 2 s+ 1

.

Heref is the warping function which has to satisfyξ(f) = 0and∆f is the Laplacian off.

Interesting result in S

Theorem (M. - 2005)

LetM =N^T ×N^⊥be a strictly proper contactCRproduct inS⁷whose second fundamental form has the norm√

6.ThenM is the Riemannian product betweenS³andS¹and, up to a rigid transformation ofR⁸ the embedding is given by

r:S³×S¹ −→S⁷

r(x₁, y₁, x₂, y₂, u, v) = (x₁u, y₁u, −y₁v, x₁v, x₂u, y₂u, −y₂v, x₂v).

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Minimal contact CR submanifolds in S ²ⁿ⁺¹ satisfying the

δ (2) -Chen’s equality

δ(2) invariant

In 1993, B.-Y. Chen:

δ(2)(p) =τ(p)−(infK)(p) where

(infK)(p)= inf

K(π)|πis a 2-dimensional subspace ofTpM andτ(p) =P

i<j

K(ei∧ej)denotes the scalar curvature

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δ(2) invariant

In 1993, B.-Y. Chen:

δ(2)(p) =τ(p)−(infK)(p)

where

(infK)(p)= inf

K(π)|πis a 2-dimensional subspace ofTpM andτ(p) =P

i<j

K(ei∧ej)denotes the scalar curvature

δ(2) invariant; δ(2) ideals

In real space formsMf(c):

δ(2)≤ ⁿ_2(n−1)²⁽ⁿ⁻²⁾∥H∥²+1

2(n−2)(n+ 1)c.

Note that, forn= 2, both sides of the above inequality are zero.

A submanifold is calledδ(2)idealif and only if at each point it realizes the equality in the above inequality.

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Recent results in odd dimensional spheres

M.I. Munteanu, L. Vrancken:

Minimal contactCRsubmanifolds inS²ⁿ⁺¹satisfying theδ(2)-Chen’s equality, Journal of Geometry and Physics,75(2014), 92–97.

LetMbe a minimal proper contactCRsubmanifold of dimensionnin the (2m+ 1)-dimensional sphereS^2m+1.

Assuming that the contactCRstructure is proper:

•q=1.

•nis an even number.

Recent results in odd dimensional spheres

Theorem (M., Vrancken - 2014)

LetMⁿbe a proper, minimal, contactCR-submanifold ofS^2m+1which is a δ(2)-ideal. Thennis even and there exists a totally geodesic SasakianSⁿ⁺¹in S^2m+1containingMas a hypersurface.

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Recent results in odd dimensional spheres

Theorem (M., Vrancken - 2014)

LetMⁿ,neven, be a proper, minimal, contactCR-submanifold ofS^2m+1which is aδ(2)-ideal. Then there exists a minimal surface inSⁿ⁺¹such that locally Mcan be considered as the unit normal bundle of this minimal surface.

Theorem (M., Vrancken - 2014)

LetMⁿ,neven, be a proper, minimal, contactCR-submanifold ofS^2m+1 which is aδ(2)-ideal. ThenM can be locally considered as the unit normal bundle of the Clifford torusS¹(^√1

2)×S¹(^√1

2)⊂S³ ⊂Sⁿ⁺¹.

Recent results in odd dimensional spheres

Theorem (M., Vrancken - 2014)

LetMⁿ,neven, be a proper, minimal, contactCR-submanifold ofS^2m+1which is aδ(2)-ideal. Then there exists a minimal surface inSⁿ⁺¹such that locally Mcan be considered as the unit normal bundle of this minimal surface.

Theorem (M., Vrancken - 2014)

LetMⁿ,neven, be a proper, minimal, contactCR-submanifold ofS^2m+1 which is aδ(2)-ideal. ThenM can be locally considered as the unit normal bundle of the Clifford torusS¹(^√1

2)×S¹(^√1

2)⊂S³ ⊂Sⁿ⁺¹.

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Contact CR

submanifolds in S ⁷

Four dimensional minimal contact

CR submanifolds in S

satisfying Chen’s equality

We find (locally) coordinates onM such that the immersionF :M −→S⁷is expressed as

F(s, t, u, v) = cosscostcosu e₁+ sinscostcosu e₂ + cosscostsinu e₃+ sinscostsinu e₄ + sintcosv e5+ sintsinv e6.

