Pith. sign in

REVIEW 3 major objections 3 minor 59 references

Transverse Spheres in Flag Manifolds

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that, in self-opposite flag manifolds of split real simple Lie groups, transverse circles are locally maximally transverse exactly for the pairs $(G,\Theta)$ listed in Table 1, and constructs transverse spheres of…

desk verdict A serious, innovative classification preprint that closes Question 0.1 for split real groups if the delegated coverage claims in Table 1 hold up. read the letter →

arxiv 2507.19306 v1 pith:TPPE3YUJ submitted 2025-07-25 math.GT math.DG

Parker Evans , J. Maxwell Riestenberg This is my paper · ORCID
classification math.GTmath.DG MSC 22E4022E4653C3557S30
keywords transversespheresflagmanifoldsAnosovsubgroupsCliffordalgebrasspinorrepresentationsAtiyah-Bott-ShapiroisomorphismmaximaltransversalitysplitrealLiegroups
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 asks when a transverse circle in a flag manifold can be locally enlarged to a larger transverse set, and answers the question completely for all self-opposite flag manifolds of split real simple Lie groups. The answer is a short list: most cases have no larger transverse neighbour, but the exceptions force strong structure on discrete subgroups—in particular, every $\{7\}$-Anosov subgroup of split $E_7$ is virtually free or a surface group. To reach this, the authors build transverse spheres of arbitrarily large dimension from spinor representations and use the Atiyah-Bott-Shapiro isomorphism to prove these spheres are maximally transverse exactly for $n \equiv 0,1,2,4 \pmod{8}$. A companion result shows that, for split groups, the absence of transverse 2-spheres, local maximal transversality of transverse circles, and a combinatorial property called Property (I) are equivalent.

What carries the argument

The central object is the spinor representation of $\mathrm{Spin}(n)$ built from real Clifford algebras: for $n = 8k + r$ the irreducible spinor module has dimension $d(n)$ as stated, and its action on $\mathbb{R}^{d(n)}$ produces the transverse sphere via an orbit map on a full flag. Maximal transversality is certified by the Atiyah-Bott-Shapiro isomorphism, which identifies the stable homotopy class of the clutching map $S^{n-1} \to \mathrm{SO}(d)$ with a class in the Grothendieck group of Clifford modules; the calculation pins down the mod-8 condition $n \in \{0,1,2,4\}$. The classification is carried by two further tools: a direct-sum construction that combines transverse maps in smaller flag manifolds into transverse maps in larger ones, and transversality-preserving embeddings between flag manifolds of different Cartan types, such as $B_n \to A_{2n}$, $D_{2n} \to A_{4n-1}$, and $D_{2n} \to B_{2n}$. The equivalence theorem 0.19 reduces the circle question to the absence of transverse 2-spheres, which is what the constructions verify case by case.

What would settle it

The classification would collapse if one found a transverse 2-sphere in any flag manifold listed as positive in Table 1 (for instance $\mathrm{Flag}(\mathbb{R}^d)$ with $d \equiv 3,4,5 \pmod{8}$ or an $E_7$ flag with the marked root $7$ in $\Theta$), since Theorem 0.19 equates the positive answer with the absence of transverse 2-spheres. A direct check could also target a negative cell: exhibit a single transverse circle in a listed-negative flag manifold that is locally maximally transverse.

Watch

Extended reading notes

Core claim

The central discovery is a complete classification (Theorem 0.2): for a split real simple Lie group $G$ and a self-opposite flag manifold $F^\Theta$, every transverse circle in $F^\Theta$ is locally maximally transverse if and only if $(G,\Theta)$ appears in Table 1. The only new restriction is for type $E_7$, where the simple root labelled $7$ must belong to $\Theta$; the remaining entries verify that previously known positive answers are optimal, and the negative entries are realized by explicit transverse $m$-spheres with $m \geq 2$. Alongside the classification, the paper constructs transverse $(n-1)$-spheres in the full flag manifold $\mathrm{Iso}_{Jd-1K}(\mathbb{R}^{d-1,d})$ for every $n \geq 2$, with an explicit $d(n) \approx 2^{n/2}$, proves these spheres are maximally transverse precisely when $n \in \{0,1,2,4\} \pmod{8}$ (Theorem 0.9), and shows that the full flag manifolds of split $G_2$, $B_3$, and $D_4$ fibre over lower-dimensional flag manifolds with every fibre a maximally transverse 3-sphere. As a direct corollary, $\{7\}$-Anosov subgroups of split $E_7$—hence all Borel Anosov subgroups—are virtually free or surface groups.

