Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres

Anne Wijffels; Joeri Van der Veken

REVIEW 1 major objections 3 minor 11 references

Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Gauss map of every hypersurface of a unit sphere is a Lagrangian immersion into the complex quadric, and conversely every Lagrangian submanifold arises locally this way.

desk verdict Solid, useful extension of the Gauss-map dictionary to arbitrary hypersurfaces of spheres; the converse proof has a repairable conjugation typo in the shape-operator formula. read the letter →

arxiv 1908.05468 v1 pith:RGDMK2BP submitted 2019-08-15 math.DG

Joeri Van der Veken , Anne Wijffels This is my paper · ORCID

classification math.DG MSC 53C4253D1253B25

keywords LagrangiansubmanifoldscomplexquadricGaussmaphypersurfacesofspheresprincipalcurvaturesanglefunctionshorizontalliftsisoparametric

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a two-way correspondence between hypersurfaces of the unit sphere $S^{n+1}(1)$ and Lagrangian submanifolds of the complex quadric $Q^n$. Its central claim is that the Gauss map $G(p)=[a(p)+ib(p)]$ of any hypersurface is a Lagrangian immersion, and that conversely every Lagrangian submanifold of $Q^n$ is locally the Gauss map of some sphere hypersurface. The explicit construction runs through horizontal lifts to the Stiefel manifold, and it comes with a quantitative statement: with the natural choice of almost product structure, the principal curvatures $\lambda_j$ and the angle functions $\theta_j$ satisfy $\lambda_j=\cot\theta_j$, and more generally $\cot(\theta_j-\theta_k)=\pm(\lambda_j\lambda_k+1)/(\lambda_j-\lambda_k)$. This matters because both sides of the relation are defined through choices, but the combination on the right is shown to be independent of those choices.

What carries the argument

The central machinery is the family $\mathcal A$ of almost product structures on the complex quadric $Q^n=\{z_0^2+\cdots+z_{n+1}^2=0\}\subset CP^{n+1}(4)$—symmetric operators with $A^2=\mathrm{Id}$ that anti-commute with $J$—together with the local angle functions $\theta_j$ they induce on a Lagrangian immersion by $A(df)e_j=\cos(2\theta_j)(df)e_j-\sin(2\theta_j)J(df)e_j$. The bridge between spheres and the quadric is the Stiefel manifold of oriented orthonormal two-frames in $\mathbb R^{n+2}$: the Gauss map lifts horizontally to $\hat G=(a+ib)/\sqrt2$, and any other lift differs by $e^{it}$, exactly the passage to a parallel hypersurface. The curvature-angle identities follow by differentiating the horizontal lift, $(d\hat G)e_j=(1-i\lambda_j)e_j/\sqrt2$, and comparing the action of the shape operator $A$ through the projection $\pi$ with the defining equation of the angle functions.

What would settle it

Recompute equation (3.6) with the formula $AX=-(d\pi)(\hat X)$ exactly as printed in Remark 2.1: the left side is $-d\hat f$ and the right side becomes $e^{-2i\theta}d\hat f$, forcing $d\hat f=0$ except for special angles. Replacing the formula by $AX=-(d\pi)(\overline{\hat X})$ restores consistency and yields $\lambda_j=\cot\theta_j$; checking which identity the geometry actually satisfies decides whether the theorem holds as stated.

