The space of all probability measures having positive density function on a connected compact smooth manifold M, denoted by P(M)$\mathcal {P}(M)$, carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of P(M)$\mathcal {P}(M)$, equipped with G. We show that a geodesic segment joining arbitrary probability measures mu(1) and mu(2) is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of P(M)$\mathcal {P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function l defined by l(mu 1,mu 2):=2arccos integral Mp1p2d lambda$\ell \!\big (\mu _1, \mu _2\big ):=2\arccos \int \nolimits _M \sqrt {p_1p_2}\,d\lambda$, mu i=pi lambda$\mu _i=p_i \lambda$, i=1,2$i=1,2$, gives the Riemannian distance function on P(M)$\mathcal {P}(M)$. It is shown that geodesics are all minimal.

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures, and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

The spherical Fourier transform on a harmonic Hadamard manifold (Xn,g) of positive volume entropy is studied. If (Xn,g) is of hypergeometric type, namely spherical functions of X are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric functions. A geometric characterization of hypergeometric type is derived in terms of volume density of geodesic spheres. Geometric properties of (Xn,g) are also discussed.

Information geometry of the space 𝒫(M) of probability measures defined on a compact smooth Riemannian manifold M and equipped with Fisher metric G, is investigated from the viewpoint of Riemannian geometry. The function ℓ : 𝒫(M)×𝒫(M)→ [0,π ) associated to the geometric mean of two probability measures is introduced. From the formulae of Levi-Civita geodesics the Riemannian distance d(· ,· ) of (𝒫(M),G) is exactly given by ℓ (· ,· ) . By applying the parametrix H(x,x_0;t) of the heat kernel of M it is shown that the diameter D satisfies D= π .

A new class of harmonic Hadamard manifolds, those spaces called hypergeometric type, is defined in terms of Gauss hypergeometric equations. A spherical Fourier transform defined on harmonic Hadamard manifolds of hypergeometric type admits an inversion formula. A characterization of the harmonic Hadamard manifold as of hypergeometric type is obtained with respect to volume density.

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling visibility axiom is presented in terms of positive definiteness of the Hessian of Busemann functions.

We give a construction which is Lie theoretic of anti-invariant Riemannian submersions from almost Hermitian manifolds, from quaternion manifolds, from para-Hermitian manifolds, from para-quaternion manifolds, and from octonian manifolds. This yields many compact Einstein examples.

A harmonic, Kahler Hadamard manifold (M-2m, g), m >= 2, with Ricci curvature Ric = -1/2(m + 1) and volume entropy rho(M, g) = m, is biholomorphically isometric to a complex hyperbolic space of holomorphic sectional curvature -1, provided (M, g) is of hypergeometric type. A similar characterization of the real hyperbolic space and the quaternionic hyperbolic space is also obtained in terms of Ricci curvature and volume entropy, without hypergeometric assumption.

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

We report Fisher information geometry of the barycenter map associated with Busemann function B-theta of an Hadamard manifold X and present its application to Riemannian geometry of X from viewpoint of Fisher information geometry. This report is an improvement of [I-Sat' 13] together with a fine investigation of the barycenter map.

Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold \(X\). We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesic. Using Fisher metric and thus obtained properties of geodesics, a fibre space structure of barycenter map and geodesical properties of each fibre are discussed. Moreover, an isometry problem on an Hadamard manifold \(X\) and its ideal boundary \(\partial X\)—for a given homeomorphism \(\Phi\) of \(\partial X\) find an isometry of \(X\) whose \(\partial X\)-extension coincides with \(\Phi\)—is investigated in terms of the barycenter map.

From geometrical study of horospheres we obtain, among asymptotically harmonic Hadamard manifolds, a rigidity theorem of the complex hyperbolic space CHm with respect to volume entropy. We also characterize CHm horospherically in terms of holomorphic curvature boundedness. Corresponding quaternionic analogues are obtained.

Using barycenter of the Busemann function we define a map, called the barycenter map from a space \(\mathcal{P}^{+}\) of probability measures on the ideal boundary ∂ X to an Hadamard manifold X. We show that the space \(\mathcal{P}^{+}\) carries a fibre space structure over X from a viewpoint of information geometry. Following the idea of [7, 9] and [8] we present moreover a theorem which states that under certain hypotheses of information geometry a homeomorphism Φ of ∂ X induces, via the push-forward for probability measures, an isometry of X whose ∂ X-extension coincides with Φ.

