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TANGENTIAL REPRESENTATIONS AT ISOLATED FIXED POINTS OF ODD-DIMENSIONAL G-MANIFOLDS
Katsuhiro Komiya 대한수학회 2008 대한수학회보 Vol.45 No.1
Let G be a compact abelian Lie group, and M an odd-dimensional closed smooth G-manifold. If the xed point set M G ≠φ and dim M G = 0, then G has a subgroup H with G/H ∼= Z₂, the cyclic groupof order 2. The tangential representation γx (M ) of G at x ∈M G is alsoregarded as a representation of H by restricted action. We show that the number of fixed points is even, and that the tangential representations at fixed points are pairwise isomorphic as representations of H.
TANGENTIAL REPRESENTATIONS AT ISOLATED FIXED POINTS OF ODD-DIMENSIONAL G-MANIFOLDS
Komiya, Katsuhiro Korean Mathematical Society 2008 대한수학회보 Vol.45 No.1
Let G be a compact abelian Lie group, and M an odd-dimensional closed smooth G-manifold. If the fixed point set $M^G\neq\emptyset$ and dim $M^G=0$, then G has a subgroup H with $G/H{\cong}\mathbb{Z}_2$, the cyclic group of order 2. The tangential representation $\tau_x$(M) of G at $x{\in}M^G$ is also regarded as a representation of H by restricted action. We show that the number of fixed points is even, and that the tangential representations at fixed points are pairwise isomorphic as representations of H.
Katsuhiro Tofuku,Hiroaki Koga,Setsuro Komiya 대한척추외과학회 2014 Asian Spine Journal Vol.8 No.3
Study Design: A retrospective study. Purpose: The purpose of this study was to evaluate outcomes in patients with pyogenic spondylitis accompanied by iliopsoas abscess who were treated by percutaneous drainage combined with hyperbaric oxygen (HBO) therapy. Overview of Literature: To the best of our knowledge, there have been no previous reports of the use of percutaneous drainage combined with HBO therapy for the treatment of this condition. Methods: Twenty-three patients (13 men, 10 women; mean age, 69.0 years; range, 45–85 years) were treated with percutaneous drainage combined with HBO therapy in addition to commonly used conservative therapy. Mean follow-up duration was 27.7 months (range, 12–48 months). Clinical outcomes and imaging examinations were retrospectively investigated. Results: Symptoms such as low back pain, radicular pain, and hip pain resolved in all patients immediately after treatment. Mean time from the start of treatment to the return of C-reactive protein levels to normal or baseline values recorded before the onset of spondylitis was 28.3 days (range, 8–56 days). In the final set of follow-up radiographic studies, all patients were free from progressive destructive changes. Follow-up magnetic resonance images or computed tomography with contrast enhancement confirmed the disappearance or near-total resolution of the iliopsoas abscess cavity with healing of the pyogenic spondylitis in all 23 patients. No recurrences were observed during follow-up. Conclusions: The present study suggests that patients with pyogenic spondylitis accompanied by iliopsoas abscess can be cured without a prolonged period of therapy or recurrence using this treatment.