{"id":"a2362a4e-a789-4b96-85da-c268bdf1dd4b","arxiv_id":"2501.00576","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Sub-Laplacian intertwining maps between sub-Riemannian Lie groups are conformal submersions, and on Carnot groups the sub-Laplacian determines the sub-Riemannian structure.","lead":"This paper proves that smooth maps between certain curved spaces with restricted allowed directions, called sub-Riemannian Lie groups, which commute with the heat-spreading sub-Laplacian operator, are exactly the maps that stretch or shrink the allowed directions by a consistent factor at each point. In the special layered spaces called Carnot groups, this forces the operator to determine the geometry completely, resolving an open question from a standard reference book.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem A is sound after a repairable proof gap, and Theorem B's Carnot rigidity rests on standard cited theorems.","rationale":"The reader's central verification of Theorem A is correct: the forward direction is a direct chain-rule computation, and the converse is recoverable from the quadratic test-function argument despite the premature use of (8). I agree with the reader that the written proof of Theorem 4.1 has a presentational gap, but the gap is immediately repairable and does not affect the validity of the theorem. The stronger Carnot and Heisenberg conclusions depend on external rigidity results, but these are standard, correctly cited, and not used in a way that appears to misstate them. Since the paper's global Theorem B can be obtained by applying the global isometry rigidity directly, the local formulations in the propositions do not create a load-bearing risk for the headline claim. Thus the reader's CONDITIONAL verdict remains appropriate, and no verdict adjustment is needed.","tokens_in":18644,"tokens_out":33062,"duration_ms":359233,"concrete_test":"Run the following analytic check: in Theorem 4.1, skip the derivation of (23); for fixed p and q=F(p), plug u_q^\\alpha into (20), note u_q^\\alpha(q)=0 and D u_q^\\alpha(q)=0, and re-derive (27) as \\sum_i <\\alpha|DF(p)X_i>^2 = \\lambda(p)^2 \\sum_j <\\alpha|Y_j>^2. Also verify that D^2 u_q^\\alpha(q)[v,v]=2<\\alpha|v>^2 holds for all v in h, not only for v in V(H). This settles whether the converse of Theorem A requires an unproved contact assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After tracing both directions of Theorem A, the only concrete defect in the written proof is in Theorem 4.1: equation (23) is derived from (22) via (8), which presupposes contact; (8) is only valid once DF(p)[V(G)] is known to lie in V(H). This is not load-bearing because the quadratic test functions used later have value and first derivative zero at the base point, so the first-order terms in (22) drop and the same computation yields (27) without using (8). One also needs the formula D^2 u_q^\\alpha(q)[v,v]=2<\\alpha|v>^2 for arbitrary v in h, not only for V(H); this is an equally harmless extension of the displayed identity. The Carnot rigidity in Theorem B is inherited from Pansu differentiability of C^1 contact maps and from affine rigidity of Carnot-group isometries; these are external results quoted from [2], [10], and [12], and I found no misstatement of them. The only residual doubt is that Propositions 5.3 and 5.4 state local versions of the rigidity conclusions; Theorem B itself has a global F, so any local-extension issue does not affect the headline claim. I therefore see no load-bearing objection that would change the reader's conditional verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper characterizes C^2 maps F between open domains of sub-Riemannian Lie groups for which the pullback of the sub-Laplacian is, up to a conformal factor and a first-order term, the sub-Laplacian on the target. Theorem A states that this happens exactly when F is a conformal submersion, with the first-order coefficient given explicitly by the trace of the second-order Lie differential plus a modular-function term. The authors then specialize to Carnot groups, proving that an intertwining map with constant conformal factor forces the target to be a Carnot quotient of the domain, and that in equal dimension F is a dilation composed with a left translation and an isometric