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Extended reading notes

Core claim

The central claim is that the Gauss map $G:M^n\to Q^n$, $p\mapsto[a(p)+ib(p)]$, of a hypersurface $a:M^n\to S^{n+1}(1)$ with unit normal $b$, is a Lagrangian immersion into the complex quadric: the complex structure $J$ of $Q^n$ sends the tangent space of the image onto its normal space. Taking the canonical horizontal lift $\hat G=(a+ib)/\sqrt2$ and the associated almost product structure $A$ specified in Remark 2.1, the paper computes $A(dG)e_j=\frac{\lambda_j^2-1}{\lambda_j^2+1}(dG)e_j-\frac{2\lambda_j}{\lambda_j^2+1}J(dG)e_j$, which is precisely the angle-function equation (2.1) with $\lambda_j=\cot\theta_j$. Conversely, every Lagrangian immersion $f:M^n\to Q^n$ is locally the Gauss map of a hypersurface of the sphere: for each point there is a neighbourhood $U$ and an immersion $a:U\to S^{n+1}(1)$ with Gauss map $f|_U$, obtained from a horizontal lift $\hat f_t=e^{it}\hat f_0$ and the parallel-hypersurface family $a_t=(\cos t)a_0-(\sin t)b_0$. For any such hypersurface the paper proves $\cot(\theta_j-\theta_k)=\pm\frac{\lambda_j\lambda_k+1}{\lambda_j-\lambda_k}$ in points where $\lambda_j\neq\lambda_k$, with the sign coming from the orientation of the normal and the right-hand side unaffected by $t$ or by the choice of $A$.

Load-bearing premise

The converse construction and the curvature-angle formula rest on the operative formula $AX=-(d\pi)(\hat X)$ in Remark 2.1; as printed, plugging it into (3.6) gives $-X=e^{-2i\theta}X$, which forces $X=0$ unless $\theta\equiv\pi/2\pmod\pi$, so the claimed derivation depends on that formula being corrected (with a complex conjugate) rather than on the text as typeset.

Editorial extensions

If this is right

Every hypersurface of a unit sphere, not only isoparametric ones, has a Lagrangian Gauss image in the complex quadric.
Every Lagrangian submanifold of $Q^n$ is locally determined by a sphere hypersurface, so local questions about one class can be translated into questions about the other.
The value of $\cot(\theta_j-\theta_k)$ is independent of the almost product structure and of the parallel hypersurface chosen, so the combination $(\lambda_j\lambda_k+1)/(\lambda_j-\lambda_k)$ records a genuine invariant of the Lagrangian submanifold.
Changing to the parallel hypersurface $a_t$ shifts all angle functions by the same constant $t$, leaving the angle differences and hence the curvature combination unchanged.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An inference from the construction is that the suspicious sign in Remark 2.1 is likely a typo: if the operator reads $AX=-(d\pi)(\overline{\hat X})$ with a complex conjugate, then equation (3.6) is consistent and the main theorem follows as stated.
A further inference: the combination $\pm(\lambda_j\lambda_k+1)/(\lambda_j-\lambda_k)$ is choice-independent, so it defines a pointwise invariant of a Lagrangian submanifold of $Q^n$ that could be computed directly and used to test whether the submanifold is locally a Gauss map.
Because the construction is purely local and uses only horizontal lifts, a testable extension is that the same correspondence should hold with the sphere replaced by other space forms, such as hyperbolic space, with a suitable quadric-like target.
In the constant-angle case the construction recovers the Gauss images of isoparametric hypersurfaces; conversely, constancy of these choice-independent ratios could be used to characterize which Lagrangian submanifolds arise from isoparametric hypersurfaces.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core result is a genuine generalization of the isoparametric case from [2]: the Gauss map of any hypersurface of a unit sphere is Lagrangian into the complex quadric, with principal curvatures related to the angle functions by lambda_j = cot theta_j, and there is a local converse. What is new is the explicit converse construction via horizontal lifts and the cotangent-difference formula for the angle functions under arbitrary choices of the almost product structure. The forward half is a direct computation and it is clean.

The one real problem is in the sign convention. Remark 2.1 states A X = -(dπ)(hat X), but the computation later requires the complex conjugate of the horizontal lift. As printed, Eq. (3.6) becomes -Y = e^{-2iθ}Y for Y = (d\hat f)e, forcing Y = 0 unless θ ≡ π/2 mod π, which would collapse the converse. The fix is exactly what the stress-test note says: insert the conjugation in Eq. (2.2), so the formula reads A X = -dπ(conjugate of hat X). With that repaired, Eq. (3.6) gives -conj(Y) = e^{-2iθ}Y, which has nonzero solutions and yields both λ = cot θ and the cot-difference formula. I have not checked every tensor index, but the rest of the proof looks coherent under that repair. This is a repairable notational slip, not a fatal flaw.