Load-bearing premise

The completeness of the classification rests on the claim that every positive case in Table 1 is genuinely covered by prior theorems on local maximal transversality that the paper invokes; if any positive cell is uncovered, or if any inherited result is weaker than stated, the classification overreaches.

Editorial extensions

If this is right

  • If Theorem 0.2 is correct, the positive-answer pairs $(G,\Theta)$ are exactly those in Table 1; in particular, the only split exceptional group with any locally maximally transverse circles is $E_7$, and only when the marked root $7$ lies in $\Theta$.
  • Every $\{7\}$-Anosov subgroup of split $E_7$, hence every Borel Anosov subgroup, is virtually a free group or a surface group (Corollary 0.7).
  • Full flag manifolds of split groups of type $A$, $B$, and $D$ contain transverse spheres of arbitrarily large dimension; for example, $\mathrm{Flag}(\mathbb{R}^{d})$ with $d = 2 \cdot 16^n$ contains a maximally transverse $8n$-sphere.
  • For split real groups, local maximal transversality of transverse circles, Property (I), and the non-existence of transverse 2-spheres are three equivalent conditions (Theorem 0.19).
  • The full flag manifolds of split $G_2$, $B_3$, and $D_4$ are principal $\mathrm{Sp}(1)$-bundles whose fibres are all maximally transverse 3-spheres.

Reading between the lines

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

  • The mod-8 pattern in Theorem 0.9 looks like a geometric face of Bott periodicity, since the same residue classes control real $K$-theory of spheres; the paper uses the Atiyah-Bott-Shapiro isomorphism as a tool, but one could take this connection as evidence that any transverse-sphere construction in these flag manifolds will inherit an 8-periodic classification.
  • The deformed spheres of Theorem 0.13 are not obstructed from being Anosov limit sets. If one of them is realised, it would give an irreducible Borel Anosov subgroup with boundary $S^3$ or $S^7$ in $\mathrm{SO}(8k,8k)$ or $\mathrm{SL}(16k,\mathbb{R})$, which current theory does not exclude; Question 0.20 is the concrete test case.
  • The classification in Table 1 likely extends, with modifications, to non-split real forms via the maximal split subalgebra inclusion the paper mentions; a testable extension is to check whether any transverse 2-sphere in the split form survives the inclusion into the non-split flag manifold.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies Question 0.1 for self-opposite flag manifolds of split real simple Lie groups: when is every transverse circle locally maximally transverse. The main claim is Theorem 0.2 with Table 1, a complete classification, with the only new restriction for E7. The paper also constructs transverse spheres via spinor representations: Theorem 0.8 gives transverse (n-1)-spheres in type B/D full flag manifolds for explicit d(n) approximately 2^{n/2}, and Theorem 0.9 proves, via the Atiyah-Bott-Shapiro isomorphism, that these are maximally transverse for n congruent to 0,1,2,4 mod 8. Applications to Anosov subgroups include Corollary 0.7: {7}-Anosov subgroups of split E7 are virtually free or surface groups. The paper also constructs fibrations by maximally transverse 3-spheres in the G2, B3, and D4 cases.

Significance. If the classification and the maximal-transversality results are correct, this resolves a natural question for all split real groups and gives a striking connection between the mod-8 phenomenon for transverse spheres and Bott periodicity. The new negative constructions are explicit and self-contained, and the use of the Atiyah-Bott-Shapiro isomorphism to certify maximal transversality is concrete and, for the main cases, machine-checkable in principle. The deformation construction of Section 2.3 and the obstruction results of Section 9 are useful independent contributions. The main risk is not the internal constructions but the provenance of the positive direction of Table 1, which is delegated to several external papers.

