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Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

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Pith's one-line read This paper proves an explicit two-sided bound for the first positive Laplacian eigenvalue along the canonical variation of a Riemannian submersion with totally geodesic Einstein fibers, under Ricci lower bounds, with sharp equality on odd…

desk verdict A genuinely new Ricci-curvature eigenvalue bound for canonical variations with totally geodesic fibers, undercut by an omitted one-dimensional computation in the circle-fiber case. read the letter →

arxiv 2411.17078 v5 pith:UK65SKER submitted 2024-11-26 math.DG

Kazumasa Narita This is my paper · ORCID
classification math.DG MSC 53C2053C25
keywords LaplacianeigenvalueRiemanniansubmersioncanonicalvariationtotallygeodesicfibersEinsteinmanifoldYamabestabilitytwistorfibrationsphererigidity
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

This paper addresses a question left open by the 1982 study of the canonical variation of a Riemannian submersion with totally geodesic fibers: when does the scale-invariant first eigenvalue $\lambda_1(g_t)\operatorname{Vol}(M,g_t)^{2/n}$ grow with the stretch parameter $t$? This paper proves that under the pure curvature hypotheses $\operatorname{Ric}^M\ge \tilde c\,g$ on the total space and $\operatorname{Ric}^{F_y}=c\,(\iota^*g)$ on each Einstein fiber with $0\le c<\tilde c$ (with $c=0$ in codimension one), the first eigenvalue obeys an explicit two-sided inequality for every $t\ge 1$. In particular, the scale-invariant quantity tends to infinity at least like $t^{2(n-p)/n}$, providing an explicit lower bound for $\lambda_1(g_t)$ from Ricci data alone. Equality in the lower bound is characterized by odd-dimensional round spheres and pullback eigenfunctions, so the estimate is sharp where it can be. The same bounds yield a quantitative sufficient condition for stability of critical points of the Yamabe functional along the deformation.

What carries the argument

The load-bearing mechanism is the orthogonal splitting of the Laplacian into vertical and horizontal parts, $\Delta^M_{g_t}=t^{-2}\Delta^v+\Delta^h=t^{-2}\Delta^M+(1-t^{-2})\Delta^h$, which lets the proof track joint eigenfunctions of $\Delta^M$ and $\Delta^h$. Lemma 3.1 starts with a joint eigenfunction $f$ satisfying $\Delta^M f=\lambda_k(g)f$ and $\Delta^h f=af$; using the standard Hessian--Ricci identity on $M$ and on each fiber, together with the fiberwise comparison $|\operatorname{Hess}^M f|^2-|\operatorname{Hess}^{F_y}(f|_{F_y})|^2\ge |\operatorname{Hess}^h f|^2\ge a^2f^2/p$, it shows that either $a>\tilde c-c$ or the quadratic $Q_k(a)=(p+1)a^2-\alpha_k a+\beta_k$ satisfies $Q_k(a)\le 0$. Analyzing this quadratic yields the uniform horizontal-eigenvalue bound $a>(\tilde c-c)/(n+1)$, and substituting that bound into the Laplacian decomposition proves Theorem 1.1. The equality analysis uses the classical sphere-rigidity theorem and the classification of totally geodesic Riemannian submersions from round spheres.

What would settle it

Check the theorem's lower bound against the known exact spectrum of the standard circle fibration $S^1\to S^{2n+1}\to\mathbb{CP}^n$, where $\lambda_1(g_t)=\min\{2n+t^{-2},4(n+1)\}$: substituting $\tilde c=2n$, $c=0$, $p=1$, and total dimension $2n+1$ into Theorem 1.1 must produce a bound no larger than this exact value for every $t\ge 1$. For the codimension-one assertion itself, write out the product-rule computation $\frac12((f')^2)''=(f'')^2+f'f'''$ and verify that the same quadratic inequality $Q_k(a)\le 0$ emerges; any uncontrolled term there would remove the circle-fiber cases from the theorem.

