pith. sign in

arxiv: 2412.19147 · v2 · pith:NSEDL7Z5new · submitted 2024-12-26 · 🧮 math.DG

The total absolute curvature of submanifolds with singularities

Yuta Yamauchi This is my paper

Pith reviewed 2026-05-23 07:17 UTC · model grok-4.3

classification 🧮 math.DG
keywords Chern-Lashof theoremtotal absolute curvaturefrontalssingular submanifoldsBetti numbersconvexityEuclidean space
0
0 comments X

The pith

An n-dimensional admissible compact frontal has total absolute curvature at least the sum of its Betti numbers.

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

This paper extends the Chern-Lashof theorem from smooth submanifolds to frontals that may contain singularities. It proves that any n-dimensional admissible compact frontal in Euclidean space of dimension n plus r has total absolute curvature greater than or equal to the sum of the Betti numbers. When the curvature value equals 2 and every singularity is of the first kind, the image must coincide with a closed convex domain lying inside an affine n-dimensional subspace. A reader would care because the result supplies a topological lower bound on curvature that continues to hold once limited singularities are allowed.

Core claim

For an n-dimensional admissible compact frontal in (n+r)-dimensional Euclidean space R^{n+r}, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to 2, and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine n-dimensional subspace of R^{n+r}.

What carries the argument

Total absolute curvature extended to admissible compact frontals, which integrates the absolute value of the appropriate curvature form over the singular object.

Load-bearing premise

The definitions of admissible compact frontal and singularities of the first kind make the total absolute curvature well-defined and allow the inequality and equality characterization to hold.

What would settle it

An explicit admissible compact frontal whose total absolute curvature is strictly smaller than the sum of its Betti numbers, or a case of equality at value 2 whose image is not a closed convex domain inside a flat affine subspace.

Figures

Figures reproduced from arXiv: 2412.19147 by Yuta Yamauchi.

Figure 1
Figure 1. Figure 1: The singular set Σf (yellow surface) and the set of singular points of the second kind Cf (blue curve) of the map germ f : R3 → R4 of the A3-type singular point. Since Cf is a regular curve in Σf , f is an admissible frontal (cf. Definition 2.7, Example 2.4). 2.2. The total absolute curvature. In this subsection, we review the notion of the Lipschitz-Killing curvature and define the total absolute curvatur… view at source ↗
Figure 2
Figure 2. Figure 2: The image fk(S 2 ) when n = 2 and k = 2 3 . 3. Proof of Theorems A and B In this section, we prove Theorems A and B by applying the Morse theory to height functions. For a frontal f : Mn → Rn+r and w ∈ S n+r−1 , we set a function hw on Mn as hw(p) = f(p) · w (p ∈ Mn ) which is called the height function with respect to w. If the derivative (dhw)p vanish at p ∈ Mn, a point p is called a critical point of hw… view at source ↗
Figure 3
Figure 3. Figure 3: The figure of closed convex domain Ω (Lemma 3.5). Finally, we prove Theorem B. Proof of Theorem B. First, we assume that the total absolute curvature τ(Mn , f) is equal to 2. By Theorem A, Mn is homeomorphic to an n-sphere. Since the singular set Σf is not empty and consists of singular points of the first kind, ¯f : Σf → Rn+r is an immersion. Thus, Fact 1.1-(1) yields that the total absolute curvature of … view at source ↗
Figure 4
Figure 4. Figure 4: The image f(S 2 ) in R3 (Example 3.6). 4. The cases of 1- and 2-dimensions In this section, we exhibit the results obtained in the cases of n = 1, 2. 4.1. In the case of n = 1. For a smooth map γ : I → Rr+1 defined on a non￾empty interval I, a point c ∈ I is a singular point of γ if and only if the derivative γ ′ (c) does vanish, where the prime ′ means d/dt. A smooth map γ : I → Rr+1 whose regular set is … view at source ↗
read the original abstract

In this paper, we give a generalization of the Chern-Lashof theorem for submanifolds with singularities called frontals in Euclidean space. We prove that, for an $n$-dimensional admissible compact frontal in $(n+r)$-dimensional Euclidean space $\boldsymbol{R}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$, and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-dimensional subspace of $\boldsymbol{R}^{n+r}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper generalizes the Chern-Lashof theorem to n-dimensional admissible compact frontals in Euclidean space R^{n+r}. It establishes that the total absolute curvature is at least the sum of the Betti numbers. When this curvature equals 2 and all singularities are of the first kind, the image of the frontal is a closed convex domain in an affine n-dimensional subspace.

