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Geometric equivalence among smooth map germs

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Pith's one-line read For connected linear Lie groups whose infinitesimal vector fields are linear, A0[G]-equivalence of map germs is exactly G-congruence, making singularity A0[G]-geometry coincide with classical G-geometry.

desk verdict A useful unifying framework for A[G]-equivalence with a true but under-proved main theorem; the gap in Theorem 7.1 is standard and fixable. read the letter →

arxiv 1908.08232 v1 pith:6SVZXGHG submitted 2019-08-22 math.DG

Shyuichi Izumiya , Masatomo Takahashi , Hiroshi Teramoto This is my paper · ORCID
classification math.DG MSC 58K4053C10
keywords G-structureA-equivalencesingularitiesofmapgermsA[G]-equivalenceG-congruenceinfinitesimaltangentspacesLiegroupactions
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 introduces A[G]-equivalence, a geometric variant of the classical right-left equivalence of smooth map germs: two germs f,g:(R^n,0)→(R^p,0) are equivalent when g can be obtained from f by a right change of coordinates in the source and a left change of coordinates in the target whose Jacobian lies in a prescribed linear Lie group G⊂GL(p,R) at every point. The central result, Theorem 7.1, is a criterion for when this new equivalence is not new at all. For a connected linear Lie group G, the following are equivalent: the Lie algebra g is isomorphic to the space θ[G]_0(p) of infinitesimal G-preserving vector fields; every vector field in θ[G]_0(p) has linear component functions; and the identity-component group Diff_0[G](p) of G-preserving diffeomorphism germs equals G itself. When any of these holds, A0[G]-equivalence coincides with G-congruence (the classical R×G-equivalence), so singularity-theoretic A0[G]-geometry is the classical G-geometry. The criterion is genuinely discriminating: it holds for the orthogonal group SO(p), making isometric A-equivalence the usual Euclidean congruence of curves and surfaces, and it fails for SL(p,R), where the equivalence is larger and carries infinite-dimensional moduli.

What carries the argument

The load-bearing object is θ[G]_0(p), the space of vector field germs on (R^p,0) that vanish at the origin and whose Jacobian matrix lies in the Lie algebra g of G at every point; it is exactly the formal tangent space T_1 Diff[G](p) of the group of G-preserving diffeomorphism germs. The argument compares two formal tangent spaces at a map germ f: the A[G]-tangent space TA[G](f)=tf(M_nθ(n))+ω_f(θ[G]_0(p)) and the R×G-tangent space T(R×G)(f)=tf(M_nθ(n))+g(f), where g(f)={X∘f : X∈g}. The embedding ι(e_ij)=y_i∂/∂y_j identifies g with the linear vector fields inside θ[G]_0(p), and Theorem 7.1 uses this identification to show that linearity of θ[G]_0(p) is equivalent to Diff_0[G](p)=G. Corollary 7.2 then turns that group equality into equality of tangent spaces and of orbits, so every A0[G]-equivalence is a G-congruence.

What would settle it

For a candidate group such as G=SO(p1,p2) with p1,p2>0, compute the space θ[G]_0(p) explicitly. If it contains a vector field with a non-linear component whose Jacobian matrix still lies in so(p1,p2) at every point, then condition (2) of Theorem 7.1 fails and the theorem predicts Diff_0[G](p)≠G; finding such a vector field, or proving none exists, would settle whether the linearity criterion is the correct dividing line. A more direct refutation would be to produce a single η∈θ[G]_0(p) whose integral flow leaves Diff_0[G](p) even though condition (1) holds.