automorphism. The paper also gives a complete classification of sub-Laplacian equivalence on Heisenberg groups in terms of the symplectic spectrum of the scalar product.","tokens_in":18681,"tokens_out":5884,"duration_ms":57807,"significance":"The main theorem provides a purely PDE characterization of conformal submersions of sub-Riemannian Lie groups, extending classical results of Helgason, Watson, Fuglede, and Ishihara to the sub-Riemannian setting. The proof of the forward direction is a direct computation, and the converse is obtained by evaluating the operator identity on quadratic test functions; there are no fitted parameters and no ad-hoc assumptions. The Carnot-group corollary answers a question from Bonfiglioli-Lanconelli-Uguzzoni and identifies sub-Laplacian equivalence with isometry, which is a strong and interesting rigidity statement. The Heisenberg classification via the symplectic spectrum is explicit and checkable. The sharp Carnot conclusions depend on two external deep results - Pansu differentiability of C^1 contact maps and affine rigidity of Carnot-group isometries - which are cited but not proved; I checked the citations and found no misstatement of those results. The technical gaps in the proof of Theorem 4.1 are local and repairable, and they do not affect the plausibility of the central claim once repaired.","major_comments":[{"comment":"The passage from (22) to (23) uses equation (8), but (8) is only stated for contact maps and contactness of F has not yet been established at that point of the proof of (i) implies (ii). This is a genuine gap in the written proof. The gap is repairable: because the quadratic test functions introduced in (24) have value and first derivative zero at the base point, the first-order terms in (22) drop and equation (27) can be derived without invoking (8); the authors should rewrite this step to avoid the circular use of contactness.","section":"Section 4, Eq. (23)"},{"comment":"The identities D u_hat(q)^alpha(hat(q))[v] = 0 and D^2 u_hat(q)^alpha(hat(q))[v,v] = 2<alpha|v>^2 are stated for v,w in V(H), but in equation (26) they are applied to DF(hat(p))X_i, which is not known to lie in V(H) at that stage. The computation of D^2 u is in fact valid for every v in h via the exponential formula, and the proof should state and prove this extension so that the application in (26) is justified.","section":"Section 4, Eqs. (24)-(26)"}],"minor_comments":[{"comment":"The abstract says 'smooth maps' while Theorem A is stated for C^2 maps; the wording should be aligned.","section":"Abstract and Theorem A"},{"comment":"In the displayed formula for the sub-Laplacian, the term 'x_i^2 + yy_i / 4' appears to contain a typo; it should likely be '(x_i^2 + y_i^2) / 4'.","section":"Section 5.5"},{"comment":"The claim that every C^1 contact map between Carnot groups is Pansu differentiable is cited to [2]; reference [17] (Warhurst) is more specific and is already in the bibliography, so it should be cited here as well.","section":"Section 5.2, Proposition 5.3"},{"comment":"The phrase 'conformal C^2 diffeomorphism' should be 'conformal C^2 local diffeomorphism' unless global invertibility is actually proved.","section":"Theorem A, final paragraph"},{"comment":"The sentence 'We don't know a corresponding characterization of conformal submersions' is confusing in light of Proposition 5.3, which characterizes when a conformal submersion exists; please rephrase to clarify that the explicit form of a general conformal submersion is not known.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of math.DG and the main claim is significant. The gaps in Section 4 are technical and repairable, so I would be willing to see a revised version. The reliance on an arXiv preprint ([2]) for Pansu differentiability is acceptable but should ideally be cross-referenced with the published article [17] already in the bibliography."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine step forward. It characterizes C^2 maps between sub-Riemannian Lie groups that intertwine sub-Laplacians up to a first-order term b and a zeroth-order term c: such maps are exactly conformal submersions, with c = 0 and b given by an explicit formula involving the second-order Lie differential, the modular functions, and the gradient of the modular function. That is a real generalization of the Riemannian theorems of Helgason, Watson, and Fuglede–Ishihara, and it answers a question that Bonfiglioli–Lanconelli–Uguzzoni left open for Carnot groups. The forward direction (Theorem 3.1) is a straightforward but correct computation; the converse (Theorem 4.1) is the interesting part, and the test-function argument is sound.