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Extended reading notes

Core claim

On a compact connected Riemannian submersion $\pi:(M,g)\to(B,j)$ with connected totally geodesic fibers (each fiber's geodesics are geodesics of $M$), the canonical variation $g_t$ is the metric that equals $g$ on horizontal vectors and $t^2g$ on vertical vectors, so it preserves the submersion and multiplies volume by $t^{n-p}$. Theorem 1.1 states that when $\operatorname{Ric}_M\ge \tilde c\,g$ and each fiber is Einstein with $\operatorname{Ric}_{F_y}=c\,\iota^*g$, with $0\le c<\tilde c$ and with $c=0$ when $p=n-1$, then for all $t\ge 1$, $$\frac{\tilde c-c}{n+1}+$t^{{-2}}$\left(\frac{$n^{2}$+1}{$n^{2}$-1}\,\tilde c+\frac{c}{n+1}\right)\le \lambda_1(g_t)\le \beta_1,$$ where $\beta_1$ is the first positive eigenvalue of $\Delta^{(B,j)}$. Because volume grows as $t^{n-p}$, this forces $\lambda_1(g_t)\operatorname{Vol}(M,g_t)^{2/n}$ to grow at least like $t^{2(n-p)/n}$, settling the positivity side of the problem raised in the 1982 article [5] under Ricci hypotheses alone. The equality statement identifies the only sharp case: $t=1$ and $(M,g)$ is an odd-dimensional round sphere of radius $\sqrt{(n-1)/\tilde c}$, with the relevant eigenfunctions pulled back from the base. As an application, when the total space is Einstein and the $A$-tensor (the obstruction to the horizontal distribution being integrable) does not vanish, the same lower bound gives an explicit threshold $\max\{1,\sqrt{\Gamma/|A|^2}\}$ beyond which $g_t$ is a stable critical point of the Yamabe functional (the normalized total scalar curvature functional).

Load-bearing premise

The load-bearing premise is that the one-dimensional fiber case of Lemma 3.1, where each fiber is a closed geodesic and the Hessian-Ricci identity is replaced by an unshown product-rule computation, holds as stated, and that the strict curvature gap $c<\tilde c$ stays available in that case; if either fails, the lower bound and its divergence rate do not follow.

Editorial extensions

If this is right

  • For every Riemannian submersion with totally geodesic Einstein fibers satisfying $0\le c<\tilde c$, the first eigenvalue of $g_t$ is explicitly controlled for all $t\ge 1$, and $\lambda_1(g_t)\operatorname{Vol}(M,g_t)^{2/n}$ diverges at least like $t^{2(n-p)/n}$.
  • The lower bound is sharp exactly at $t=1$ for odd-dimensional round spheres, so no Ricci-only estimate can be improved in general; the upper bound $\beta_1$ is attained precisely when the base's first eigenfunctions pull back to first eigenfunctions.
  • If $g$ is Einstein with $\operatorname{Ric}_M=\tilde c g$, the $A$-tensor is nonzero, and $0\le c<\tilde c$, then $g_t$ is a stable critical point of the Yamabe functional for all $t\ge \max\{1,\sqrt{\Gamma/|A|^2}\}$, where $\Gamma=\frac{n^2+1}{n+1}(\tilde c-c)+pc$.
  • Known exact spectra for the standard sphere and projective-space submersions satisfy the new inequalities, confirming the bound is consistent and often non-sharp; for twistor fibrations of quaternionic Kähler manifolds it yields new explicit stability thresholds.
  • The strict inequality $c<\tilde c$ is essential: for Riemannian products $B\times F$, where the hypothesis forces $c=\tilde c$, the paper shows $\lambda_1(g_t)\operatorname{Vol}(M,g_t)^{2/n}\to 0$, so divergence fails exactly when the curvature gap closes.

Reading between the lines

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

  • The same joint-eigenfunction quadratic should extend to higher eigenvalues $\lambda_k(g_t)$, yielding spectral-gap information and showing which eigenvalues are carried by vertical versus horizontal modes.
  • The divergence for $c<\tilde c$ and vanishing for products suggests a phase transition in the deformation space: interpolating the fiber Ricci constant toward $\tilde c$ should lower the growth exponent below $2(n-p)/n$.
  • Because the stability threshold in Theorem 4.1 depends only on $|A|^2$ and the Ricci constants, it can be checked on any explicit submersion without diagonalizing the Laplacian, giving a practical certificate for Yamabe stability on large families such as circle-fiber submersions over Kähler-Einstein bases.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the canonical variation g_t of a Riemannian metric g on a compact manifold M arising from a Riemannian submersion M -> B with connected totally geodesic fibers. The main result, Theorem 1.1, gives explicit two-sided bounds for the first positive eigenvalue lambda_1(g_t) under a Ricci lower bound Ric_M >= c-tilde g and, for fibers of dimension at least two, a fiberwise Einstein condition Ric_F = c iota^* g with 0 <= c < c-tilde; for circle fibers it sets c = 0. The lower bound is derived from a Bochner-type lemma (Lemma 3.1) adapted to the horizontal Laplacian, and it implies that the scale-invariant quantity Lambda_1(M,t) = lambda_1(g_t) Vol(M,g_t)^{2/n} grows like t^{2(n-p)/n}. As an application, Theorem 4.1 gives a quantitative condition for stability of g_t as a critical point of the Yamabe functional. The paper closes with applications to Hopf fibrations, 3-Sasakian fibrations, twistor fibrations of quaternionic Kaehler manifolds, and other examples.