Remark thatM lies in a 5-dimensional sphere in a 6-dimensionalJ invariant subspace ofR⁸:e2 =J e1,e4=J e3,e6=J e5.

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New problem:

LetMbe a contactCRsubmanifold in the Sasakian sphereS⁷. Recall that:

T(M) =H(M)⊕E(M)⊕[ξ]

T^⊥(M) =φE(M)⊕ν(M)

Problem.Find all proper contactCRsubmanifolds inS⁷such that h(H(M), H(M)) = 0 & h(E(M), E(M)) = 0.

New problem:

LetMbe a contactCRsubmanifold in the Sasakian sphereS⁷. Recall that:

T(M) =H(M)⊕E(M)⊕[ξ]

T^⊥(M) =φE(M)⊕ν(M)

Problem.Find all proper contactCRsubmanifolds inS⁷such that h(H(M), H(M)) = 0 & h(E(M), E(M)) = 0.

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Work in progress

We have:

s+q+r=3

where2s= dim(H(M)),q = dim(E(M)),2r= dim(ν(M)).

Then:

I. s=q=r= 1, hencedim(M) = 4 II. s= 1,q = 2,r= 0hencedim(M) = 5

III. s= 2,q = 1,r= 0henceM is a hypersurface inS⁷

4-dimensional contact CR submanifolds in S

Problem solved: with M. Djorić and L. Vrancken;

Math. Nachr. 290 (2017) 16, 2585–2596.

Theorem (Djorić, M., Vrancken, 2017)

LetMbe a4-dimensional nearly totally geodesic contactCR-submanifold in S⁷. TheM is locally congruent with one of the following immersions:

➀

F(u, v, s, t) =

cosssinte^iλu,costsinve^iµu,

−sinssinte^iλu,costcosve^iµu

➁ F :S³×R−→R⁸,F(y, t) = (cost y,sint y).

➂ F(v, u, t, s) =e^iv e^iscostcost0cosu+e^−issintsint0,

−ie^iscostsint₀cosu+ie^−issintcost₀, e^iscostsinu,0 .

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4-dimensional contact CR submanifolds in S

Theorem (Djorić, M., Vrancken, 2017)

LetMbe a4-dimensional nearly totally geodesic contactCR-submanifold in S⁷. TheM is locally congruent with one of the following immersions:

➀

F(u, v, s, t) =

cosssinte^iλu,costsinve^iµu,

−sinssinte^iλu,costcosve^iµu

➁ F :S³×R−→R⁸,F(y, t) = (cost y,sint y).

➂ F(v, u, t, s) =e^iv e^iscostcost₀cosu+e^−issintsint₀,

−ie^iscostsint₀cosu+ie^−issintcost₀, e^iscostsinu,0 .

5-dimensional contact CR submanifolds in S

Problem solved:with M. Djoric – 2020

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5-dimensional contact CR submanifolds in S

Theorem (Djoric and M., 2020)

LetM be a five-dimensional proper nearly totally geodesic contactCR−submanifold of seven-dimensional unit sphere. ThenMis locally congruent to warped products S³×f₁S¹×f₂S¹andS³×fS²via the immersions

➀

F(x, y, z, u, v) =

cosycosucos(z+^x₂),cosycosusin(z+^x₂), sinysinvcos(z−^x₂),sinysinusin(z−^x₂), sinycosvcos(z−^x₂),sinycosvsin(z−^x₂), cosysinucos(z+ ^x₂),cosysinusin(z+ ^x₂)

fory∈(0,^π₂),x, z, u, v∈R.

5-dimensional contact CR submanifolds in S

Theorem (Djoric and M., 2020)

LetM be a five-dimensional proper nearly totally geodesic contactCR−submanifold of seven-dimensional unit sphere. ThenMis locally congruent to warped products S³×f₁S¹×f₂S¹andS³×fS²via the immersions

➁

F(. . .) =−cosy(1−T) cos z+ ^x₂

,sin z+^x₂

,0,0,0,0,0,0 + cosyT u 0,0,cos z+^x₂

,sin z+^x₂

,0,0,0,0, + cosyT v 0,0,0,0,0,0,cos z+^x₂

,sin z+^x₂ + siny 0,0,0,0,cos z−^x₂

,sin z−^x₂ ,0,0

. whereT = 2/(1 +u²+v²).