\n\nI verified the main line myself. The one flaw in the written proof is in Theorem 4.1: equation (23) is derived from (22) using (8), which presupposes that F is a contact map, and that has not yet been shown at that point. The repair is trivial—the quadratic test functions have value and first derivative zero at the base point, so the first-order terms in (22) drop and the same computation gives (27) directly. The companion issue is that identity (24) needs D^2 u at the base point evaluated on arbitrary v in the Lie algebra h, not just on V(H); the displayed formula extends to that case without difficulty. So the gap is presentational, not load-bearing.\n\nThe Carnot rigidity results (Theorem B and Propositions 5.3–5.4) rely on two external theorems: Pansu differentiability of C^1 contact maps, and affine rigidity of Carnot-group isometries. The paper cites these clearly and they are standard. I did not re-derive them from first principles, but they are standard and the paper states them accurately. If either failed, the quotient conclusion would weaken, but Theorem A is independent of them. The Heisenberg classification via the symplectic spectrum is clean and gives the promised concrete picture.\n\nNo fitted parameters, no invented entities, no circular reasoning that I can see. Citation pattern is appropriate; the overlap with [1] is explicitly discussed. The paper is written with care, though the small proof gap should be flagged to the authors.\n\nWho is this for? Sub-Riemannian and PDE readers who care about symmetry characterization of sub-Laplacians. It deserves a serious referee; I would send it to review and ask for the gap in Theorem 4.1 to be fixed before publication.","headline":"A sound, genuinely general characterization of sub-Laplacian intertwining maps as conformal submersions; the proof has one small repairable gap and the Carnot consequences rest on standard external rigidity theorems.","tokens_in":19440,"tokens_out":2398,"would_cite":true,"duration_ms":23041,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B06","53C17","35H20","53C30","22F30","22E25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Smooth maps between sub-Riemannian Lie groups commute with sub-Laplacians exactly when they are conformal submersions.","keywords":["sub-Laplacian","sub-Riemannian Lie group","polarized group","Carnot group","Heisenberg group","conformal submersion","sums of squares","symplectic spectrum"],"falsifier":"In the Heisenberg example of Section 5.5, choose two scalar products on $R^{5}$ whose symplectic spectra are not proportional, such as (1,2) and (1,3). The paper says the two sub-Laplacians are inequivalent. Checking directly whether any C² change of coordinates F satisfies Δ_{(1,2)}(u∘F)=λ²(Δ_{(1,3)}u)∘F for all u would settle the rigidity claim; the predicted answer is that no such F exists.","tokens_in":18246,"feed_emoji":"📐","tokens_out":11513,"duration_ms":100869,"temperature":0.7,"pith_summary":"This paper characterizes the smooth maps between sub-Riemannian Lie groups (called polarized groups) that commute with the sub-Laplacian, the horizontal Laplace operator built from the group's left-invariant metric and Haar measure. Theorem A states that a C² map F satisfies the intertwining identity Δ_G(u∘F) = λ²(Δ_H u)∘F plus a first-order drift term for every test function u if and only if F is a conformal submersion of factor λ: at every point its horizontal derivative is, up to the factor λ, an isometric projection onto the whole horizontal space of the target. The drift term is forced to be the trace of the second Lie differential plus contributions from the modular functions of the two groups. In the Carnot case the rigidity sharpens: if such a map exists with constant λ, the target is a Carnot quotient of the domain, and in equal dimension the map is a dilation, a left translation, and an isometric automorphism composed. Consequently