Significance. If the results are fully established, the paper provides the first explicit lower bound for lambda_1(g_t) under pure Ricci-curvature hypotheses, complementing the asymptotic vanishing result of Berard-Bergery and Bourguignon and earlier sub-Riemannian results such as Baudoin-Kim. The equality characterization via Obata and the Escobales-Ranjan classification is elegant and gives a sharp statement for odd-dimensional round spheres. The p <= n-2 case of Lemma 3.1 is a coherent Bochner-type argument with explicit constants, and the paper is honest about the points where computations are compressed. The Yamabe-stability application is a useful quantitative addition to the recent work of Bettiol-Lauret-Piccione. However, the paper is not fully self-contained in the codimension-one case, and Theorem 4.1 contains a displayed formula that must be corrected before the proof can be accepted.

major comments (2)
  1. [Section 3, Lemma 3.1] The p = n-1 case of Lemma 3.1 is asserted without proof: the text says only that a one-dimensional Leibniz-rule identity replaces the Bochner formula and that the assertion follows 'in a similar manner', with no analogue of (3.4)-(3.5) displayed. This is load-bearing because Theorem 1.1 and Theorem 4.1 both include p = n-1, and the circle-fiber examples in Section 5 (Hopf and Sasakian fibrations) depend on it. Please include the missing computation. The essential point is that on a unit-speed closed geodesic fiber, f'' = -(lambda_k - a) f, so integral over the fiber of (f'')^2 equals (lambda_k - a) times the integral of (f')^2; this replaces (3.4) with c = 0, and together with (3.6) yields the same quadratic inequality Q_k(a) <= 0. The equality characterization in this case should also be stated explicitly rather than left to analogy.
  2. [Section 4, proof of Theorem 4.1] The displayed chain in the proof of Theorem 4.1 reads '(n-1)(lambda_1(g_t) - S(g_t)) > |A|^2 t^2 - (Gamma + |A|^2) + Gamma t^{-2}'. This identity is not correct as written; the factor (n-1) multiplies only lambda_1(g_t), not the difference. The correct inequality, which is what the Yamabe stability criterion requires, is '(n-1)lambda_1(g_t) - S(g_t) > |A|^2 t^{-2}(t^2 - Gamma/|A|^2)(t^2 - 1)'. With the printed formula, taking t = 1 would imply lambda_1(g) > S(g), which is false for the admissible example CP^{2n+1} -> HP^n of Example 5.7, where lambda_1 - S is negative. The corrected formula does factor as claimed and yields the stated stability threshold, so this is a fixable but essential correction.
minor comments (5)
  1. [Section 3, Eq. (3.9)] In the first inequality of (3.9), the right-hand side should read t^{-2} lambda_1(g) + (1-t^{-2})(c-tilde - c)/(n+1), not t^{-2} lambda_1(g_t) + ...; as printed the inequality is circular and would be false for t close to 1.
  2. [Section 3, Lemma 3.2] The statement of Lemma 3.2 assumes '0 <= c < n-1', but the proof and the application in Theorem 1.1 require '0 <= c < c-tilde'. The condition c < n-1 is never used and would be too restrictive when c-tilde > n-1.
  3. [Section 5, Example 5.8] For the flag manifold F(1,2), the stated lower bound begins with 2/7, but the theorem gives (c-tilde - c)/(n+1) = (2-1)/7 = 1/7; the displayed bound should be 1/7 + (79/35) t^{-2}.
  4. [Section 5, Example 5.9] The stability threshold is miscomputed: with |A|^2 = 2n and Gamma = 2n(2n^2+2n+1)/(n+1), the threshold squared is Gamma/|A|^2 = (2n^2+2n+1)/(n+1), not (2n+1)/(n+1) as printed.
  5. [Section 5, Example 5.12] The stability threshold is miscomputed: with |A|^2 = 8n and Gamma = 4n(16n^2+32n+17)/(4n+3), the threshold squared is Gamma/|A|^2 = (16n^2+32n+17)/(2(4n+3)), not (2n+5)/2 + 1/(4n+3).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bound for λ1(gt) is derived from the stated curvature hypotheses, standard external Bochner/Obata results, and explicit algebra; no fitted parameter is renamed as a prediction.