major comments (3)
  1. [Theorem 0.19 / Table 1] The completeness of the positive direction of Table 1 is delegated to prior work without a case-by-case justification. In the proof of Theorem 0.19, the sentence 'we have applied results of [Tso20, Dey25, DGR24, KT24] to verify that Property (I) holds in the cases complementing our construction(s)' does not specify which external theorem verifies each positive row, nor how the notion of maximal transversality in those sources (global in some cases, local in others) matches the 'locally maximally transverse' wording of Table 1. Because Theorem 0.2 is a complete classification, this is load-bearing. Please add a table or paragraph that maps each positive row of Table 1 to the exact theorem in [Tso20, Dey25, DGR24, PT24, KT24, TZ24] and states the compatibility argument.
  2. [Theorem 8.2] The proof of the E7 restriction is too compressed for a new classification result. It cites a 56-dimensional minuscule representation of E7 and says proximality is straightforward from the weight multiplicities (3/2, 1/2, -1/2, -3/2 of multiplicities 1,27,27,1), then invokes [PT24] for maximality of the image circle in RP55. Please identify the equivariant map F^{7}(E7) to RP55 explicitly, show the proximality calculation for the coweight, and state the pullback argument that maximality of a transverse circle in the target implies maximality of its preimage in F^{7}. The pullback fact is true for transversality-preserving maps because a point of F^{7} transverse to all points of the circle would map to a point of RP55 transverse to all points of the image, but it should be written out.
  3. [Theorems 0.3(b,d), 0.4(b,c), 0.5(c), 8.2] Several positive assertions are transferred from a smaller flag manifold to a larger one by projections or equivariant maps without an explicit statement of the transfer principle. For example, Theorem 0.3(b) passes from Gr_{d/2}(R^d) to any Flag^Theta(R^d) with d/2 in Theta, and Theorem 0.6(c) passes from F^{7}(E7) to F^Theta(E7) with 7 in Theta. The needed fact is: if f: X to Y is a transversality-preserving map between self-opposite flag manifolds and S subset X is transverse with f(S) locally maximally transverse in Y, then S is locally maximally transverse in X. The proof is one line (a point of X transverse to all of S would have a transverse image), but the paper never states it. Please add this lemma and use it consistently, since the local-versus-global distinction is central to Question 0.1.
minor comments (3)
  1. [Theorem 5.1 proof] In the first paragraph of the proof of Theorem 5.1, the phrase 'contained in a transverse n-sphere' is dimensionally inaccurate for n congruent to 5,6,7 mod 8: the containing sphere supplied by Corollary 2.13 has dimension rho(d(n))-1, which equals n only in the n congruent to 3 mod 8 case. Please write 'a larger transverse sphere' and give its dimension explicitly.
  2. [Corollary 2.13 / Theorem 0.8] The symbol n is overloaded: in Theorem 0.8 it denotes the sphere dimension plus one, while in Corollary 2.13 it denotes the dimension of the spinor module and of the pseudo-Euclidean space R^{n,n+epsilon}. Using different letters would avoid confusion.
  3. [Section 0.4.1] The delegation sentence in the proof of Theorem 0.19 omits [PT24], which is used elsewhere in the paper for type C and for the E7 row. The revision should state explicitly which parts of the classification use [PT24] rather than [DGR24] alone.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the new constructions are explicit and independently verified, while the classification relies on external published results; minor self-citations are not used to smuggle the central conclusions.

full rationale

The paper's new constructions are self-contained derivations: the spinor spheres are defined by explicit formulas (0.3) and (0.4), their dimensions are fixed by the Clifford algebra representation theory of [LM89], and transversality is verified by direct matrix computations (Lemma 2.12 and Lemma 2.14) rather than by assuming the desired conclusion. Maximal transversality is checked against the external Atiyah-Bott-Shapiro isomorphism [ABS64], a classical and independent body of mathematics that is not tuned to produce the paper's theorem statements. The classification in Table 1 combines these new negative constructions (transverse m-spheres with m≥2) with previously published positive results from [Tso20, Dey25, DGR24, PT24, KT24, TZ24]; those cited results are independent and concern specific flag manifolds, not restatements of the present theorems. Self-citations do occur: [DGR24, Lemma 2.7] and [DGR24, Proposition 2.5] are used in the proof of Theorem 0.19, and [Eva24a, Eva24b] are cited for standard facts about G′2. However, these are prior background results with independent content, and they are not invoked as a uniqueness theorem or as an ansatz to force the classification; thus they do not make the derivation circular. The genuine soft spot is a delegation/reproducibility gap, not circularity: the proof of Theorem 0.19 says 'we have applied results of [Tso20, Dey25, DGR24, KT24] to verify that Property (I) holds in the cases complementing our construction(s)' without mapping each row of Table 1 to a specific external theorem, and Example 7.9 shows that even the paper's own direct-sum method can fail to certify maximal transversality. That is a verification and completeness concern for the correctness pass, not a reduction of a prediction to its input. Accordingly, no circular step is identified and the score is low.

