On the existence of nonlinear Dirac-geodesics on compact manifolds

Isobe, Takeshi
January 2012
Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p83
Academic Journal
We show that for super-linear nonlinearity, there exists a 'non-trivial' nonlinear Dirac-geodesic on $${\mathbb{T}^n_{\Gamma}=\mathbb{R}^n/\Gamma}$$, a flat tori, in each 'bosonic' sector. We also show that for any compact Riemannian manifold with 'bumpy' metric, there exists a non-trivial nonlinear Dirac-geodesic in each bosonic sector if the nonlinearity is cubic or super-cubic and 'large'. Our proof is based on critical point theory, in particular, a generalized linking argument applied to a strongly indefinite functional on a fibered Hilbert manifold.


Related Articles

  • Geodesic uniqueness in the whole of some generally recurrent Riemannian spaces. Sinyukova, H. // Journal of Mathematical Sciences;Sep2011, Vol. 177 Issue 5, p710 

    In this paper, we present a detailed proof of two theorems of geodesic uniqueness in the whole of compact, in some sense generally recurrent, Riemannian spaces with a positively defined metric. Our studies are based on the H. Hopf theorem.

  • Multiple Brake Orbits and Homoclinics in Riemannian Manifolds. Giambò, Roberto; Giannoni, Fabio; Piccione, Paolo // Archive for Rational Mechanics & Analysis;May2011, Vol. 200 Issue 2, p691 

    Let ( M, g) be a complete Riemannian manifold, $${\Omega\subset M}$$ an open subset whose closure is homeomorphic to an annulus. We prove that if ∂Ω is smooth and it satisfies a strong concavity assumption, then there are at least two distinct geodesics in...

  • Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications. Matveev, Vladimir S.; Mounoud, Pierre // Annals of Global Analysis & Geometry;Oct2010, Vol. 38 Issue 3, p259 

    We extend the Gallot-Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold admits a parallel symmetric (0, 2)-tensor then it is Riemannian. Applications of this result to the existence of metrics with distinct Levi-Civita connections but...

  • The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. Kouranbaeva, Shinar // Journal of Mathematical Physics;Feb99, Vol. 40 Issue 2, p857 

    Shows that the Camassa-Holm (CH) equation for the case of k=0 is the geodesic spray of the weak Riemannian metric on the diffeomorphism group. Use of the right-trivialization technique to verify that the Euler-Poincare theory for Lie groups can be applied to diffeomorphism groups;...

  • Non-Contraction of Heat Flow on Minkowski Spaces. Ohta, Shin-ichi; Sturm, Karl-Theodor // Archive for Rational Mechanics & Analysis;Jun2012, Vol. 204 Issue 3, p917 

    We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in...

  • Riemann Extension of Minkowski Line Element in the Rindler Chart. Nagaraja, H. G.; Harish, D. // General Mathematics Notes;Sep2015, Vol. 30 Issue 1, p21 

    In this paper, we discuss Riemannian extension of Minkowski metric in Rindler coordinates and its geodesics.

  • Extension of a theorem of Shi and Tam. Eichmair, Michael; Miao, Pengzi; Wang, Xiaodong // Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p45 

    In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79-125, ): Let (O, g) be an n-dimensional ( n = 3) compact Riemannian manifold, spin when n > 7, with non-negative scalar curvature and mean convex boundary. If every boundary component S has...

  • The characterization of eigenfunctions for Laplacian operators. Xiang Gao // Balkan Journal of Geometry & Its Applications;2012, Vol. 17 Issue 2, p46 

    In this paper, we consider the characterization of eigenfunctions for Laplacian operators on some Riemannian manifolds. Firstly we prove that for the space form (MKn, g K) with the constant sectional curvature K, the first eigenvalue of Laplacian operator λ1 (MKn) is greater than the limit of...

  • On fundamental equations of geodesic mappings and their generalizations. Hinterleitner, I.; Mikĕ, J. // Journal of Mathematical Sciences;May2011, Vol. 174 Issue 5, p537 

    No abstract available.


Read the Article


Sign out of this library

Other Topics