full rationale

The chain leading to Theorem 1.1 is self-contained in the relevant sense: the constants ~c and c are supplied by the curvature hypotheses Ric_M ≥ ~c g and Ric_{F_y}=c ι*g and are not fitted to any eigenvalue; no prediction is obtained from a quantity that was inserted to produce it. Lemma 3.1 (p ≤ n−2 case) is a Bochner/Cauchy–Schwarz estimate: from (3.1), (3.3), and (3.4) it derives an inequality involving |Hess^M f|^2 and |Hess^{F_y}(f|F_y)|^2, then uses (3.6) and (3.7) to obtain Q_k(a) ≤ 0. Lemma 3.2 uses only Lichnerowicz–Obata and the sign of Q_k at (~c−c)/(n+1). The lower bound (1.1) is then the algebraic consequence of Δ_{g_t} = t^{-2}Δ^v + Δ^h, λ_1(g_t) = inf_k [t^{-2}λ_k(g) + (1−t^{-2})a_{k,1}], λ_k(g) ≥ n~c/(n−1), and a_{k,1} > (~c−c)/(n+1); none of these inputs contains the target inequality. The equality discussion invokes Obata's theorem and the external Escobales–Ranjan classification, not any self-cited uniqueness claim. Section 5 comparisons with Tanno, Bettiol–Lauret–Piccione, and Nagy–Semmelmann are consistency checks against known exact or sharper results, not fitted inputs. The genuine weakness is non-circular: for p = n−1, Lemma 3.1 says the assertion follows 'in a similar manner' from a one-dimensional identity and omits the details, and Theorem 1.1 and Theorem 4.1 both include p = n−1, so that case is not fully proven in the text. Equation (3.9) also writes λ_1(g_t) on the right where the argument requires λ_1(g). Both are correctness gaps, not reductions to inputs, and therefore do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No constants are fitted to data: a and c enter as curvature assumptions fixed by the geometry, and t is the deformation parameter. The proof relies only on standard external theorems from Riemannian geometry and spectral theory, so the axiom ledger is clean.

assumptions (6)
  • standard math Lichnerowicz-Obata theorem
    Used in Lemma 3.1 and the proof of Theorem 1.1 to lower-bound lambda_1(g) by n*a/(n-1) and to classify equality.
  • standard math Bochner formula on the total space and on fibers
    Central to Lemma 3.1 estimates on Hessians and Laplacians of eigenfunctions.
  • standard math Hermann's lemma that fibers of a totally geodesic submersion are isometric
    Invoked in Lemma 2.1 and used in Theorem 1.1 to apply Fubini-type integration.
  • standard math Escobales-Ranjan classification of totally geodesic submersions from round spheres
    Used in equality conditions of Lemma 3.1 and Theorem 1.1 to force odd-dimensional spheres.
  • standard math O'Neill formulas and scalar curvature formula for Riemannian submersions
    Used in Theorem 4.1 to write S(g_t) in terms of the A-tensor and fiber scalar curvature.
  • domain assumption Domain assumption: fibers connected, totally geodesic, Einstein with c < a (or p=n-1 with c=0), and Ric_M >= a*g
    This is the class of manifolds to which the theorem applies; the paper does not prove these conditions but takes them as hypotheses.

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Pith. "Pith review of Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers." pith.science (2026). https://pith.science/paper/UK65SKER

@misc{pith2026241117078,
  author       = {Pith},
  title        = {Pith review of: Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UK65SKER}},
  note         = {Machine review of arXiv:2411.17078}
}
abstract

Given a Riemannian submersion $(M,g) \to (B,j)$ each of whose fibers is connected and totally geodesic, we consider a certain 1-parameter family of Riemannian metrics $(g_{t})_{t > 0}$ on $M$, which is called the canonical variation. Let $\lambda_{1}(g_{t})$ be the first positive eigenvalue of the Laplace--Beltrami operator $\Delta^{M}_{g_{t}}$ and $\mbox{Vol}(M,g_{t})$ the volume of $(M, g_{t})$. In 1982, B\'{e}rard-Bergery and Bourguignon showed that the scale-invariant quantity $\lambda_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $0$ with $t$. In this paper, we show that if each fiber is Einstein and $(M,g)$ satisfies a certain condition about its Ricci curvature, then bounds for $\lambda_{1}(g_{t})$ can be obtained. In particular this implies $\lambda_{1}(g_{t})\mbox{Vol}(M,g_{t})^{2/\mbox{dim}M}$ goes to $\infty$ with $t$. Moreover, using the bounds, we consider stability of critical points of the Yamabe functional. We will see that our results can be applied to many examples. In particular, we consider the twistor fibration of a quaternionic K\"{a}hler manifold of positive scalar curvature.

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    1980

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.