Riemannian connection

Now we fix a Riemannian metric g which is invariant under the T-action. The symplectic form co together with the Riemannian metric g gives an almost complex structure I defined by co(v, x) = g(Iv, w). With this almost complex structure, we regard the tangent space TxX as a complex vector space. Let XT = Cv be the decomposition into the connected components. For each x G C , we have the weight decomposition... 

We have assumed that the geometric connection coefficients can be defined in some sensible way. To do this, we simply note that, in order to define conservation laws (i.e., to do physics) in a Riemannian space, it is necessary to be have a generalized form of Gauss divergence theorem in the space. This is certainly possible when the connections are defined to be the metrical connections, but it is by no means clear that it is ever possible otherwise. Consequently, the connections are assumed to be metrical and so gai, given in (3), can be written explicitly as... 

This cone is real in the case of relativity theory, while the quadratic form gij is indefinite. Prom the point of view stressed by E. Cartan (bibb 1928,1) the Riemannian geometry of the underlying world is to be considered as the theory of these connected Euclidean tangent spaces. The generalization that we have in mind now consists of the following ... 

In an underlying space the tensor g j gives a Riemannian metric. Since this metric is invariantly connected with our quadric we suspect that the Riemannian metric is exactly the non-Euclidean metric that appears as absolute image of our quadric. 

Theorem 5.2.1 can be reformulated as follows. Let be a real-analytic two-dimensional compact closed connected manifold endowed with an arbitrary real-analytic Riemannian metric. If the genus of a manifold is higher than unity then the geodesic flow of this metric (as the Hamiltonian flow on a four-dimensional manifold T M ) is not completely Liouville integrable, that is, does not have an additional (second) integral which is independent of the energy integral and is in involution with it. 

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 5.2.4 (Bolotin). Let be a connected Riemannian compact two-dimensional real-analytic manifold with a locally geodesically convex boundary and such that x( ) < 0. Then the geodesic Sow of the Riemannian metric on the three-dimensional manifold of constant energy h = 1 does not have an analytic Brst integral (which is independent of the energy integral and is in involution with it on T M). 

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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