{"id":"b08bac03-d01e-408e-87a9-b4f3701ade3e","arxiv_id":"2411.17078","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under positive Ricci curvature on the total space and Einstein fibers with constant less than the total Ricci bound, the first eigenvalue of the canonical variation satisfies an explicit lower bound of the form constant plus t^{-2}, so the scale-invariant product tends to infinity.","lead":"This paper proves explicit lower and upper bounds for the first nonzero eigenvalue of the Laplacian on a family of metrics obtained by rescaling the fiber directions of a Riemannian submersion with totally geodesic fibers. The bounds imply that the scale-invariant eigenvalue-volume product grows at a predictable rate, and they yield stability thresholds for critical points of the Yamabe functional.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Circle-fiber case (p=n-1) is asserted without the one-dimensional computation on which Theorem 1.1 and Theorem 4.1 rely; the gap should be filled before those statements are taken as proven.","rationale":"The reader's weakest-assumption analysis already identifies the p=n−1 case as the fragile point, and my read agrees. The p≤n−2 argument is a genuine Lichnerowicz–Obata-style derivation with no obvious algebraic error: the trace Cauchy–Schwarz estimates (3.6)–(3.7) lead correctly to the quadratic Q_k, and the final simplification of the lower bound is consistent with the stated formula. The concern is not that the theorem is false; it is that the codimension-one case is essential to the announced generality and is left as an omitted computation. Filling that computation is a necessary verification before the claim should be regarded as fully established. The known Hopf-fibration eigenvalues provide a concrete test case that would almost certainly confirm the expected inequality, which is why I do not recommend moving from the reader's conditional verdict to a rejection; the same reason also means the burden is modest. No ad hominem or theatrical language is intended; the review is on the argument's completeness.","tokens_in":17358,"tokens_out":16694,"duration_ms":147419,"concrete_test":"Re-derive Lemma 3.1 for p=n−1 from (3.1)–(3.4) by setting c=0, writing each fiber as a unit-speed geodesic, and using f''=-(λ−a)f together with the identity (1/2)((f')^2)'' = (f'')^2+f'f''' to obtain the analogue of (3.5). Check that this yields exactly Q_{n−1}(a)≤0 for a≤ c~. Then test the resulting Lemma 3.2 threshold against the solvable Hopf fibration S^{2m+1}→CP^m, where Tanno's formula gives λ1(g_t)=min{2m+t^{-2},4(m+1)}; the derived bound must be compatible with this known value and strict where required. If the derivation cannot be completed or disagrees with the solvable example, Theorem 1.1 must be restricted to p≤n−2 or corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The p≤n−2 proof of Lemma 3.1 is internally coherent: it combines the vertical and total Bochner inequalities with the Cauchy–Schwarz bounds (3.6)–(3.7) and obtains Q_k(a)≤0. For p=n−1, however, the paper says only that a one-dimensional Leibniz-rule identity replaces the fiber Bochner formula and that the assertion follows 'in a similar manner'; no analogue of (3.4)–(3.5) is displayed. Since Theorem 1.1 and Theorem 4.1 both include p=n−1, the central claim depends on an unshown computation at exactly the codimension-one boundary. This is not an inconsistency in the p≤n−2 part, but it is load-bearing: if the one-dimensional estimate has a different sign or an extra term, the lower bound (1.1) would be unproved for all circle-fiber submersions, including the Hopf and Sasakian examples in §5. The equality characterization also invokes the Escobales–Ranjan classification after the unshown case, so the gap is not isolated to one line. There is additionally a typo in (3.9) where λ1(g_t) appears where λ1(g) is meant; this is minor by comparison but should be corrected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17607,"tokens_out":34864,"duration_ms":267304,"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":[{"comment":"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.","section":"Section 3, Lemma 3.1"},{"comment":"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.","section":"Section 4, proof of Theorem 4.1"}],"minor_comments":[{"comment":"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.","section":"Section 