Watch

Extended reading notes

Core claim

The core discovery is Theorem 7.1 together with Corollary 7.2. For a connected linear Lie group G⊂GL(p,R), let θ[G]_0(p) be the R-vector space of vector field germs η=Ση_i ∂/∂y_i on (R^p,0) whose Jacobian matrix (∂η_i/∂y_j)(y) lies in the Lie algebra g for every y and which vanish at the origin. The following are equivalent: (1) g≅θ[G]_0(p) as R-vector spaces; (2) the component functions η_i of every element of θ[G]_0(p) are linear functions; (3) Diff_0[G](p)=G, where Diff_0[G](p) is the group of diffeomorphism germs isotopic to the identity through diffeomorphisms whose Jacobians stay in G. Under any of these conditions, the formal tangent space of the A0[G]-orbit equals the tangent space of the R×G-orbit for every map germ f, and two map germs are A0[G]-equivalent exactly when they are G-congruent. Thus, precisely in this situation, A0[G]-geometry and classical G-geometry are identical.

Load-bearing premise

The load-bearing premise is that every vector field whose derivative matrix lies in the Lie algebra g integrates to a flow of diffeomorphism germs whose Jacobians remain in G; without that integration step, the identification T_1 Diff[G](p)=θ[G]_0(p) cannot force Diff_0[G](p)=G, and the paper proves only one direction of it.

Editorial extensions

If this is right

  • For G=SO(p), Euclidean curve and surface geometry—curvatures, Frenet-type invariants, Monge normal forms—is exactly the corresponding A0[SO(p)]-singularity theory; the paper makes the classical identification precise.
  • Whenever the theorem's conditions hold, the relative infinitesimal moduli space M(A[G];R×G)(f) vanishes for every germ f, so no functional moduli separate the two geometries.
  • For G=SL(p,R), the conditions fail; θ[SL(p,R)] is infinite-dimensional, A[SL] is not a geometric subgroup in the sense used in the paper, and the relative moduli space is infinite-dimensional, so volume-preserving (unimodular) classification is genuinely broader than equi-affine congruence.
  • For H<G with G satisfying the theorem, the quotient M(A[G];A[H])(f) equals M(R×G;R×H)(f) and its dimension is bounded by dim G−dim H; the gap between two such geometries is finite-dimensional and computable from the Lie algebras.
  • The paper's proposed semi-finite determinacy problem follows: when the conditions fail, finite Taylor jets can still encode all geometric invariants of interest, but the classical finite-determinacy theorems no longer apply, and the tangent-space structure described here is offered as the guide.

Reading between the lines

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

  • The same infinitesimal linearity test could be applied to the indefinite orthogonal groups SO(p1,p2), which the paper mentions but does not fully analyze; deciding whether θ[G]_0(p) is linear for those groups would map out exactly where A[G]-geometry collapses into classical pseudo-Riemannian congruence.
  • A natural extension is to source-side G'-structures—the R[G'], A[G';G], and K[G';G] equivalences the introduction mentions but leaves aside; one would expect an analogue of Theorem 7.1 to characterize when those equivalences reduce to the corresponding classical actions on the source.
  • The relative moduli space M(A[G];R×G)(f) could serve as a quantitative measure of how many independent geometric invariants a singular germ carries; computing it for the known normal forms of cuspidal edges and swallowtails would connect the formalism to existing curvature calculations.
  • If the criterion holds broadly, singularity classification and geometric classification become the same task for the affected groups; a practical consequence is that existing lists of G-congruence normal forms can be read as complete A0[G]-classifications, and vice versa.
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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

3 major / 5 minor

Summary. The paper proposes A[G]-equivalence for smooth map germs (R^n,0)→(R^p,0), where G⊂GL(p,R) is a linear Lie group viewed as a G-structure on the target; A[GL(p,R)] recovers classical A-equivalence and A[{I_p}] recovers R-equivalence. It introduces the space θ[G](p) of vector-field germs whose Jacobian lies in the Lie algebra g, the algebra E_p[G] of functions compatible with θ[G]_0(p), and defines infinitesimal tangent spaces for A[G]- and R×G-equivalences. The paper computes these objects for SO(p), SL(p,R), Sp(2,R), block-diagonal and block-triangular groups, and relates them to existing classifications (isometric A-equivalence, unimodular geometry, bi-A-equivalence, Lagrangian equivalence). The main theorem (Thm. 7.1) asserts that, for connected linear G, three conditions are equivalent: (1) g≅θ[G]_0(p) as vector spaces, (2) every vector field in θ[G]_0(p) has linear components, and (3) Diff_0[G](p)=G; Corollary 7.2 concludes that under these conditions A0[G]-equivalence coincides with G-congruence and the relative infinitesimal moduli space vanishes.