Significance. If the extension of the total absolute curvature via the Gauss map (or equivalent) to the singular locus is rigorously defined and the admissible frontal conditions are appropriately chosen, the result provides a controlled generalization of a classical lower bound to singular submanifolds. The equality-case characterization mirrors the smooth convex case and may be useful for applications involving wavefronts and Legendre immersions. The manuscript ships a parameter-free topological bound together with a convexity conclusion under an explicit singularity hypothesis.

minor comments (2)
  1. [§1] §1: A one-sentence reminder of the classical Chern-Lashof statement (including the role of the Gauss map) would help readers situate the extension.
  2. [Definition 2.3] Definition 2.3 (admissible frontal): the precise list of conditions that make the curvature measure well-defined and the lower bound hold should be cross-referenced when first used in the main theorem statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The referee's summary accurately captures the main results on the generalization of the Chern-Lashof theorem to admissible compact frontals.

Circularity Check

0 steps flagged

No significant circularity; extension of Chern-Lashof theorem is self-contained

full rationale

The paper presents a generalization of the classical Chern-Lashof lower bound on total absolute curvature to the setting of admissible compact frontals, with an equality case under an additional first-kind singularity hypothesis. The abstract and claim structure frame the result as a direct extension relying on well-defined notions of frontals and curvature measure extension, without any visible reduction of the central inequality or equality characterization to a fitted parameter, self-citation chain, or definitional tautology. No load-bearing step is shown to collapse by construction to its own inputs. The derivation is therefore treated as independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; no free parameters, invented entities, or non-standard axioms are visible in the given text.

axioms (1)
  • standard math Standard axioms and definitions of differential geometry and algebraic topology apply to frontals and their total absolute curvature.
    The statements rely on the extension of classical notions (Betti numbers, total absolute curvature) to the singular frontal setting.

pith-pipeline@v0.9.0 · 5618 in / 1302 out tokens · 23353 ms · 2026-05-23T07:17:33.644690+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    B. -Y. Chen, On the total absolute curvature of manifolds immersed in Rie mannian manifold, K¯ odai Math. Semin. Rep.19 (1967), 299–311

    work page 1967

  2. [2]

    B. -Y. Chen, On the total absolute curvature of manifolds immersed in Rie mannian manifold II, K¯ odai Math. Semin. Rep. 22 (1970), 89–97

    work page 1970

  3. [3]

    S. -S. Chern and R. K. Lashof, On the total curvature of immersed manifolds , Am. J. Math. 79 (1957), 306–318

    work page 1957

  4. [4]

    S. -S. Chern and R. K. Lashof, On the total curvature of immersed manifolds. II , Mich. Math. J. 5 (1958), 5–12

    work page 1958

  5. [5]

    Fenchel, ¨Uber Kr¨ ummung und Windung geschlossener Raumkurven , Math

    W. Fenchel, ¨Uber Kr¨ ummung und Windung geschlossener Raumkurven , Math. Ann. 101 (1929), 238–252

    work page 1929

  6. [6]

    Fenchel, On the differential geometry of closed space curves , Bull

    W. Fenchel, On the differential geometry of closed space curves , Bull. Am. Math. Soc. 57 (1951), 44–54

    work page 1951

  7. [7]

    Ghomi and J

    M. Ghomi and J. Spruck, Total curvature and the isoperimetric inequality in Cartan - Hadamard manifolds , J. Geom. Anal. 32 (2022), 54

    work page 2022

  8. [8]

    J. A. Hoisington, On the total curvature and Betti numbers of complex projecti ve manifolds , Geom. Topol. 26 (2022), 1–45

    work page 2022

  9. [9]

    Honda, Isometric immersions with singularities between space for ms of the same positive curvature, J

    A. Honda, Isometric immersions with singularities between space for ms of the same positive curvature, J. Geom. Anal. 27 (2017), 2400–2417

    work page 2017

  10. [10]

    Honda, K

    A. Honda, K. Naokawa, K. Saji, M. Umehara, and K. Yamada, Duality on generalized cuspidal edges preserving singular set images and first fundamental f orms, J. Singul. 22 (2020), 59–91

    work page 2020

  11. [11]

    Honda and K

    A. Honda and K. Saji, Geometric invariants of 5/2-cuspidal edges, Kodai Math. J. 42 (2019), 496–525

    work page 2019

  12. [12]

    Honda, C

    A. Honda, C. Tanaka, and Y. Yamauchi, The total absolute curvature of closed curves with singularities, to appear in Advances in Geometry (arXiv:2403.00487)

    work page arXiv

  13. [13]