3, Eq. (3.9)"},{"comment":"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.","section":"Section 3, Lemma 3.2"},{"comment":"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}.","section":"Section 5, Example 5.8"},{"comment":"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.","section":"Section 5, Example 5.9"},{"comment":"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).","section":"Section 5, Example 5.12"}],"recommendation":"major_revision","confidential_remarks":"The main theorems appear defensible after filling the p=n-1 computation and correcting the displayed formula in the proof of Theorem 4.1. The examples contain several numerical slips that should be rechecked before publication. I recommend major revision rather than rejection, because the gaps are local and the core Bochner argument for p <= n-2 is sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the p ≤ n−2 part of Theorem 1.1 is new, non-circular, and looks correct. The p = n−1 part is asserted on an omitted computation, and the paper should not be accepted until that is written out.\n\nWhat is actually new: Bérard-Bergery and Bourguignon only got decay as t→0; Urakawa, Tanno, and Bleecker handled special Sasakian/Hopf cases; Baudoin and Kim need an H-type sub-Riemannian structure. Theorem 1.1 gives an explicit two-sided bound for λ1(g_t) under pure Ricci hypotheses, with equality characterization on odd spheres, and Theorem 4.1 turns it into a checkable Yamabe stability criterion. The proof for p ≤ n−2 is a clean Bochner/Lichnerowicz–Obata argument; the Cauchy–Schwarz estimates (3.6)–(3.7) are reasonable, and the quadratic inequality checks out. The examples, especially the twistor fibration, are appropriate and the stability thresholds are concrete.\n\nThe soft spots, in order of importance:\n\n1. p = n−1 is genuinely load-bearing. Lemma 3.1 says the one-dimensional Leibniz-rule identity lets the assertion follow “in a similar manner,” but no analogue of the vertical Bochner estimate (3.4) is displayed. Since Theorem 1.1 and Theorem 4.1 both include p = n−1, the Hopf and Sasakian examples in §5 rest on this unshown computation. I do not think it is a wrong sign—the one-dimensional estimate is probably a straightforward adaptation—but it is not in the paper. The equality characterization also uses Escobales–Ranjan after this gap.\n\n2. Lemma 3.2 is compressed, especially the step where n^{−l} is set so that Q_k(n^{−l}) is positive. That deserves a few lines of expansion.\n\n3. Display (3.9) has λ1(g_t) on the right-hand side where λ1(g) is meant. As written it looks circular; it is a typo, but in a central inequality it will confuse readers and should be fixed.\n\nI agree with the reader's conditional verdict. The circularity burden is zero—the constants a and c come from curvature assumptions, not from the desired bound. The result is new and the p ≤ n−2 part is solid. This paper is for people working on eigenvalue bounds under Riemannian submersions and on Yamabe stability; they will want it, but they should not cite the circle-fiber case until it is actually proved.\n\nRecommendation: send it to peer review, but make clear the referee should demand the p = n−1 computation in full.","headline":"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.","tokens_in":18175,"tokens_out":3220,"would_cite":true,"duration_ms":28290,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C20","53C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Laplacian eigenvalue","Riemannian submersion","canonical variation","totally geodesic fibers","Einstein manifold","Yamabe stability","twistor fibration","sphere rigidity"],"falsifier":"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.","tokens_in":17143,"feed_emoji":"📐","tokens_out":18230,"duration_ms":156124,"temperature":0.7,"pith_summary":"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.","feed_headline":"Ricci bounds pin down the first eigenvalue under fiber stretching","feed_subtitle":"A two-sided first-eigenvalue bound follows from Ricci curvature alone, with sharp equality on odd round spheres.