The paper leans on [2] for the angle-shift lemma when changing the almost product structure. That is legitimate: [2] is by the same group, the lemma is cited precisely, and it is not the target theorem. The citation pattern is otherwise standard (Reckziegel, Palmer, Smyth). The geometric claims are plausible and the explicit constructions are genuinely useful.

This is a modest but solid contribution for people working on submanifolds of the complex quadric and on structural approaches to isoparametric hypersurfaces. The printed version should not be accepted without a referee flagging the typo and checking the corrected derivation, but the paper deserves serious peer review. I would send it out and ask for the conjugation fix plus a short verification of the tensor conventions. It is not a breakthrough, but it is honest, useful work.

Referee Report

1 major / 3 minor

Summary. The paper studies the Gauss map of a hypersurface a: M^n -> S^{n+1}(1) into the complex quadric Q^n, defined by G(p)=[a(p)+ib(p)]. The main claims are: (1) G is a Lagrangian immersion; (2) with the canonical horizontal lift \hat G=(a+ib)/\sqrt{2} and the associated shape operator A from Remark 2.1, the principal curvatures λ_j and the angle functions θ_j from (2.1) satisfy λ_j = cot θ_j; (3) conversely, every Lagrangian immersion f: M^n -> Q^n is locally the Gauss map of an immersion a into S^{n+1}(1), and for any such hypersurface the angle functions and principal curvatures satisfy cot(θ_j-θ_k) = ±(λ_jλ_k+1)/(λ_j-λ_k). The proof uses horizontal lifts of Lagrangian immersions and derives the curvature–angle correspondence by direct calculation. The paper generalizes earlier results for isoparametric hypersurfaces in spheres to arbitrary hypersurfaces, and it clarifies the dependence of the correspondence on the choice of horizontal lift and almost product structure.

Significance. If the results stand, the paper provides a clean and explicit correspondence between the extrinsic geometry of hypersurfaces of spheres and the intrinsic Lagrangian geometry of the complex quadric. The relation λ_j = cot θ_j is simple, coordinate-free, and does not depend on the isoparametric assumption, so it is a genuine structural extension of the work in [2]. The converse construction, producing all parallel hypersurfaces with a given Gauss map, is natural and is presented with explicit formulas for the derivatives in (3.8)–(3.9). The derivation is direct and does not appear circular: the only reliance on the prior work [2] is the standard angle-shift formula under change of the almost product structure, which is a lemma rather than the target theorem. However, the proof as printed has a load-bearing algebraic inconsistency in the shape-operator formula, so the manuscript is not yet in publishable form.

major comments (1)

[Remark 2.1 and Section 3, Eq. (3.6)] The shape-operator formula in Remark 2.1 is stated as AX = -(dπ)_{\hat f(p)}(\hat X) (Eq. (2.2) in the special case). This formula is used in both directions of Theorem 3.1, and as printed it makes the proof inconsistent. In the forward direction, combining (2.2) with (3.4) would give A(dG)e_j = -(dπ)((1-iλ_j)e_j/\sqrt{2}), but the text substitutes (1+iλ_j)e_j/\sqrt{2} before applying dπ; this step is only correct if the shape operator is defined with a complex conjugation, i.e., AX = -(dπ)(\overline{\hat X}). In the converse direction, lifting (2.1) by this same formula yields Eq. (3.6): -(d\hat f_t)e_j = e^{-2iθ_j^{(t)}}(d\hat f_t)e_j. Since d\hat f_t is injective, this equation forces e^{-2iθ_j^{(t)}} = -1 and hence θ_j^{(t)} ≡ π/2 (mod π) for every j, contradicting the subsequent derivation of λ_j^{(t)} = cot(θ_j^{(0)}+t) in Eq. (3.9). With the conjugated formula, Eq. (3.6) becomes -\overline{(d\hat f_t)e_j} = e^{-2iθ_j^{(t)}}(d\hat f_t)e_j, which has nonzero solutions and reproduces the claimed relation. This is a load-bearing gap: the central theorem depends on a formula that the text states without the conjugation it later uses. I suspect this is a typographical omission, but the proof must be corrected for the theorem to be valid.