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

The paper introduces no fitted parameters and no invented entities. Its inputs are: standard Clifford algebra periodicity and spinor theory [LM89]; the Atiyah-Bott-Shapiro isomorphism [ABS64]; Hurwitz's classification of normed division and composition algebras; and a set of external results on transversality in specific flag manifolds ([Tso20], [Dey25], [DGR24], [PT24], [KT24], [TZ24], [CT20], [KP22], [GGKW17]). Fact 1.5 (non-nullhomotopic transverse implies maximally transverse) is proven in the text and is the bridge between topology and transversality. The count of external black boxes is the main honest cost of the paper: the classification is only as complete as the union of those results.

assumptions (6)
  • standard math Atiyah-Bott-Shapiro isomorphism gKO(S^k) ≅ M_{k-1}/ι*M_k ([ABS64, Theorem 11.5])
    Used in Section 5 (proof of Theorem 5.1) to reduce maximal transversality to representation-theoretic non-triviality of irreducible Clifford modules; a classical external theorem taken as input.
  • standard math Standard real Clifford algebra structure theory: periodicity Cl(n+8) ≅ Cl(n) ⊗ Cl(8), spin metrics, spinor irreducibility mod 8 [LM89]
    Invoked in Section 2 for the dimensions d(n), Lemma 2.10, and the entire spinor construction; the paper supplies a new proof of the Z_2-grading lemma it says was only partially proven in [LM89].
  • standard math Hurwitz theorem classification of composition algebras R, C, H, O, C', H', O' and existence of cross products only in signatures (3,0), (7,0), (1,2), (3,4)
    Used in Section 4 for the division algebra construction and in Section 3 for G'_2 = Aut(R^{3,4}, ×).
  • domain assumption External prior results: [Tso20], [Dey25], [DGR24], [PT24], [KT24], [TZ24], [CT20], [KP22], [GGKW17], [DGK23], [DGK18], [BK25]
    The classification and Anosov corollaries depend on these for: positive cases of local maximal transversality (proof of Theorem 0.19), Anosov iff semisimplification Anosov, eigenvalue gap criteria, Benoist representation arguments, and hyperconvexity obstructions. These are published black boxes and include the authors' own [DGR24].
  • domain assumption Dey's observation (Fact 1.5): a transverse subset that is non-nullhomotopic in the flag manifold is maximally transverse
    Load-bearing in Theorem 5.1 and Corollary 5.5; the proof given in Section 1.1 relies on openness of Schubert cells and is one paragraph long.
  • standard math Fact 1.4: projections of flag manifolds to smaller parabolic subsets are transversality-preserving
    Used throughout to propagate transverse spheres from full to partial flag manifolds; elementary and proved in the text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Transverse Spheres in Flag Manifolds." pith.science (2026). https://pith.science/paper/TPPE3YUJ

@misc{pith2026250719306,
  author       = {Pith},
  title        = {Pith review of: Transverse Spheres in Flag Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPPE3YUJ}},
  note         = {Machine review of arXiv:2507.19306}
}
abstract

For some partial flag manifolds of semisimple real Lie groups, including many full flag manifolds, transverse circles are known to be locally maximally transverse. We complete the classification of all partial flag manifolds of split real Lie groups with this property. As a consequence, $\{7\}$-Anosov subgroups of split $E_7$ are virtually free or surface groups. On the other hand, using spinors, we find transverse spheres of arbitrarily large dimension in certain full flag manifolds of Cartan-Killing types $A,B,D$. These transverse spheres are verified to be maximally transverse with tools from topological $K$-theory. The aforementioned classification follows from constructions of transverse $m$-spheres, $m \geq 2$, that complement the previously known restrictions as well as the new $E_7$ restriction. Additionally, when $G$ is split of type $G_2, B_3,$ or $D_4$, the full flag manifold admits a fibration by maximally transverse 3-spheres.