the sub-Laplacian of a Carnot group is a complete invariant of its sub-Riemannian structure: two such operators are equivalent by a change of coordinates exactly when the underlying Carnot geometries are isometric.","feed_headline":"Maps that commute with sub-Laplacians are conformal submersions","feed_subtitle":"On Carnot groups, the sub-Laplacian alone determines the sub-Riemannian structure up to isometry.","key_machinery":"The proof machinery is a family of localized test functions u^q_α(x)=φ($q^{{-1}}$x)⟨α|log($q^{{-1}}$x)⟩², built from a bump function φ and a covector α; at the point q these functions have zero horizontal derivative but their second Lie derivatives in horizontal directions equal 2⟨α|v⟩². Substituting these into the intertwining identity at a point p with F(p)=q yields Σ_i⟨α|DF(p)X_i⟩² = λ(p)² Σ_j⟨α|Y_j⟩² for every α∈h*, which is precisely the condition that the adjoint of DF(p)|_{V(G)} is a homothetic embedding of factor λ(p). By Proposition 2.4 this is equivalent to DF(p)|_{V(G)} being a homothetic projection, i.e., F being a conformal submersion. The lower-order drift b is then identified by the chain rule from the same identity, giving the formula with trace_G(D²F) and modular-function gradients.","core_discovery":"The central discovery is that, on sub-Riemannian Lie groups, the sub-Laplacian intertwining condition is a geometric condition. Theorem A proves that for a C² map F between open domains of sub-Riemannian Lie groups G and H, the identity Δ_G(u∘F)=λ²(Δ_H u)∘F+⟨b,(∇_H u)∘F⟩_H+c(u∘F) holds for all C² functions u exactly when F is a conformal submersion of factor λ, c≡0, and b(p)=trace_G(D²F(p))+DF(p)[∇_G μ_G(p)]−λ(p)²∇_H μ_H(F(p)). In particular, if the lower-order terms vanish and λ is constant, F is a homothety; if λ≡1 and dimensions agree, F is an isometry. Theorem B then shows that in Carnot groups the sub-Laplacian determines the sub-Riemannian structure: a C² map satisfying the pure scaling identity Δ_G(u∘F)=λ²(Δ_H u)∘F forces H to be a Carnot quotient of G, and in equal dimension F is a dilation, a left translation, and an isometric automorphism composed, so G and H are isomorphic as Carnot groups.","pith_inferences":["Because Theorem A is proved by pointwise test functions, the same conformal-submersion characterization should hold for sub-Laplacians on general sub-Riemannian manifolds with a smooth measure, not only on Lie groups; the left-invariant structure only fixes the displayed form of the operator.","The explicit drift formula suggests a quantitative rigidity device: measuring how far a map commuting with the sub-Laplacian's principal part is from a homothety reduces to computing trace_G(D²F) and the modular gradients, both of which are directly computable in coordinates.","Since Remark 1.1 upgrades C² intertwiners on Carnot groups to C∞ via hypoellipticity, and Pansu differentiability is known under weaker regularity, one could hope that the rigidity in Theorem B persists for C¹ or even continuous intertwiners; this is a natural testable weakening.","The Heisenberg classification by symplectic spectrum up to scale suggests that for higher-step Carnot groups the equivalence classes of sub-Laplacians will be parameterized by analogous invariants of the horizontal metric under the automorphism group of the nilpotent group; computing such invariants for the first non-Heisenberg case would be a concrete test."],"forward_implications":["For any two sub-Riemannian Lie groups, a C² map whose pullback action on the sub-Laplacian has principal symbol λ²(Δ_H u)∘F must be a conformal submersion; there is no other way to commute with the principal part.","On a Carnot group, two sub-Laplacians given by sums of squares of left-invariant fields are equivalent by a coordinate change if and only if the corresponding sub-Riemannian Carnot groups are isometric, and any such coordinate change is itself an isometry and a group automorphism.","A sub-Laplacian-commuting map of constant factor forces the target Carnot group to be a quotient of the domain, so the sub-Laplacian detects the whole hierarchy of Carnot quotients.","In Heisenberg groups, non-isometric sub-Riemannian structures yield non-equivalent sub-Laplacians, and the equivalence classes are parameterized by the symplectic spectrum of the