Significance. If Theorem 7.1 is established, the paper gives a clean algebraic criterion for when the newly introduced A[G]-geometry collapses to classical G-geometry, and it explains why for SO(p) the two coincide while for SL(p,R) they differ. The examples connect the framework to substantial existing work (Domitrz–Rieger, Dufour, Ishikawa–Janeczko, Lagrangian singularity theory) and the paper explicitly flags when A[G] is or is not a geometric subgroup in Damon's sense. The manuscript contains a number of correct and useful computations (e.g., the finite-dimensionality of θ[SO(p)]_0 and the identification of θ[SL(p,R)] with exact (p−1)-forms). However, the proof of the central theorem has two real gaps — the unproved reverse inclusion in the identification T_1 Diff[G](p)=θ[G]_0(p) and a non-sequitur in the proof of (2)⇒(3) — so the central claim is not yet established as written.

major comments (3)
  1. [§4] The equality T_{1_{R^p}} Diff[G](p) = θ[G]_0(p) is asserted after proving only the inclusion 'tangent vectors lie in θ[G]_0(p)'. The reverse inclusion — that every germ η with η(0)=0 and Dη(y)∈g for all y is tangent to a curve in Diff[G](p) — is not shown. This matters because Theorem 7.1(3)⇒(1) and the identification of A[G]-infinitesimal data with g(f) in Corollary 7.2 both rely on this equality. The missing argument is standard: integrate the time-dependent vector field η to get a flow φ_t with φ_0=id; then D_y φ_t solves X'(t)=Dη(φ_t(y))X(t), X(0)=I, and since Dη(φ_t(y))∈g and G is a connected Lie subgroup of GL(p,R) with Lie algebra g, the solution X(t) lies in G for all t. Please add this proof.
  2. [§4 and §7, Theorem 4.6 and Theorem 7.1] In the proof of Theorem 4.6, the step 'so that (dh_t/dt)|_{t=t0}(y)=0' is a non-sequitur. If dφ_t/dt|_{t=t0} = η∘φ_{t0} with η(y)=B(t0)y linear, and φ_t(y)=A(t)y+h_t(y), then comparing linear and higher-order parts gives A'(t0)=B(t0)A(t0) and h'_{t0}(y)=B(t0)h_{t0}(y), not h'_{t0}=0. The conclusion φ_t∈SO(p) still follows because h_0=0 and the linear ODE h'=B h has the unique solution h≡0, but the proof must be corrected. The same flawed step is invoked in Theorem 7.1 for the implication (2)⇒(3) via 'the same method'. A cleaner proof uses the right logarithmic derivative v_t=dφ_t/dt∘φ_t^{-1}; then D v_t∈g, condition (2) forces v_t(y)=B(t)y, and hence dφ_t/dt=B(t)φ_t, so φ_t∈G.
  3. [§7, proof of Theorem 7.1] The implication (3)⇒(1) is dismissed with the phrase 'If we consider the formal tangent space of Diff_0[G](p)=G, we can easily show...'. This is load-bearing and should be written out: from Diff_0[G](p)=G one obtains T_1 Diff_0[G](p)=g (embedded in θ(p) by linear vector fields X y·∂/∂y), and combining this with the equality T_1 Diff[G](p)=θ[G]_0(p) (whose proof is incomplete; see the first comment) gives θ[G]_0(p)≅g. Please provide the details, including the identification of the tangent space of Diff_0[G](p) at the identity.
minor comments (5)
  1. [§2] The definition of A[G]-equivalence uses f∘φ=ψ∘g, while A-equivalence was defined by ψ∘f=g∘φ. Since φ and ψ range over all diffeomorphisms the two conventions are equivalent, but the reversal should be flagged for the reader.
  2. [Example 4.10(2)] The notation θ[N]_0(p)=M_pθ(π_{p1}) is ambiguous; it should be clarified whether the module is over E_{p1} and what the subscript p denotes.
  3. [Proposition 6.9] The entry '(x1x1x2+x4_2+Q)' appears to contain a typo; it should probably read '(x1,x1x2+x4_2+Q)'.
  4. [After Corollary 7.2] The assertion that condition (2) of Theorem 7.1 holds for G=SO(p1,p2) is stated without proof; either give the proof or mark it as conjectural.
  5. [Theorem 4.6] The phrase 'η_i(y) are linear function germs' for SO(p) relies on Example 4.4; for the benefit of the reader, the argument that θ[SO(p)]_0 is spanned by the angular vector fields should be restated briefly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the main equivalence is derived from definitions and standard singularity theory; the only self-citation is not load-bearing.