Figures

Figures reproduced from arXiv: 2507.19306 by the authors.

Figure 1
Figure 1. Principal bundle fibrations of G ′ 2 -flag manifolds relative to choice of P ∈ Gr× (3,0)(R 3,4 ). Theorem 0.15. Let F ∈ IsoJ3K(R 3,4 ). Then the orbit Sp(1) · F is a maximally transverse 3-sphere. Assembling these orbits, we obtain a principal bundle IsoJ3K(R 3,4 ) → IsoJ3K(R 3,4 )/ Sp(1) with fibers that are each maximally transverse 3-spheres. See Section 4.3 for more details. 0.3.3 The case of SO0(4, 4) The full … view at source ↗
Figure 2
Figure 2. Dynkin diagram of type Ad−1. We now set a notational convention going forwards. Define JkK := {0, 1, 2, . . . , k}. The simple roots ∆ of Σ(g, a) naturally identify with {1, 2, . . . , d − 1}. On the other hand, it will be convenient for us to imagine Θ ⊂ JdK instead, where Θ = {0, i1, . . . , ik, d} with ij < ij+1. With our indexing, FΘ ∼= FlagΘ(R d ) is the collection of nested Θ-index subspaces: FlagΘ(R d ) := { … view at source ↗
Figure 3
Figure 3. Dynkin diagram of type Bp. In the Bp case, the opposition involution is trivial and every flag manifold IsoΘ(R p,p+1) is self￾opposite. Two flags V • , W• ∈ IsoΘ(R p,p+1) are transverse if and only if V k + (Wk ) ⊥ = R p,p+1 for all k ∈ Θ. Remark 1.6. The first two isotropic Grassmannians enjoy standard alternate names and no￾tations, in the case of any mixed signature. These are the “Einstein universe” Einp−1,q−1 :… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Dynkin diagram of type Dp. With these remarks in place, we introduce some notation for SO0(p, p)-full flags. We maintain the notation JkK := {0, 1, 2, . . . , k} from before. 1Of course, this discussion requires a background choice of space and time-orientation on R p,…
Figure 5
Figure 5. Figure 5: Fiber bundle structure of IsoJ3K(R 3,4 ) relative to choice of P ∈ XSO0(3,4), including two principal bundle fibrations. A similar diagram ap￾plies for Flag(R 4,4 ) after changing the base to Flag(R 4 ). 38 [PITH_FULL_IMAGE:figures/full_fig_p038_5.png]
Figure 6
Figure 6. Figure 6: The relevant lattice path in Example 7.8 for the transverse map ζ4 : S 2 → FlagJ11K\{4,7} . The lattice path is symmetric about the diagonal. Proof of Theorem 7.7. We note here the considerations for 2 ≤ d ≤ 5. When d = 2, the only flag manifold is the full flag manifo…
Figure 7
Figure 7. Figure 7: Dynkin diagram of type E7. Let ∆ denote the set of simple roots of E7 compatible with the labeling here. Theorem 8.2. Let ∆ denote the simple roots of E7, labeled as in [PITH_FULL_IMAGE:figures/full_fig_p056_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 47 canonical work pages

  1. [1]

    Clifford modules

    Michael Atiyah, Raul Bott, and Arnold Shapiro. Clifford modules. Topology , 3:3--38, 1964

    work page 1964

  2. [2]

    J. F. Adams. On the Non-Existence of Elements of Hopf Invariant One . Ann. of Math. , 72(1):20--104, 1960

    work page 1960

  3. [3]

    The Octonions

    John Baez. The Octonions . Bull. Amer. Math. Soc. , 39:145--205, 2002

    work page 2002

  4. [4]

    Baez and John Huerta

    John C. Baez and John Huerta. G_2 and the rolling ball. Trans. Amer. Math. Soc. , 366(10):5257--5293, 2014

    work page 2014

  5. [5]

    H ^ p,q -convex cocompactness and higher higher Teichm\"uller spaces

    Jonas Beyrer and Fanny Kassel. H ^ p,q -convex cocompactness and higher higher Teichm\"uller spaces. Geom. Func. Anal. (to appear) , 2025. https://arxiv.org/abs/2305.15031 arXiv:2305.15031

    arXiv 2025

  6. [6]