    Ishikawa

    G. Ishikawa. Singularities of frontals , in: Singularities in Generic Geometry, in: Advanced Studies in Pure Mathmatics (ASPM), vol. 78, Math, Soc, Japan , 2018, pp. 55–106

    work page 2018

  14. [14]

    Izumiya, Total lightcone curvatures of spacelike submanifolds in Lo rentz-Minkowski space, Differ

    S. Izumiya, Total lightcone curvatures of spacelike submanifolds in Lo rentz-Minkowski space, Differ. Geom. Appl. 34 (2014), 103–127

    work page 2014

  15. [15]

    Kossowski and M

    M. Kossowski and M. Scherfner, Total curvature for C∞-singular surfaces with limiting tangent bundle , Ann. Global Anal. Geom. 28 (2005), 179–199

    work page 2005

  16. [16]

    Koike, The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type , Result

    N. Koike, The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type , Result. Math. 39 (2001), 230–244. 15

    work page 2001

  17. [17]

    Koike, Theorems of Gauss-Bonnet and Chern-Lashof types in a simply connected sym- metric space of non-positive curvature , Tokyo J

    N. Koike, Theorems of Gauss-Bonnet and Chern-Lashof types in a simply connected sym- metric space of non-positive curvature , Tokyo J. Math. 26 (2003), 527–539

    work page 2003

  18. [18]

    Koike, The Gauss-Bonnet and Chern-Lashof theorems in a simply conn ected symmetric space of compact type , Tokyo J

    N. Koike, The Gauss-Bonnet and Chern-Lashof theorems in a simply conn ected symmetric space of compact type , Tokyo J. Math. 28 (2005), 483–497

    work page 2005

  19. [19]

    N. H. Kuiper, Convex immersions of closed surfaces in E3. Non-orientable closed surfaces in E3 with minimal total absolute Gauss-curvature , Comment. Math. Helv. 35 (1961), 85–92

    work page 1961

  20. [20]

    N. H. Kuiper, Minimal total absolute curvature for immersions , Invent. Math. 10 (1970), 209–238

    work page 1970

  21. [21]

    N. H. Kuiper and W. Meeks. III, Total curvature for knotted surfaces , Invent. Math. 77 (1984), 25–69

    work page 1984

  22. [22]

    K. Saji, M. Umehara, and K. Yamada, The geometry of fronts , Ann. Math. 169-2 (2009), 491–529

    work page 2009

  23. [23]

    K. Saji, M. Umehara, and K. Yamada, Ak singularities of wave fronts , Math. Proc. Camb. Philos. Soc. 146 (2009), 731–746

    work page 2009

  24. [24]

    Teufel, Eine Differentialtopologische Berechnung der totalen Kr¨ ummung und totalen Abso- lutkr¨ ummung in der sph¨ arischen Differentialgeometrie, Manuscr

    E. Teufel, Eine Differentialtopologische Berechnung der totalen Kr¨ ummung und totalen Abso- lutkr¨ ummung in der sph¨ arischen Differentialgeometrie, Manuscr. Math. 31 (1980), 119–147

    work page 1980

  25. [25]

    Teufel, Differential topology and the computation of total absolute curvature, Math

    E. Teufel, Differential topology and the computation of total absolute curvature, Math. Ann. 258 (1982), 471–480

    work page 1982

  26. [26]

    Teufel and G

    E. Teufel and G. Solanes, Horo-tightness and total (absolute) curvatures in hyperbo lic spaces, Isr. J. Math. 194 (2013), 427–459

    work page 2013

  27. [27]

    Umehara, K

    M. Umehara, K. Saji, and K. Yamada, Differential geometr y of curves and surfaces with singularities, Series in Algebraic and Differential Geomet ry 1. W orld Scientific Publishing Co. Pte. Ltd. Hackensack, NJ, xvi, 370 p. (2022)

    work page 2022

  28. [28]

    T. J. Willmore and B. A. Saleemi, The total absolute curvature of immersed manifolds , J. Lond. Math. Soc. 41 (1966), 153–160

    work page 1966

  29. [29]

    Wintgen, On total absolute curvature of nonclosed submanifolds , Ann

    P. Wintgen, On total absolute curvature of nonclosed submanifolds , Ann. Global Anal. Geom. 2 (1984), 55–87. Graduate School of Engineering Science, Yokohama National Uni versity, Hodogaya, Yokohama 240-8501, Japan Email address : yamauchi-yuta-hj@ynu.jp 16

    work page 1984