full rationale

The central claim is Theorem 7.1 / Corollary 7.2: for a connected linear Lie group G, the conditions g ≅ θ[G]_0(p), linearity of all component functions of elements of θ[G]_0(p), and Diff_0[G](p)=G are equivalent, and then A0[G]-equivalence coincides with G-congruence. The proof of (1)⇔(2) is a direct argument using the inclusion ι(g)⊂θ[G]_0(p) and the linearity assumption; it does not import a fitted parameter or an earlier result of the same authors. The step (3)⇒(1) uses the formal tangent space identification T_1 Diff[G](p)=θ[G]_0(p), which is asserted in Section 4; this is a mathematical claim about tangent spaces, and although the paper only explicitly proves one inclusion, that is a proof-completeness concern rather than a circular reduction. The proof of Corollary 7.2 is essentially a consequence of the definition of A0[G]-equivalence once Diff_0[G](p)=G is known, but the content lies in Theorem 7.1 and Theorem 4.6. The in-preparation self-citation [20] appears only in a list of applications of K[G]-equivalence and is not load-bearing for the derivation. The examples in Section 6 quote earlier classifications as applications rather than as inputs to the proof. No prediction is fitted and then renamed as a result, and no load-bearing argument reduces to a self-citation. Thus no step is circular by construction; score 0.

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

No empirical parameters appear; the paper is pure mathematics. The central theorem rests on standard singularity theory (Mather, Damon, Malgrange), on the Poincare lemma, and on an implicit integrability assertion for vector fields with Jacobian in g.

assumptions (4)
  • domain assumption Diffeomorphisms with Jacobian in G are exactly the local automorphisms of the G-structure; in particular the formal tangent space of Diff[G](p) at the identity is theta[G]_0(p).
    Stated in Section 4 and used in the proof of Theorem 7.1. The forward inclusion is shown, but the reverse inclusion rests on an unstated integrability argument for vector fields with Jacobian in g.
  • standard math Mather's criterion: if p > 1 and theta(f)/tf(theta(n)) has finite R-dimension, then f is a submersion germ.
    Used in Proposition 5.2 through the unpublished note [29, Proposition 1.11].
  • standard math Malgrange preparation theorem holds for DA-algebras and DA-modules.
    Invoked in Section 4 when treating Ep[G] as a DA-subalgebra; this is standard in singularity theory.
  • standard math Poincare lemma for germs of differential forms at the origin.
    Used in Proposition 4.8 to identify divergence-free vector fields with exact (p-1)-forms.

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Pith. "Pith review of Geometric equivalence among smooth map germs." pith.science (2026). https://pith.science/paper/6SVZXGHG

@misc{pith2026190808232,
  author       = {Pith},
  title        = {Pith review of: Geometric equivalence among smooth map germs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SVZXGHG}},
  note         = {Machine review of arXiv:1908.08232}
}
read the original abstract

We consider equivalence relations among smooth map germs with respect to geometry of G-structures on the target space germ. These equivalence relations are natural generalization of right-left equivalence (i.e., A-equivalence) in the sense of Thom-Mather depending on geometric structures on the target space germ. Unfortunately, these equivalence relations are not necessarily geometric subgroups in the sense of Damon (1984). However, we have interesting applications of these equivalence relations.

Discussion (0). Continue with ORCID to comment.

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