minor comments (3)

[Remark 2.1] The definition of ζ as ζ_{f(p)} = (dπ)_{\hat f(p)}(\hat f(p)) is imprecise because dπ acts on tangent vectors, not on points of V^{2n+1}; the intended construction should be clarified, for example by writing the normal vector field explicitly in terms of \overline{\hat f} or a tangent representative.
[Section 3, Eqs. (3.6), (3.8), (3.9)] The notation θ_j^(t) (with a caret before the parenthesized t) is typographically confusing; please use θ_j^{(t)} consistently throughout the proof.
[Section 3, proof of Theorem 3.1] The step 'This implies that the frame {e_1^{(t)},...,e_n^{(t)} does not depend on t' is stated without justification. When the eigenvalues of the shape operator A_0 have multiplicities, the adapted frame in (2.1) is not unique; a short argument (e.g., by choosing an orthonormal eigenframe at a point and extending by continuity, or by noting that the relation (3.3) is only needed where the λ_j are distinct) would make the proof complete.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Gauss-map/angle-function correspondence is a direct calculation, with only a supporting (non-load-bearing) self-citation to [2]; the apparent sign/conjugation issue in Remark 2.1 is a correctness typo, not a circular step.

full rationale

The central relation lambda_j = cot(theta_j) is derived by an explicit computation, not by fitting or by definition: the canonical horizontal lift (3.1) gives (d hat G)e_j = (1 - i lambda_j)e_j/sqrt(2), and the proof then computes A(dG)e_j and compares the coefficients with the angle-function representation (2.1). This is a direct calculation, so Theorem 3.1's forward direction is self-contained modulo the cited structural lemma. The converse direction constructs horizontal lifts using [7] (an external reference) and derives lambda_j^(t) = cot(theta_j^(0)+t) from equations (3.6)-(3.9); it does not assume the result it is proving. The only same-group dependency is reference [2] (which includes the first author), used for the existence of angle functions satisfying (2.1) and for the angle-shift formula theta_j = theta_j^(0) - phi/2 under changes of A. These are supporting facts about the family of almost product structures on Q^n, not the Gauss-map correspondence itself, and they are parameter-free with assumptions that do not include the target theorem, so they do not amount to circularity. A separate correctness concern: Remark 2.1's formula A X = -(d pi)(hat X) appears to omit a complex conjugation; as printed it makes equation (3.6) force e^{-2i theta} = -1 and would block the derivation unless the intended conjugated formula is restored. This is an apparent typo in the operative formula, not a circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities appear. The paper's central claim is a correspondence theorem about known geometric objects; the only nontrivial imported input is the horizontal-lift shape-operator formula in Remark 2.1, which is listed as an axiom because the printed formula appears to miss a complex conjugation.

assumptions (4)

standard math Almost product structures A on Q^n from shape operators are involutive, symmetric, anti-commute with J, and satisfy the derivative formulas in Lemma 1.2.
Cited from Smyth [10] and Reckziegel [8]; used throughout Section 1 and in the definition of angle functions.
standard math Every Lagrangian immersion into Q^n locally admits a horizontal lift into V^{2n+1}, and any two horizontal lifts differ by e^{it}.
From Reckziegel [7]; the converse of Theorem 3.1 is built on this.
ad hoc to paper The shape operator associated to a horizontal lift is A X = -(dπ)(conjugate of hat X); the submitted text writes A X = -(dπ)(hat X) without the conjugation.
Remark 2.1, Eq. (2.2); the proof of Theorem 3.1 uses the conjugate implicitly, and without it Eq. (3.6) is algebraically inconsistent.
standard math Changing A changes the angle functions by θ_j = θ_j^0 - φ/2.
Taken from [2], an unpublished same-group manuscript; used to show θ_j - θ_k is independent of A in Eq. (3.3).

