Clerc and Ørsted [CØ03] expressed the sympectic area of a geodesic triangle in terms of the Bergman kernel kD of D. There is a very rich structure theory derived from Harish-Chandra embedding that realizes a Hermitian symmetric space of the non-compact type as a bounded symmetric domain. Furthermore, the right side of (A) depends only on the vertices of the geodesic triangle.
In Chapter IV, the results for CP1 are generalized to symmetric Hermitian spaces of compact type. The setup is as follows: Let M = U/K0 be an irreducible symmetric Hermitian space of compact type; hereU is a. The last part of the thesis deals with simple pre-Hermitian symmetric complex spaces, among which SL(2,C)/C∗ is the basic example.
Then we say that J is an invariant complex or paracomplex structure of the symmetric Lie algebra (g,h, σ). To be more specific, we say this. g,h, σ, J) is a semisimple parahermitic symmetric Lie algebra if J is paracomplex. Finally, if J is complex and h is not compactly embedded, we say that (g,h, σ, J) is a semisimple pseudohermitian symmetric Lie algebra.
All simple symmetric Lie algebras (gk,h∩gk, σ|gk) have an invariant. para)complex structure given by Jk. If (g,h, σ, J) is a simple symmetric Lie algebra and J is a paracomplex or complex structure and the center of h is 1-dimensional over R, then gC is simple. It follows that (g,g0, σ) is a complex simple symmetric Lie algebra and J0 (resp. iJ0) is an invariant paracomplex (resp. complex) structure.
Since 1−zw¯ and its inverse belong to the right half-plane for all z and w in the unit disk, we can define a continuous argument for c by adding the main argument of each of the factors in the expression for C. The oriented area of ∆ isπ or −π depending on the orientation of the vertices; the area is π when one reaches ζ2. For the classical setting and definition of the Maslov index, see the book [GS77, Ch.
From this formula we can derive a number of properties of the Maslov index ι, such as its invariance under the action of SU(1,1) on∂D. There are a number of similarities between the geometry of the unit disk D and that of the Riemann sphere CP1. The difference between the area of ∆0 and the area of ∆00 is the area of the disc bounded by the geodesic door zand w, i.e.
From a geometric point of view, we look for the maximum area of a spherical triangle on the two-sphere whose sides are shortest geodesic arcs that do not pass through the antipodes of the vertices. To estimate the maximum area of a geodesic triangle ∆(z0, z1, z2) constructed from a triple (z0, z1, z2) where each of the pairs (zi, zj) belongs to S, first note that there is a simple upper bound given by the total surface area of the two-sphere, i.e. The paracomplex numbers A are a two-dimensional associative unit algebra over R consisting of elements of the form
The next result, which is straightforward to verify, shows that k captures some geometry of Σ. In the case of a unit disc, the theorem was proved using the group invariance of both sides of formula (3.4) and the Gauss-Bonnet theorem. In the previous sections, we studied the three homogeneous spaces SU(1,1)/U(1), SU(2)/U(1), SL(2,R)/R∗, (6.1) and in each case the relationship between the geometry space and corresponding core function.
And this space is an orbit of the adjacency group SL(2,C) acting on the three-dimensional complex vector space sl(2,C). Restricting Q to one of the three subspaces (6.1) gives (up to a sign in the case of the Riemann sphere) the invariant metric on each of these spaces. Then each of the constant curvature metrics on D, C and B in X are just the restrictions of H (up to a sign on C).
Thus Ω is a Hermitian symmetric space and can also be shown to be of non-compact type. In contrast, the Harish–Chandra embedding shows that every Hermitian symmetric space of noncompact type is holomorphically isometric to a bounded symmetric domain with Bergman metric. By choosing the appropriate order, we can assume that p+ is the sum of positive non-compact root spaces.
Using the K0exp(a0)K0-decomposition of G0, we saw (7.1) that D eliminated K0 acting on a 'cube' in p+ consisting of all elements of the form Pr. The G0 operation extends to the topological closure of the DofD and the decomposition of the boundary ∂D into G0-orbits was established by Wolf and Korányi in [WK65]. It is clear from the definition that the Šilo limit D consists of G0-orbits.
In both cases, the Weyl group of the bounded root system consists of all signed permutations of strongly orthogonal roots Γ. If any of the above conditions are met, we say that G0/K0 and D is of tube type. It can be shown that G0,T/K0,T = DT is of non-compact type and a tube-type domain.
With an oriented geodesic triangle ∆ = ∆(z0, z1, z2) in D with vertices z0, z1, and z2 in D we mean the broken geodesic curve consisting of the three geodesic segments connecting the vertices z0, z1, and z2 and in go through that sequence. To connect ζ0 and ζ1 by a geodesic it is necessary that both points in the same G0 orbit lie on the boundary. If we let D2> denote the set of transversal pairs in D×D, then it is clear that D2> is star-shaped with respect to 0 and thus the argument argh(z, w) extends to D2>.