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4-dimensional contact CR submanifolds in S

Example 1.

F˜:R⁴×R² −→R⁸

F˜(x1, y1, x2, y2;u, v) = (x1u, y1u, x2u, y2u, x1v, y1v, x2v, y2v), is not an immersion.

Its restriction F :M =S³×S¹−→S⁷

F(x1, y1, x2, y2;u, v) = (x1u, y1u, x2u, y2u, x1v, y1v, x2v, y2v) (with standard metrics) is an isometric immersion.

the characteristic vector field

ξ= (−y₁, x1,−y₂, x2) =J pforp= (x1, y1, x2, y2)ofS³ F∗ξ =ξ

ζ = (−x₂, y2, x1,−y₁), µ= (−y₂,−x₂, y1, x1) andψ= (−v, u) SetH(M) = span{ζ, µ}andE(M) = span{ψ}

M becomes anearly t.g.contactCR-submanifold ofS⁷

4-dimensional contact CR submanifolds in S

Example 1.

Its restriction F :M =S³×S¹−→S⁷

F(x₁, y₁, x₂, y₂;u, v) = (x₁u, y₁u, x₂u, y₂u, x₁v, y₁v, x₂v, y₂v) (with standard metrics) is an isometric immersion.

the characteristic vector field

ξ= (−y₁, x₁,−y₂, x₂) =J pforp= (x₁, y₁, x₂, y₂)ofS³ F∗ξ =ξ

ζ = (−x₂, y2, x1,−y₁), µ= (−y₂,−x₂, y1, x1) andψ= (−v, u) SetH(M) = span{ζ, µ}andE(M) = span{ψ}

M becomes anearly t.g.contactCR-submanifold ofS⁷

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4-dimensional contact CR submanifolds in S

Example 1.

Its restriction F :M =S³×S¹−→S⁷

F(x₁, y₁, x₂, y₂;u, v) = (x₁u, y₁u, x₂u, y₂u, x₁v, y₁v, x₂v, y₂v) (with standard metrics) is an isometric immersion.

the characteristic vector field

ξ= (−y₁, x₁,−y₂, x₂) =J pforp= (x₁, y₁, x₂, y₂)ofS³ F∗ξ =ξ

ζ = (−x₂, y2, x1,−y₁), µ= (−y₂,−x₂, y1, x1) andψ= (−v, u) SetH(M) = span{ζ, µ}andE(M) = span{ψ}

Mbecomes anearly t.g.contactCR-submanifold ofS⁷

4-dimensional contact CR submanifolds in S

Example 1.

Its restriction F :M =S³×S¹−→S⁷ Chen type inequality F(x₁, y₁, x₂, y₂;u, v) = (x₁u, y₁u, x₂u, y₂u, x₁v, y₁v, x₂v, y₂v)

(with standard metrics) is an isometric immersion.

the characteristic vector field

ξ= (−y₁, x₁,−y₂, x₂) =J pforp= (x₁, y₁, x₂, y₂)ofS³ F∗ξ =ξ

ζ = (−x₂, y2, x1,−y₁), µ= (−y₂,−x₂, y1, x1) andψ= (−v, u) SetH(M) = span{ζ, µ}andE(M) = span{ψ}

Mbecomes anearly t.g.contactCR-submanifold ofS⁷

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4-dimensional contact CR submanifolds in S

Example 2.

F :M =S³×S¹−→S⁷

F(x1, y1, x2, y2;u, v) = (x1u, y1u, x1v, y1v, x2, y2,0,0), with the warped metric g_M =g_S3 +f²g_S2 where

f :D⊂S³ →R, f(x1, y1, x2, y2) =p

x²₁+y₁² F is an isometric immersion and we have:

(i) Misnearly totally geodesic;

(ii) Mis minimal and satisfies the equality in two Chen type inequalities from my thesis;

(iii) Mis aδ(2)-ideal inS⁷.[M. and Vrancken - 2014]

4-dimensional contact CR submanifolds in S

Example 2.

