Then the quotient space M/ G (defined in Exercise 4.43) is a differentiable manifold, and the natural projection tt M M/ G is a differentiable function. [Pg.370]

We call this quotient space a hyper-Kahler quotient. [Pg.34]

A simple singularity is a quotient space C /P, where P is a finite subgroup of SU(2). This singularity has been studied from various point of view. See e.g. [5], [76]. [Pg.42]

The interference factor therefore depends on the spacing and the quotient R/p, which is determined by the anode dimensions according to Table 24-1. Figure 9-8 contains examples of various practical cases. [Pg.545]

The projective plane arises as a quotient space of the sphere, the required group being C,-. It is obtained by identifying antipodal points of the spherical surface in other words, it is the antipodal quotient of the sphere (see Section 1.2.2). P2 is the simplest compact non-orientable surface in the sense that it can be obtained from the sphere by adding just one cross-cap. [Pg.41]

Xq, aside from its generic point, is the quotient space by conjugation [Pg.101]

P.B. Kronheimer, The construction of ALE spaces as a hyper-Kdhler quotients, J. Differential Geom. 29 (1989) 665-683. [Pg.114]

Since the G-action on is free, the slice theorem implies that the quotient space [Pg.36]

Example 3.40. The moduli space of instantons on a 4-dimensional hyper-Kahler manifold X can be considered as a hyper-Kahler quotient. Let us take a smooth vector bundle E over X with a Hermitian metric. Let us denote the space of metric connections on E by A. Its tangent space at A G M can be identified with [Pg.37]

Since the G-action on // 1(C) is free, the slice theorem implies that the quotient space gTl((,)/G has a structure of a C°°-manifold such that the tangent space TaAtt-HO/G) at the orbit G x is isomorphic to the orthogonal complement of Vx in Txg 1( ). Hence the tangent space is the orthogonal complement of Vx IVX JVX KVX in TxX, which is invariant under I, J and K. Thus we have the induced almost hyper-complex structure. The restriction of the Riemannian metric g induces a Riemannian metric on the quotient g 1(()/G. In order to show that these define a hyper-Kahler structure, it is enough to check that the associated Kahler forms u>[, u) 2 and co z are closed by Lemma 3.32. [Pg.36]

We put Z = G/S

Show that is an equivalence relation. If the action cr is clear from the context, then the quotient space S/ is often denoted S/G. [Pg.152]

This is the moduli space of the anti-self-dual connections. (Note that -action is not necessarily free. Hence the moduli spaces may have singularities.) The spaces /r (0) and G are both infinite dimensional, but its quotient, that is, the moduli space of the anti-selfdual connections is finite dimensional. The proof for Theorem 3.30 works even in this case if one uses the appropriate analytical packages, i.e. the Sobolev space, etc. [Pg.38]

Furthermore, as we also noted last time, at sufficiently large spacings (AX) the numerator term ceased to increase. As we noted before, at this point the various points used for the computation are each individually tracing out the shape of the underlying curve. However, as AX in denominator continues to increase, we can expect that the quotient, Ay/AX will decrease, and this is the behavior we observe. [Pg.356]

The basic principle in this technique is to replace derivatives by finite differences, i.e., dy/dx is replaced by Ay/Ax. The differential equation is then rewritten using these difference quotients in place of the derivatives and the boundary conditions of the problem introduced. The equations can then be solved analytically. Space and time [Pg.444]

It is easy to see a matrix has a closed orbit if and only if it is diagonalizable. Hence the set of closed orbits can be identified with the set of eigenvalues. On the other hand, a matrix B with [B, B — 0 (i.e. a normal matrix) can be diagonalizable by a unitary matrix. Hence the quotient space is also identified with the set of eigenvalues. The identification can be seen directly in this example. [Pg.29]

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