Riemannian manifold

J. P. Vigier, Possible external and internal motions on elementary particles on Riemannian manifolds, Phys. Rev. Lett. 17(1) (1966). 

Within this context, ordinary differential equations are viewed as vector fields on manifolds or configuration spaces [2]. For example, Newton s equations are second-order differential equations describing smooth curves on Riemannian manifolds. Noether s theorem [4] states that a diffeomorphism,3 < ), of a Riemannian manifold, C, indices a diffeomorphism, D< >, of its tangent4 bundle,5 TC. If 4> is a symmetry of Newton s equations, then Dt(> preserves the Lagrangian o /Jc ) = jSf. As opposed to equations of motion in conventional... 

In summary, the principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. The electromagnetic field is the compensating field of local phase transformation and the gravitational field is the compensating field of local Lorentz transformations. 

A set of points M is said to be a -dimensional manifold if each point of M has an open neighborhood, which has a continuous 1 1 map onto an open set of of R , the set of all w-tuples of real numbers. Consider an w-dimensional Riemannian manifold with metric G. In an arbitrary coordinate system x, .. . , x", the volume -form is generally given by u> = dx a a dx . Here, g is the determinant of the metric in this basis, and a denotes the wedge or antisymmetric tensor product. For a flow field on the manifold prescribed by x = x) with density f x, t), a continuity equation for f x, t) can be obtained by considering the number of ensemble members >T t) within a volume Q of phase space given by... 

N. Ogawa, N. Chepilko, and A. Kobushkin, Quantum mechanics in Riemannian manifold II. Prog. Theor. Phys. 85, 1189-1201 (1991). 

All CS manifolds in based on the one-particle group U 2s) are families of AGP states, some of these manifolds are irreducible Riemannian manifolds and correspond to cosets formed by the maximal compact subgroups U 2M) XU 2s - 2M) and USp(25), while others are reducible and correspond to non-maximal compact subgroups USp(2basic physical properties, e.g., U 2M) X U 2s - 2M) invariant manifold describes uncorrelated IPSs, the USp(2s) invariant manifold describes highly correlated extreme AGP states that are superconducting, while the USp(2a>i) X X USp(2wp) X SU(2) X SU(2 ) invariant manifold for general (Oj,..., cr, describe intermediate types of correlation and linear response properties, see, for a particular example. Ref. [35], most of which have not been explored in any depth. 

A highly important role in geometry in played by Riemannian manifolds, that is, smooth n-dimensional manifolds M supplied with a Riemannian metric It is known that setting this metric is equivalent to setting in each tangent space TxM at a point x G M a, bilinear symmetric nondegenerate positive definite scalar product which smoothly depends on the point x. V a, b G TxM is an arbitrary pair of tangent vectors, this scalar product can be written in the form... 

In the case of a two-dimensional Riemannian manifold Af with a Riemannian metric gij and with the form of the Riemannian area w = y/det(gij)dx A dy as a symplectic structure (see above), the condition that the group (3 of diffeomorphisms gt preserve the form cj is equivalent to the condition that the domain areas be preserved on the surface when these domains are shifted by the diffeomorphisms gt. Thus, the shifts along integral trajectories of a Hamiltonian field on a two-dimensional symplectic manifold preserve the domain areas. 

FVom the results of Anosov, Klingenberg, and Takens, it follows that in the set of all geodesic flows on smooth Riemannian manifolds there exists an open everywhere dense subset of flows without closed stable integral trajectories [170], 17l. This means that the property of a geodesic flow to have no stable trajectories is the property of general position. Recall once again that we mean strong stability (see Definition 2.1.2). 

Let be a connected compact two-dimensional analytic Riemannian manifold with the boundary dM homeomorphic to a circle (that is, the boundary is connected). The way of obtaining such a manifold is well known from elementary topology. One should remove from a sphere with g handles a certain number of sufEciently small two-dimensional open nonintersecting disks (Fig. 77). In other words, such a manifold is homeomorphic to a two-dimensional flat domain (whose boundary is connected) with g handles. 

Theorem 5.3.1 (Taymanov). If the fundamental group ni(M ) of a closed Riemannian manifold is not almost commutative, that is, does not contain a commutative subgroup of hnite index, then the geodesic Bow on does not admit a geometrically simple set of Brst integrals. 

Jf the configuration space is two-dimensional, then Theorem 5.2.2 obviously follows. Note that Theorems 5.3.1 and 5.3.2 are applicable to geodesic flows on cotangent bundles to Riemannian manifolds. It would be of interest to deduce analogous statements for an arbitrary symplectic manifold, that is, to find out what topological invariants hamper the existence of a complete set of involutive integrals. We will develop the results of Ch. 2 in this direction. 

By the geodesic flow of the Riemannian manifold M with the metric ds = Y Gijdqidqj we mean a Lagrangian system in the tangent bundle TM with the Langrange function L — Identifying TM with T M by means of the... 

Theorem 5.4.3 (see [307]). The geodesic Sow in T M, where M is a two-dimensional Riemannian manifold, has an additional integral quadratic in momenta if and only if in any isothermic coordinates x, y the function X setting the metric satisSes equation (4), where R = R - 2iR2 is a holomorphic function of the variable z = X + ty, which is not identical zero and under transition to other isothermic coordinates is transformed in accordance with formula (7). 

A Riemannian manifold M is termed an SCi-manifold if all the geodesics on M are simple (that is, non-self-intersecting) closed curves of length L Of course, we deal here with single-path geodesics only. 

The first to discuss the question of constructing nontrivial surfaces with closed geodesics was evidently Darboux [189]. We will say that a Riemannian manifold satisfies the 5C-property if there exists a number / > 0 such that any geodesic on Af is a simple closed curve of length / ( or its multiplicities). 

Without going into details of this problem for manifolds of dimension larger that two, where there exist many exquisite results but, as was observed by Besse, many fundamental questions remain still open, we only point out the single general condition necessary for a Riemannian manifold to satisfy the SC-condition. Namely, as has been shown by Weinstein [195], the total volume of a Riemannian... 

The base manifold of internal motion is a Riemannian manifold (Refs.70,71). This conformation space of a molecule has some good properties, but one conspicuously missing property is the general local structure. It is rather complicated and not as well-behaved as one might hope. The case of a triatomic is still simple. But it is to be noted that even though the base manifold here is a trivial bundle, the connection has nonvanishing curvature, i.e., it is not flat with respect to this connection (cf. also Refs.72-74). 

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