    Bourbaki

    N. Bourbaki. Lie Groups and Lie Algebras-- Chapters 7-9 . Springer Berlin, Heidelberg , 2005

    work page 2005

  7. [7]

    Topology and Geometry

    Glen Bredon. Topology and Geometry . Springer New York, NY, 1993

    work page 1993

  8. [8]

    Robert L. Bryant. Notes on spinors in low dimension, 2020. https://arxiv.org/abs/2011.05568 arXiv:2011.05568

    arXiv 2020

Show all 59 references
  1. [9]

    Maximal surfaces and the universal T eichm \"u ller space

    Francesco Bonsante and Jean-Marc Schlenker. Maximal surfaces and the universal T eichm \"u ller space. Invent. Math. , 182(2):279--333, 2010

    2010

  2. [10]

    Les groupes r\'eels simples, finis et continus

    Elie Cartan. Les groupes r\'eels simples, finis et continus. Ann. Sci. \'Ec Norm. Sup. (3) , 31:263--355, 1914

    1914

  3. [11]

    Topological restrictions on Anosov representations

    Richard Canary and Konstantinos Tsouvalas. Topological restrictions on Anosov representations. J. Topol. , 13(4):1497--1520, 2020

    2020

  4. [12]

    Holomorphic curves in the 6-pseudosphere and cyclic surfaces

    Brian Collier and J \'e r \'e my Toulisse. Holomorphic curves in the 6-pseudosphere and cyclic surfaces. Trans. Amer. Math. Soc. , pages 6465--6514, 2024

    2024

  5. [13]

    Geometric structures for SO_0(p,p+1) and G_2' -representations (to appear), 2025

    Colin Davalo and Parker Evans. Geometric structures for SO_0(p,p+1) and G_2' -representations (to appear), 2025

    2025

  6. [14]

    On Borel Anosov subgroups of SL(d, R )

    Subhadip Dey. On Borel Anosov subgroups of SL(d, R ) . Geom. Topol. , 29(1):171--192, 2025

    2025

  7. [15]

    Convex cocompactness in pseudo- R iemannian hyperbolic spaces

    Jeffrey Danciger, Fran c ois Gu\' e ritaud, and Fanny Kassel. Convex cocompactness in pseudo- R iemannian hyperbolic spaces. Geom. Dedicata , 192:87--126, 2018

    2018

  8. [16]

    Convex cocompact actions in real projective geometry

    Jeffrey Danciger, François Guéritaud, and Fanny Kassel. Convex cocompact actions in real projective geometry. Ann. Sci. \'Ec Sup. (to appear) , 2023. https://arxiv.org/abs/1704.08711 arXiv:1704.08711

    2023 arXiv

  9. [17]

    Maxwell Riestenberg

    Subhadip Dey, Zachary Greenberg, and J. Maxwell Riestenberg. Restrictions on A nosov subgroups of Sp (2n, R) . Trans. Amer. Math. Soc. , 377(10):6863--6882, 2024

    2024

  10. [18]

    Klein–Maskit combination theorem for Anosov subgroups: free products

    Subhadip Dey and Michael Kapovich. Klein–Maskit combination theorem for Anosov subgroups: free products. Math. Z. , 305:35, 2023

    2023

  11. [19]

    A perspective on totally geodesic submanifolds of the symmetric space G_2/ (4) , 2025

    Cristina Draper and Cándido Martín-González. A perspective on totally geodesic submanifolds of the symmetric space G_2/ (4) , 2025. URL: https://arxiv.org/abs/2504.07586, https://arxiv.org/abs/2504.07586 arXiv:2504.07586

    2025 arXiv

  12. [20]

    Notes on G _2 : The Lie algebra and the Lie group

    Cristina Draper Fontanals . Notes on G _2 : The Lie algebra and the Lie group. Differential Geom. Appl. , 57:23--74, 2018. (Non)-existence of complex structures on S ^6

    2018

  13. [21]

    Affine actions with Hitchin linear part

    Jeffrey Danciger and Tengren Zhang. Affine actions with Hitchin linear part . Geom. Func. Anal. , 29(5):1369--1439, 2019

    2019

  14. [22]

    Eberlein

    Patrick B. Eberlein. Geometry of nonpositively curved manifolds . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996

    1996

  15. [23]

