Free Circle Action Research Articles

A note on the circle actions on Einstein manifolds

A fundamental result in the theory of black holes due to Hawking asserts that the event horizon of a black hole in the stationary space-time is a 2-sphere topologically. In this article we prove the Riemannian analogue of Hawking's result. In other words, we prove that each bolt of a 4-dimensional complete noncompact Einstein manifold of zero scalar curvature admitting a semifree isometric circle action is a 2-sphere topologically. We also study the structure of the orbit space of an Einstein manifold admitting a free isometric circle action.

The Computation of η-Invariants on Manifolds with Free Circle Action

We present an explicit procedure to compute η-invariants. This also yields a topological formula for adiabatic limits and simplifies the calculation of Kreck–Stolz-invariants detecting components of the space of positive scalar curvature metrics.

FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

Compact symplectic manifolds with free circle actions, and Massey products.

Compact symplectic manifolds with free circle actions, and Massey products.

Extending free circle actions on spheres to 𝑆³ actions

Let X X be a PL homotopy C P 2 k + 1 C{P^{2k + 1}} corresponding by Sullivan’s classification to the element ( N 1 , α 2 , N 2 , ⋯ , α k , N k ) ({N_1},{\alpha _2},{N_2}, \cdots ,{\alpha _k},{N_k}) of Z ⊕ Z 2 ⊕ Z ⊕ ⋯ ⊕ Z 2 ⊕ Z Z \oplus {Z_2} \oplus Z \oplus \cdots \oplus {Z_2} \oplus Z . Theorem 1. The topological circle action on S 4 k + 3 {S^{4k + 3}} with orbit space X X is the restriction of an S 3 {S^3} action with a triangulable orbit space iff α i = 0 , i = 2 , ⋯ , k {\alpha _i} = 0,i = 2, \cdots ,k ; and N 1 ≡ 0 mod 2 {N_1} \equiv 0\bmod 2 ; and ∑ ( − 1 ) i N i = 0 \sum {( - 1)^i}{N_i} = 0 . If X X admits a smooth structure and satisfies the hypotheses of Theorem 1, a certain smoothing obstruction arising from the integrality theorems vanishes for the corresponding S 3 {S^3} action.

Extending Free Circle Actions on Spheres to S 3 Actions

Extending Free Circle Actions on Spheres to S 3 Actions

Strong deformation retraction of the space of Zoll Finsler projective planes

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the Zoll Finsler metrics along the retraction.