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the canonical variation, the vertical/horizontal Laplacian decomposition, the volume formula, and the vanishing of the scale-invariant eigenvalue as $t\\to 0$ that this paper extends.","marker":"[5]"},{"why":"Proves that totally geodesic fibers of a Riemannian submersion are isometric, a fact used throughout the fiberwise Einstein hypothesis.","marker":"[21]"},{"why":"Supplies the sphere-rigidity theorem used to characterize equality in Lemma 3.1 and hence in Theorem 1.1.","marker":"[29]"},{"why":"Classifies Riemannian submersions with totally geodesic fibers from round spheres, forcing the equality case to have odd dimension.","marker":"[15]"},{"why":"Provides the companion classification of totally geodesic fibrations of spheres used together with [15] to pin down the equality case.","marker":"[31]"},{"why":"Computes the exact first eigenvalue for compact rank-one symmetric space submersions, serving as the benchmark that the new inequalities must satisfy and as the stability context for Theorem 4.1.","marker":"[8]"},{"why":"Gives a prior sharp lower bound under an H-type sub-Riemannian hypothesis, the setting the paper replaces with pure Ricci-curvature assumptions.","marker":"[3]"},{"why":"Supplies the O'Neill formula for scalar curvature along the canonical variation and the constancy facts for Einstein submersions used in the Yamabe-stability application.","marker":"[7]"}],"fun_headline_variants":["Ricci curvature yields two-sided Laplace eigenvalue bounds","Eigenvalue growth under fiber stretching tied to Ricci bound","Two-sided eigenvalue bound from Ricci and Einstein fibers","Stable Yamabe points via Ricci condition on submersions","Odd spheres sharpen eigenvalue-volume limit in submersions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ricci curvature yields two-sided Laplace eigenvalue bounds","Eigenvalue growth under fiber stretching tied to Ricci bound","Two-sided eigenvalue bound from Ricci and Einstein fibers","Stable Yamabe points via Ricci condition on submersions","Odd spheres sharpen eigenvalue-volume limit in submersions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000391,"raw_usage":{"total_tokens":2215,"prompt_tokens":1261,"completion_tokens":954,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":877,"completion_tokens_details":{"reasoning_tokens":877}},"tokens_in":877,"tokens_out":954,"duration_ms":7136,"temperature":1.0,"reasoning_tokens":877,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:33:37.856003+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"B´ erard-Bergery and J","cited_arxiv_id":null,"evidence_quote":"Introduces the canonical variation, the vertical/horizontal Laplacian decomposition, the volume formula, and the vanishing of the scale-invariant eigenvalue as $t\\to 0$ that this paper extends."},{"cited_title":"Hermann, A sufficient condition that a mapping of Riemannian ma nifolds be a fibre bundle, Proc","cited_arxiv_id":null,"evidence_quote":"Proves that totally geodesic fibers of a Riemannian submersion are isometric, a fact used throughout the fiberwise Einstein hypothesis."},{"cited_title":"Obata, Certain conditions for a Riemannian manifold to be isome tric with a sphere, J","cited_arxiv_id":null,"evidence_quote":"Supplies the sphere-rigidity theorem used to characterize equality in Lemma 3.1 and hence in Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies Riemannian submersions with totally geodesic fibers from round spheres, forcing the equality case to have odd dimension."},{"cited_title":"Ranjan, Riemannian submersions of spheres with totally geod esic fibres, Osaka","cited_arxiv_id":null,"evidence_quote":"Provides the companion classification of totally geodesic fibrations of spheres used together with [15] to pin down the equality case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the exact first eigenvalue for compact rank-one symmetric space submersions, serving as the benchmark that the new inequalities must satisfy and as the stability context for Theorem 4.1."},{"cited_title":"Baudoin and B","cited_arxiv_id":null,"evidence_quote":"Gives a prior sharp lower bound under an H-type sub-Riemannian hypothesis, the setting the paper replaces with pure Ricci-curvature assumptions."},{"cited_title":"Besse, Einstein Manifolds , Springer-Verlag, 1987","cited_arxiv_id":null,"evidence_quote":"Supplies the O'Neill formula for scalar curvature along the canonical variation and the constancy facts for Einstein submersions used in the Yamabe-stability application."}],"review_version":1}