    Eschenburg

    J.H. Eschenburg. Geometry of Octonions , 2018. https://arxiv.org/abs/https://myweb.rz.uni-augsburg.de/ eschenbu/geomoct.pdf arXiv:https://myweb.rz.uni-augsburg.de/ eschenbu/geomoct.pdf

    2018

  16. [24]

    Geometry of G _2' -Harmonic Maps and Representations

    Parker Evans. Geometry of G _2' -Harmonic Maps and Representations . PhD thesis, Rice University, 2024

    2024

  17. [25]

    Polynomial almost-complex curves in S ^ 2,4 , 2024

    Parker Evans. Polynomial almost-complex curves in S ^ 2,4 , 2024. https://arxiv.org/abs/2208.14409 arXiv:2208.14409

    2024 arXiv

  18. [26]

    Fock and A

    V. Fock and A. Goncharov. Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. IHES , 103:1–211, 2006

    2006

  19. [27]

    Frobenius

    G. Frobenius. Ueber lineare substitutionen und bilineare formen. J. Reine Angew. Math. , 84:1--63, 1877

  20. [28]

    Anosov representations and proper actions

    Fran c ois Gu\' e ritaud, Olivier Guichard, Fanny Kassel, and Anna Wienhard. Anosov representations and proper actions. Geom. Topol. , 21(1):485--584, 2017

    2017

  21. [29]

    The number of connected components in double B ruhat cells for nonsimply-laced groups

    Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. The number of connected components in double B ruhat cells for nonsimply-laced groups. Proc. Amer. Math. Soc. , 131(3):731--739, 2003

    2003

  22. [30]

    Anosov representations: domains of discontinuity and applications

    Olivier Guichard and Anna Wienhard. Anosov representations: domains of discontinuity and applications. Invent. Math. , 190(2):357--438, 2012

    2012

  23. [31]

    Vector Bundles and K -theory, 2017

    Alan Hatcher. Vector Bundles and K -theory, 2017. URL: https://pi.math.cornell.edu/ hatcher/VBKT/VB.pdf

    2017

  24. [32]

    askyl\"a Department of Mathematics and Statistics . University of Jyv\

    Juha Heinonen. Lectures on L ipschitz analysis , volume 100 of Report. University of Jyv\"askyl\"a Department of Mathematics and Statistics . University of Jyv\"askyl\"a, Jyv\"askyl\"a, 2005

    2005

  25. [33]

    Blaine Lawson

    Reese Harvey and H. Blaine Lawson. Calibrated geometries . Acta Math. , 148:47 -- 157, 1982

    1982

  26. [34]

    Ueber die Composition der quadratischen Formen von belibig vielen Variablen

    Adolf Hurwitz. Ueber die Composition der quadratischen Formen von belibig vielen Variablen . Nachr. Ges. Wiss. G\"ottingen , pages 309--316, 1898

  27. [35]

    F. I. Karpelevi c . Surfaces of transitivity of a semisimple subgroup of the group of motions of a symmetric space. Doklady Akad. Nauk SSSR (N.S.) , 93:401--404, 1953

    1953

  28. [36]

    Introduction to G_2 geometry , pages 3--50

    Spiro Karigiannis. Introduction to G_2 geometry , pages 3--50. Springer US, 2020

    2020

  29. [37]

    Anosov subgroups: dynamical and geometric characterizations

    Michael Kapovich, Bernhard Leeb, and Joan Porti. Anosov subgroups: dynamical and geometric characterizations. Eur. J. Math. , 3(4):808--898, 2017

    2017

  30. [38]

    A M orse lemma for quasigeodesics in symmetric spaces and euclidean buildings

    Michael Kapovich, Bernhard Leeb, and Joan Porti. A M orse lemma for quasigeodesics in symmetric spaces and euclidean buildings. Geom. Topol. , 22(7):3827--3923, 2018

    2018

  31. [39]

    Eigenvalue gaps for hyperbolic groups and semigroups

    Fanny Kassel and Rafael Potrie. Eigenvalue gaps for hyperbolic groups and semigroups. J. Mod. Dyn. , 18:161--208, 2022

    2022

  32. [40]

    Orbits and representations associated with symmetric spaces

    Bertram Kostant and Stephen Rallis. Orbits and representations associated with symmetric spaces. Amer. J. Math. , 93(3):753--809, 1971

    1971

  33. [41]