Let X be a Hadamard manifold. By applying the stable Jacobi tensor along horospherical foliations of X, a characterization of the real space form is obtained by means of the second fundamental form of the horospheres. The complex space form, the quaternionic space form and the other rank-one symmetric spaces of non-compact type are also similarly characterized. Geometrical characterization of horospheres are also given.

Fisher Information Geometry of the Barycenter of Probability Measures Mitsuhiro Itoh and Hiroyasu Satoh Institute of Mathematics, University of Tsukuba, Japan and Tokyo Denki University, Japan Motivation. Consider the following character- ization problem. Let (Xo, go) be a Damek-Ricci space. Let (X, g) be an Hadamard manifold, a simply connected complete Riemannian manifold of nonpositive curvature. Assume (X, g) ∼= (Xo, go) (quasi-isometric). Then, is (X, g) itself Damek- Ricci ? Here (Xo, go) is Damek-Ricci, an R-extention of a generalized Heisenberg group N. A Damek- Ricci space is a solvable Lie group with a left invariant metric. A Damek-Ricci space is Riemannian homogeneous and of nonpos- itive curvature. Moreover, a Damek-Ricci space is harmonic, namely, mean curvature of a geodesic sphere is a function of radius. A Damek-Ricci space is a rank one symmetric space of noncompact type, when it is strictly negative curvature. RHn, CHn, HHn and a Cayley hyperbolic space QH2 exhaust the rank one symmetric spaces of noncompact type. §1 Barycenter and barycenter-isometric maps Denote by ∂X the ideal boundary of (X, g). Let P+(∂X) = P+(∂X, dθ) be the space of probability me

Let (X,g) be an Hadamard manifold with ideal boundary ∂X. We can then define the map φ:X→P(∂X) associated with Poisson kernel on X, where P(∂X) is the space of probability measures on ∂X, together with the Fisher information metric G. We make geometrical investigation of homothetic property and minimality of this map with respect to the metrics g and G. The map φ is shown to be a minimal homothetic embedding for a rank one symmetric space of noncompact type as well as for a nonsymmetric Damek–Ricci space. The following is also obtained. If φ is assumed to be homothetic and minimal, then, (X,g) turns out to be an asymptotically harmonic, visibility manifold with the Poisson kernel being expressed in terms of the Busemann function.

1.1 3-D Seiberg-Witten equations. 3-D Seiberg-Witten equations are defined over a closed oriented Riemannian 3-manifold M , as giving a 3-D geometric invariant. These equations are similar to the four-dim Seiberg-Witten equations. Let W −→ M be an Spin(3)-complex vector bundle over M associated to the orthonormal frame bundle O(M) −→ M induced naturally from a Riemannian metric h. The group Spin(3) = (Spin(3)× U(1))/Z2 is isomorphic to U(2) so that W is considered as a rank 2 complex vector bundle over M with a Hermitian fiber metric, and actually W is written as W = W0 ⊗ L1, where W0 is the product bundle and L1 is a complex line bundle ( det W = L1 is called the determinant line bundle of W and is denoted by L ). Notice that L carries a Hermitian fibre metric induced from the Spin(3)-structure, and L1 is L. Let A be a unitary connection on L and Φ be a smooth section of W , called a spinor field. Then the Seiberg-Witten equations are { DAΦ = 0 c(∗FA) = Φ⊗ Φ∗ − 12 |Φ|idW (1.1)

For Damek-Ricci spaces (X, g) we compute the exact form of the Busemann function which is needed to represent the Poisson kernel of (X, g) in exponential form in terms of the Busemann function and the volume entropy. From this fact, we show that the Poisson kernel map phi : (X, g) -> (P(partial derivative X), G) is a homothetic embedding. Here P(partial derivative X) is the space of probability measures having positive density function on the ideal boundary partial derivative X of X, and G is the Fisher information metric on P(partial derivative X).

We introduce on a Kähler manifold the covariant Dolbeault operator ∂ ∇ and show that ∂ ∇ -closedness of the Bochner curvature B is equivalent to ∂ ∇∗ B = 0. Further we reformulate the remarkable result of [Ki-Kim] under the ∂ ∇ -closedness by using Kähler convention.

The moduli space of the solutions to the monopole equations over ail oriented closed 3-manifold M carrying the geometric structure R x H-2 is studied. Solving the parallel spinor equation, we obtain an explicit solution to the monopole equations. The moduli space consists of a single point with the Seiberg-Witten invariant +/-1. Further, the (anti-) canonical line bundle K-M(+/-1) gives a monopole class of M.