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Pith. "Pith review of Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres." pith.science (2026). https://pith.science/paper/RGDMK2BP

@misc{pith2026190805468,
  author       = {Pith},
  title        = {Pith review of: Lagrangian submanifolds of the complex quadric as Gauss maps of hypersurfaces of spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGDMK2BP}},
  note         = {Machine review of arXiv:1908.05468}
}

abstract

The Gauss map of a hypersurface of a unit sphere $S^{n+1}(1)$ is a Lagrangian immersion into the complex quadric $Q^n$ and, conversely, every Lagrangian submanifold of $Q^n$ is locally the image under the Gauss map of several hypersurfaces of $S^{n+1}(1)$. In this paper, we give explicit constructions for these correspondences and we prove a relation between the principal curvatures of a hypersurface of $S^{n+1}(1)$ and the local angle functions of the corresponding Lagrangian submanifold of $Q^n$. The existence of such a relation is remarkable since the definition of the angle functions depends on the choice of an almost product structure on $Q^n$ and since several hypersurfaces of $S^{n+1}(1)$, with different principal curvatures, correspond to the same Lagrangian submanifold of $Q^n$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[2]

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work page Pith review arXiv
[1]

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work page 1969
[3]

Ma and Y

H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics and is opara- metric hypersurfaces in spheres , Math. Z. 261 (2009), 749–785. MR 2480757

work page 2009
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work page 2014
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[9]

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MR 0206881. KU Leuven, Department of Mathematics, Celestijnenlaan 200 B - Box 2400, 3001 Leuven, Belgium E-mail address : joeri.vanderveken@kuleuven.be KU Leuven, Department of Mathematics, Celestijnenlaan 200 B - Box 2400, 3001 Leuven, Belgium E-mail address : anne.wijffels@k...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

H. Li, H. Ma, J. Van der Veken, L. Vrancken and X. W ang, Minimal Lagrangian submanifolds of the complex hyperquadric , Sci. China Math., to appear. arXiv:1812.07888

work page Pith review arXiv

[2] [1]

G. R. Jensen, Homogeneous Einstein spaces of dimension four , J. Diﬀerential Geom. 3 (1969), 309–349. MR 0261487

work page 1969

[3] [3]

Ma and Y

H. Ma and Y. Ohnita, On Lagrangian submanifolds in complex hyperquadrics and is opara- metric hypersurfaces in spheres , Math. Z. 261 (2009), 749–785. MR 2480757

work page 2009

[4] [4]

Ma and Y

H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous is opara- metric hypersurfaces. I , J. Diﬀerential Geom. 97 (2014), 275–348. MR 3263508

work page 2014

[5] [5]

Ma and Y

H. Ma and Y. Ohnita, Hamiltonian stability of the Gauss images of homogeneous is opara- metric hypersurfaces II , Tohoku Math. J. (2) 67 (2015), 195–246. MR 3365370

work page 2015

[6] [6]

Palmer, Hamiltonian minimality and Hamiltonian stability of Gauss maps, Diﬀerential Geom

B. Palmer, Hamiltonian minimality and Hamiltonian stability of Gauss maps, Diﬀerential Geom. Appl. 7 (1997), 51–58. MR 1441918

work page 1997

[7] [7]

H. Reckziegel, Horizontal lifts of isometric immersions into the bundle sp ace of a pseudo- Riemannian submersion , Global Diﬀerential Geometry and Global Analysis 1984, Lec ture Notes in Mathematics, vol. 1156, Springer, Berlin, Heidelb erg, 1985, 264–279. MR 0824074

work page 1984

[8] [8]

Reckziegel, On the geometry of the complex quadric , Geometry and Topology of Subman- ifolds, VIII (Brussels, 1995 / Nordfjordeid, 1995), W orld S ci

H. Reckziegel, On the geometry of the complex quadric , Geometry and Topology of Subman- ifolds, VIII (Brussels, 1995 / Nordfjordeid, 1995), W orld S ci. Publ., River Edge, NJ, 1996, pp. 302–315. MR 1434581

work page 1995