horizontal scalar product up to a common scale.","Every sub-Laplacian on a Carnot group is induced from a free Carnot group by the quotient submetry, giving a normal form in which all Carnot sub-Laplacians are quotients of a free one."],"supporting_citations":[{"why":"Provides the Carnot-group background and the Chapter 16 question on equivalence of sums-of-squares operators that the paper answers.","marker":"[1]"},{"why":"Cited for the theorem that every C¹ contact map between Carnot groups is Pansu differentiable at every point, used in Proposition 5.3 to produce a Carnot morphism from a conformal submersion.","marker":"[2]"},{"why":"Cited for the rigidity that isometries of nilpotent metric groups are affine, used in Theorem 5.6 and Proposition 5.4.","marker":"[10]"},{"why":"Cited for the definition and basic properties of Carnot groups, including dilations, used in Section 5.2 for the quotient statement.","marker":"[11]"},{"why":"Cited for the rigidity that isometries of Carnot groups are left translations composed with automorphisms, used to identify F in the equal-dimension case of Theorem B.","marker":"[12]"},{"why":"Supplies the symplectic-spectrum classification lemma (Lemma 5.5) that underlies Theorem 5.6 for Heisenberg groups.","marker":"[14]"}],"fun_headline_variants":["Sub-Laplacians force conformal submersions","Sub-Laplacian intertwining maps are conformal","Carnot sub-Laplacian determines sub-Riemannian structure","Sub-Laplacian reveals sub-Riemannian geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharp Carnot conclusions rest on two imported facts: that a smooth-enough map sending horizontal directions to horizontal directions between Carnot groups has a derivative-like approximation at every point, and that every distance-preserving map of a Carnot group is a rigid affine map; if either of these fails, the conclusion that the target is a quotient of the domain could fail even though the conformal-submersion characterization from Theorem A would survive.","fun_headline_variants_meta":{"raw":{"variants":["Sub-Laplacians force conformal submersions","Sub-Laplacian intertwining maps are conformal","Carnot sub-Laplacian determines sub-Riemannian structure","Sub-Laplacian reveals sub-Riemannian geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001102,"raw_usage":{"total_tokens":4555,"prompt_tokens":861,"completion_tokens":3694,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":3628}},"tokens_in":477,"tokens_out":3694,"duration_ms":35440,"temperature":1.0,"reasoning_tokens":3628,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:50:41.142726+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the Heisenberg example of Section 5.5, choose two scalar products on $R^{5}$ whose symplectic spectra are not proportional, such as (1,2) and (1,3). The paper says the two sub-Laplacians are inequivalent. Checking directly whether any C² change of coordinates F satisfies Δ_{(1,2)}(u∘F)=λ²(Δ_{(1,3)}u)∘F for all u would settle the rigidity claim; the predicted answer is that no such F exists.","supporting_citations":[{"cited_title":"Bonfiglioli, E","cited_arxiv_id":null,"evidence_quote":"Provides the Carnot-group background and the Chapter 16 question on equivalence of sums-of-squares operators that the paper answers."},{"cited_title":"Isometries of nilpotent met ric groups","cited_arxiv_id":null,"evidence_quote":"Cited for the rigidity that isometries of nilpotent metric groups are affine, used in Theorem 5.6 and Proposition 5.4."},{"cited_title":"A primer on Carnot groups: homogenous grou ps, Carnot- Carathéodory spaces, and regularity of their isometries","cited_arxiv_id":null,"evidence_quote":"Cited for the definition and basic properties of Carnot groups, including dilations, used in Section 5.2 for the quotient statement."},{"cited_title":"Isometries of Carnot groups and sub-Finsler homogeneous manifolds","cited_arxiv_id":null,"evidence_quote":"Cited for the rigidity that isometries of Carnot groups are left translations composed with automorphisms, used to identify F in the equal-dimension case of Theorem B."},{"cited_title":"McDuff and D","cited_arxiv_id":null,"evidence_quote":"Supplies the symplectic-spectrum classification lemma (Lemma 5.5) that underlies Theorem 5.6 for Heisenberg groups."}],"review_version":1}