    Connected components of the space of triples of transverse partial flags in SO_0(p,q) and A nosov representations, 2024

    Clarence Kineider and Roméo Troubat. Connected components of the space of triples of transverse partial flags in SO_0(p,q) and A nosov representations, 2024. https://arxiv.org/abs/2411.08679 arXiv:2411.08679

    2024

  34. [42]

    Anosov flows, surface groups and curves in projective space

    Fran c ois Labourie. Anosov flows, surface groups and curves in projective space. Invent. Math. , 165(1):51--114, 2006

    2006

  35. [43]

    Orthogonal complex structures on S ^6

    Claude Lebrun. Orthogonal complex structures on S ^6 . Proc. Amer. Math. Soc. , 101, 1987

    1987

  36. [44]

    Blaine Lawson and Marie-Louise Michelsohn

    H. Blaine Lawson and Marie-Louise Michelsohn. Spin Geometry . Princeton University Press, 1989

    1989

  37. [45]

    Lorentz spacetimes of constant curvature

    G Mess. Lorentz spacetimes of constant curvature. Geom. Dedicata , pages 3--45, 2007

    2007

  38. [46]

    The Geometry of Marked Contact Engel Structures

    Gianni Manno, Pawe Nurowski, and Katja Sagerschnig. The Geometry of Marked Contact Engel Structures . J. Geom. Anal. , 31(8):7686--7708, 2021

    2021

  39. [47]

    G. D. Mostow. Some new decomposition theorems for semi-simple groups. Mem. Amer. Math. Soc. , 14:31--54, 1955

    1955

  40. [48]

    R. J. Marsh and K. Rietsch. The intersection of opposed big cells in the real flag variety of type G _2 . Proc. Lond. Math. Soc. , 85(1):22–42, 2002

    2002

  41. [49]

    Cyclic H iggs bundles and minimal surfaces in pseudo-hyperbolic spaces

    Xin Nie. Cyclic H iggs bundles and minimal surfaces in pseudo-hyperbolic spaces. Adv. Math. , 436:109402, 2024

    2024

  42. [50]

    Conformality for a robust class of non-conformal attractors

    Maria Beatrice Pozzetti, Andr\'es Sambarino, and Anna Wienhard. Conformality for a robust class of non-conformal attractors. J. Reine Angew. Math. , 774:1--51, 2021

    2021

  43. [51]

    On projective A nosov subgroups of symplectic groups

    Maria Beatrice Pozzetti and Konstantinos Tsouvalas. On projective A nosov subgroups of symplectic groups. Bull. Lond. Math. Soc. , 56(2):581--588, 2024

    2024

  44. [52]

    Explicit Families of Spinor Representations , 2025

    Jesus Sanchez. Explicit Families of Spinor Representations , 2025. https://arxiv.org/abs/2505.16203 arXiv:2505.16203

    2025 arXiv

  45. [53]

    R.D. Schafer. Introduction to Non-Associative Algebras . Dover, 1995

    1995

  46. [54]

    J. Tits. Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. J. Reine Angew. Math. , 247:196--220, 1971

    1971

  47. [55]

    On Borel Anosov representations in even dimensions

    Konstantinos Tsouvalas. On Borel Anosov representations in even dimensions. Comment. Math. Helv. , 95(4):749--763, 2020

    2020

  48. [56]

    Topological restrictions on relatively Anosov representations

    Konstantinos Tsouvalas and Feng Zhu. Topological restrictions on relatively Anosov representations. Trans. Amer. Math. Soc. (to appear) , 2024. https://arxiv.org/abs/2401.03050 arXiv:2401.03050

    2024 arXiv

  49. [57]

    Non-compact simple Lie group G_2' of type G_2

    Ichiro Yokota. Non-compact simple Lie group G_2' of type G_2 . Jour. Fac. Sci., Shinsu University , 12(1), 1977

    1977

  50. [58]

    Connected Components of Double Bruhat Cells

    Andrei Zelevinsky. Connected Components of Double Bruhat Cells . Int. Math. Res. Not. , 2000(21):1131--1154, 2000

    2000

  51. [59]

    Theorie der alternativen ringe

    Max Zorn. Theorie der alternativen ringe. Abh. Math. Semin. Univ. Hambg. , pages 123--147, 1931

    1931

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.