F :M =S³×S¹−→S⁷

F(x1, y1, x2, y2;u, v) = (x1u, y1u, x1v, y1v, x2, y2,0,0), with the warped metric g_M =g_S3 +f²g_S2 where

f :D⊂S³ →R, f(x1, y1, x2, y2) =p

x²₁+y₁² F is an isometric immersion and we have:

(i) M isnearly totally geodesic;

(ii) M is minimal and satisfies the equality in two Chen type inequalities from my thesis;

(iii) M is aδ(2)-ideal inS⁷.[M. and Vrancken - 2014]

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5-dimensional contact CR submanifolds in S

Example 1.[Djoric and M., 2020]

F :M =S³×S²−→S⁷

F(x1, y1, x2, y2;u, v, w) = (x1u, y1u, x1v, y1v, x1w, y1w, x2, y2).

isometric immersion: warped metric onM

g_M =g_S3 +f²g_S2,where f :D⊂S³ →R, f(x₁, y₁, x₂, y₂) = q

x²₁+y₁².

Proposition.

(i) Misnearly totally geodesic;

(ii) Mis minimal and satisfies the equality in two Chen type inequalities from my thesis.

5-dimensional contact CR submanifolds in S

Example 1.[Djoric and M., 2020]

F :M =S³×S²−→S⁷

F(x1, y1, x2, y2;u, v, w) = (x1u, y1u, x1v, y1v, x1w, y1w, x2, y2).

isometric immersion: warped metric onM

g_M =g_S3 +f²g_S2,where f :D⊂S³ →R, f(x₁, y₁, x₂, y₂) = q

x²₁+y₁².

Proposition.

(i) M isnearly totally geodesic;

(ii) M is minimal and satisfies the equality in two Chen type inequalities from my thesis.

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5-dimensional contact CR submanifolds in S

Example 2.[Djoric and M., 2020]

F :S³×_f1 S¹×_f2 S¹ −→S⁷

F(x₁,y₁,x₂,y₂;u,v;a,b) = (ux₁,uy₁,vx₁,vy₁,ax₂,ay₂,bx₂,by₂), where the warping functionsf1, f2:D⊂S³ →(0,∞)are given by

f1(x1,y1,x2,y2) =p

x²₁+y²₁ f₂(x₁,y₁,x₂,y₂) =p

x²₂+y²₂.

Main References

[thesis] M. I. Munteanu,Warped product contactCR-submanifolds of Sasakian space forms, Publ. Math. Debrecen661-2 (2005), 75–120.

[MV14] M.I. Munteanu and L. Vrancken,Minimal contactCRsubmanifolds inS²ⁿ⁺¹ satisfying theδ(2)-Chen’s equality, J. Geom. Phys.75(2014), 92–97.

[DMV17] M. Djoric, M.I. Munteanu and L. Vrancken, Four-dimensional contact CR-submanifolds inS⁷(1), Math. Nachr.,290(2017) 16, 2585–2596.

[DM20a] M. Djoric and M. I. Munteanu,On certain contactCR-submanifolds inS⁷, Contemporary Mathematics,Geometry of submanifolds, Eds. (J. van der Veken et al.)756(2020) 111–120.

[DM20b] M. Djoric and M.I. Munteanu,Five-dimensional contactCR-sub- manifolds inS⁷(1), Mathematics, Special IssueRiemannian Geometry of Submanifolds, Guest Editor: L. Vrancken,8(2020) 8, art. 1278.

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Other References

[ADV07] M. Antić, M. Djorić and L. Vrancken, 4-dimensional minimal CR

submanifolds of the sphereS⁶satisfying Chen’s equality, Diff. Geom. Appl.,25 (2007) 3, 290–298.

[AV15] M. Antić and L. Vrancken,Three-Dimensional MinimalCRSubmanifolds of the SphereS⁶Contained in a Hyperplane, Mediterr. J. Math.12(2015) 4, 1429–1449.

[DV06] M. Djorić and L. Vrancken,Three-dimensional minimalCRsubmanifolds inS⁶ satisfying Chen’s equality, J. Geom. Phys.,56(2006) 11, 2279–2288.

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