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arxiv: 2606.11168 · v1 · pith:BLCBDHLJnew · submitted 2026-06-09 · 🧮 math.DG

Sasakian manifolds and spin-c Killing spinors

Alejandro Gil-Garc\'ia , C. S. Shahbazi This is my paper

Pith reviewed 2026-06-27 11:39 UTC · model grok-4.3

classification 🧮 math.DG
keywords Sasakian manifoldsspin-c Killing spinorsodd-dimensional manifoldsRiemannian geometryKilling spinorscomplex spinorial forms
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The pith

An odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with real Killing constant α if and only if it is α-Sasakian.

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

The paper proves an if-and-only-if relation between two structures on odd-dimensional Riemannian manifolds. A manifold carries a pure spin-c Killing spinor with nonzero real constant α precisely when it is α-Sasakian. This characterization extends an earlier result by removing the requirements of simple connectivity and completeness. The proof relies on complex spinorial forms. Readers interested in spinorial methods in geometry will see a cleaner link between spinors and Sasakian structures.

Core claim

Using the theory of complex spinorial forms, an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with a real Killing constant α in R* if and only if it is α-Sasakian. This extends Moroianu's result under the purity assumption without needing simple connectivity or completeness.

What carries the argument

Complex spinorial forms, which establish the equivalence between the existence of a pure spin-c Killing spinor and the α-Sasakian condition.

If this is right

  • The equivalence holds on any odd-dimensional Riemannian manifold.
  • The result applies without assuming the manifold is complete.
  • The result applies without assuming the manifold is simply connected.
  • α-Sasakian manifolds are characterized by the existence of such spinors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This could enable the study of Sasakian geometry on incomplete manifolds using spinorial techniques.
  • Similar characterizations might exist for other contact or CR structures in odd dimensions.
  • Applications to physics models involving Sasakian manifolds could benefit from this spinorial view.

Load-bearing premise

The equivalence requires that the spin-c Killing spinor is pure.

What would settle it

A counterexample would be an odd-dimensional Riemannian manifold that is not α-Sasakian yet has a pure spin-c Killing spinor with real constant α, or the converse.

read the original abstract

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with a real Killing constant $\alpha\in\mathbb{R}^{\ast}$ if and only if it is $\alpha$-Sasakian, thereby obtaining an extension of a well-known result by A. Moroianu that, under the purity assumption, does not require simple connectivity or completeness.

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 / 3 minor

Summary. The paper proves that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with real nonzero Killing constant α if and only if it is α-Sasakian. The proof relies on the theory of complex spinorial forms and removes the simple-connectivity and completeness hypotheses from Moroianu's earlier result.

Significance. If the equivalence holds, the result gives a spinorial characterization of α-Sasakian structures that applies without topological or metric-completeness restrictions. This strengthens the link between spin-c geometry and Sasakian/contact geometry and may enable applications on a wider class of manifolds.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of the main theorem should explicitly record the purity assumption on the spinor, as this is essential to the equivalence and is only implicit in the abstract.
  2. The notation for the complex spinorial forms (e.g., the precise bundle and the action of the Clifford multiplication) should be introduced with a short self-contained paragraph before the main argument, to aid readers unfamiliar with the technique.
  3. [§2] Ensure that the definition of α-Sasakian structure (contact form, Reeb field, transverse Kähler condition) is recalled verbatim in §2 so that the equivalence can be checked without external references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The report provides no specific major comments to address point by point.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a direct if-and-only-if equivalence between the existence of a pure spin-c Killing spinor with real nonzero Killing constant and the manifold being α-Sasakian, via the theory of complex spinorial forms. This extends an external result by Moroianu without relying on self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations; the central claim is a self-contained mathematical proof independent of the paper's own fitted quantities or prior author-specific uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.1-grok · 5586 in / 938 out tokens · 17594 ms · 2026-06-27T11:39:41.924536+00:00 · methodology

discussion (0)

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Reference graph

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