The stabilizer of the three vertices is a compact subgroup of G0 whose fixed points are exactly g−1(ρ(D)).
Note that a0 and a+R are isomorphic under the constraint of the Ad(K0)-equivariant R isomorphism X 7→ 12(X−iad(H0)X) between p0 and p+. Repeating this procedure for each of the factors of (CP1)r shows that we can arrange that p,q and γ lie in Ξ(a+). We introduce the compact analog of the kernel K and show that it contains some information about the geometry of U/K0.
We will make heavy use of the same ideas that worked in the non-compact case. Proposition 11.3 Letx, y be points in p+.Kc(x, y) is defined if and only if there is a unique shortest geodesic curve between Ξ(x) and Ξ(y). It follows from Lemma 10.2 and the transformation rule (11.3) that we can assume that x andy belongs to a+.
A priorigz is only an element of G, but in case z =Pri=1ziXi ∈a+ the element gz is explicitly given by. This proof is a variation of the proof of Theorem 9.1 in which a similar statement played a key role. Now if we get a triple (z0, z1, z2) of points inp+ such that each of the pairs (zi, zj) belongs to S, then we can form an oriented geodesic triangle. z0, z1, z2) as follows: The triangle ∆ consists of the three unique shortest geodesic segments connecting the three vertices z0, z1 and z2 with orientation given by crossing the boundary in the order z0 →z1 →z2 →z0.
We haveω = 12ρ, so the result follows from an application of Stokes' theorem and (12.5) to each of the three geodesic segments of. Hoping to do so, we will now describe how to construct a set R⊂p+×p+ with certain similarities to S. Proposition 13.2 Suppose that zT ∈p+T lies in the maximal tube-type subdomain and w ∈ p+ is any point .
The coefficients of px,y(X) are K0-invariant polynomials in p+×p+ which are holomorphic and antiholomorphic in the first and second variables, respectively.
The group G acts on this product by acting on each factor simultaneously and the G orbit of (o−, o+) is G/H. The cover space SL(2,R)/R>0∗ is embedded in two copies of the space of rays inR2 radiating at the origin. A parahermitic kernel function Remark 14.3 The additive action of H leaves each m± invariant and thus defines the Adm± characters of H.
Then the action of g maps a neighborhood of x into m+ and its differential dxg:m+→m+ is given by. The image of ξ is open and closed, but generally does not contain the orbit G.(o−, o+). In fact, g(o−, o+) lies in the image ofξ if and only if g lies in both expm+Hexpm− and expm−Hexpm+.
Since every z∈M has the form z=g(0) for someeg∈G, it follows from the invariance that Hz gives a perfect pairing of m+ andm−. We can now state and prove a result connecting the mixed kernel κ defined by (14.6) with curve integrals of the 1-form dJlogk. If g is any element of G such that the action of g is defined on all points of γ, then.
We can assume that γ has the form γ(t) = exp(tX).0 for someX ∈q and fast in some intervalI containing 0. By definition pt= exp(γ(t)+) and if we useσ and taking inverses on both sides of (16.2) we get Assume that there exists an element g∈Gsuch that the action of g is defined at all points of γ, and such that gγ passes through 0.
Since ω is exact, we can talk about the integral R∆ω instead of RΣω, where Σ⊂ Mis any smooth surface with boundary ∂Σ.
17.3) The complex structure comes into view through the complex linear action of Ad(HC) on eachg±1 and the holomorphic characters detCAdg±1:HC →C∗, which we will use to define the complex kernel functions kC and κC. It is proved that the right-hand side of (18.1) is invariant under the partial action of GC and that equality holds in (18.1) at 0∈M. The analogs of Lemma 16.1 and Theorem 16.2, as well as their corollaries, hold in this setting.
If g is any element of GC such that the action g is defined on all points of γ, then. Suppose there exists an element g∈GC such that the action g is defined at all points. Now h is a real form of g0 and thus has a one-dimensional center over R and contains an element of the form αZ0 with non-zero complex number α.
Since qr is a real form of the complex vector space q and invariant with respect to α(adZ0), we must have α2 ∈R. Now suppose that the conjugation τr:g → g of g with respect to gr defines the involution of GC, also denoted by τr. If (gr,hr, σ|gr) is parahermitian, we have instead τrg±1 =g±1, from which it follows that τrz+=z+ and similarly τrz−=z−.
Then θZ0 = −Z0 since adqZ0 has real eigenvalues and so (u,k, σ|u) is a simple compact-type Hermitian symmetric Lie algebra. It should be noted that the action of GC on g1 is the same as the action defined in Chapter III. Then p consists of all elements of the form x−θx where x is an element of g1 and ip consists of all elements of the form x+θx,x again ing1.
Now write KC(z) for the canonical kernel defined by (17.3) and calculate its top andip limit.
