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Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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TL;DR: In this paper, two vector fields of horizontal Liouville type are introduced and it is shown that these vector fields are Killing if and only if the base Finsler manifold is of positive constant curvature.
3 citations
01 Jan 2008
TL;DR: In this paper, it was shown that each g-natural metric on a linear frame bundle LM over a Riemannian manifold is invariant with respect to a lifted map of a (local) isometry of the base manifold.
Abstract: In this paper we prove that each g-natural metric on a linear frame bundle LM over a Riemannian manifold (M,g) is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define g-natural metrics on the orthonormal frame bundle OM and we prove the same invariance result as above for OM . Hence we see that, over a space (M,g) of constant sectional curvature, the bundle OM with an arbitrary g-natural metric ˜ G is locally homogeneous.
3 citations
TL;DR: In this paper, the authors generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds and show that if the square of length of the second fundamental form in M n is not more than, then either M n n is totally geodesic, or M n m is the Veronese surface in S 4 (1) or M m n is the Clifford torus.
Abstract: Let M n be an n -dimensional Riemannian manifold minimally immersed in the unit sphere S n+p (1) of dimension n + p . When M n is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖ h ‖ 2 of length of the second fundamental form h in M n is not more than , then either M n is totally geodesic, or M n is the Veronese surface in S 4 (1) or M n is the Clifford torus . In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.
3 citations
TL;DR: In this article, it was shown that the divergence of the right-hand side of the system over a period of the attack angle is zero, and that the system is, in a sense, "semiconservative".
Abstract: 2. The system of equations has variable dissipation with zero mean. The system of equations (1)–(3) is a dynamical system having variable dissipation with zero mean (in this case the mean is taken with respect to the angle α) [2]. This expresses the fact that the integral of the divergence of the right-hand side of the system over a period of the attack angle is zero (after an appropriate reduction of the system, this integral reflects the variation of the phase volume). So the system is, in a sense, ‘semiconservative’. If z1 = Ωy cos β1 + Ωz sin β1, z2 = −Ωy sin β1 + Ωz cos β1, σh1/I2 = H1, β = σAB/I2, and σzk = νwk, k = 1, 2 (here α ′ = να•/σ, and so on), then we obtain a supporting system of the form
3 citations
TL;DR: In this paper, a stochastic process is formulated in the tangent bundle of a Riemann manifold, where the vector fibre portion of the process is a jump process, and a suitable modification of some results for estimation in linear spaces can be used to solve the aforementioned estimation problem.
Abstract: A stochastic process is formulated in the tangent bundle of a Riemann manifold where the vector fibre portion of the process is a jump process. Since the tangent spaces change as the process in the base manifold evolves, it is necessary to define a jump process in the fibres of the tangent bundle with respect to the process in the base manifold. An estimation problem is formulated and solved for a process obtained from the jump process in the fibres of the tangent bundle where the observations include the process in the base manifold and the jump times. Since each fibre of the tangent bundle is a linear space, a suitable modification of some results for estimation in linear spaces can be used to solve the aforementioned estimation problem.
3 citations