GEOMETRY OF THE FISHER RAO METRIC ON THE SPACE OF SMOOTH DENSITIES ON A COMPACT MANIFOLD

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1 GEOMETRY OF THE FISHER RAO METRIC ON THE SPACE OF SMOOTH DENSITIES ON A COMPACT MANIFOLD MARTINS BRUVERIS, PETER W. MICHOR Abstract. It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form G µ(α, β) = C (µ(m)) M α β µ µ µ + C (µ(m)) α β M M for some smooth functions C, C of the total volume µ(m). Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.. Introduction. The Fisher Rao metric on the space Prob(M) of probability densities is invariant under the action of the diffeomorphism group Diff(M). Restricted to finite-dimensional submanifolds of Prob(M), so-called statistical manifolds, it is called Fisher s information metric []. A uniqueness result was established [4, p. 56] for Fisher s information metric on finite sample spaces and [3] extended it to infinite sample spaces. The Fisher Rao metric on the infinite-dimensional manifold of all positive probability densities was studied in [7], including the computation of its curvature. In [4] it was proved that any Diff(M)-invariant Riemannian metric on the space Dens + (M) of smooth positive densities on a compact manifold M without boundary is of the form α β () G µ (α, β) = C (µ(m)) M µ µ µ + C (µ(m)) α β M M for some smooth functions C, C of the total volume µ(m). This implies that the Fisher Rao metric on Prob(M) is, up to a multiplicative constant, the unique Diff(M)-invariant metric. By Cauchy Schwarz the metric () is positive definite if and only if C (m) > m C (m) for all m >.. The setting. Let M m be a smooth compact manifold. It may have boundary or it may even be a manifold with corners; i.e., modelled on open subsets of quadrants in R m. For a detailed description of the line bundle of smooth densities we refer to [4] or [,.]. We let Dens + (M) denote the space of smooth positive densities on M, i.e., Dens + (M) = {µ Γ(Vol(M)) : µ(x) > x M}. Let Prob(M) be the subspace of positive densities with integral on M. Both spaces are smooth Fréchet manifolds; in particular they are open subsets of the affine spaces of all densities and densities of integral respectively. For µ Dens + (M) we have Date: May 9, 8. Mathematics Subject Classification. Primary 58B, 58D5. Key words and phrases. Fisher Rao Metric; Information Geometry; Invariant Metrics; Space of Densities; Surfaces of Revolution. MB was supported by a BRIEF award from Brunel University London.

2 MARTINS BRUVERIS, PETER W. MICHOR T µ Dens + (M) = Γ(Vol(M)) and for µ Prob(M) we have T µ Prob(M) = {α Γ(Vol(M)) : α = }. The Fisher Rao metric, given by G FR µ (α, β) = M µ is a Riemannian metric on Prob(M); it is invariant under the natural action of the group Diff(M) of all diffeomorphisms of M. If M is compact without boundary of dimension, the Fisher-Rao metric is the unique Diff(M)-invariant metric up to a multiplicative constant. This follows, since any Diff(M)-invariant Riemannian metric on Dens + (M) is of the form () as proved in [4]. M α µ β µ 3. Overview. We will study four different representations of the metric G in (). The first representation is G itself on the space Dens + (M). Next we fix a density µ Prob(M) and consider the mapping µ R : Dens + (M) C (M, R > ), R(µ) = f =. µ This map is a diffeomorphism with inverse R (f) = f µ, and we will denote the induced metric by G = ( R ) G; it is given by the formula G f (h, k) = 4C ( f ) h, k + 4C ( f ) f, h f, k, with f = M f µ denoting the L (µ )-norm, and this formula makes sense for f C (M, R). See Sect. 5 for calculations. Next we take the pre-hilbert space (C (M, R),, L (µ )) and pass to polar coordinates. Let S = {ϕ L (M, R) : M ϕ µ = } denote the L -sphere. Then ( ) Φ : C (M, R > ) R > (S C>) f, Φ(f) = (r, ϕ) = f,, f is a diffeomorphism, where C > = C (M, R > ); its inverse is Φ (r, ϕ) = r.ϕ. We set Ḡ = ( Φ ) G; the metric Ḡ has the expression Ḡ r,ϕ = g (r) dϕ, dϕ + g (r)dr, with g (r) = 4C (r )r and g (r) = 4 ( C (r ) + C (r )r ). Finally we change the coordinate r diffeomorphically to s = W (r) = r Then, defining a(s) = 4C (r(s) )r(s), we have g (ρ) dρ. Ḡ s,ϕ = a(s) dϕ, dϕ + ds. We will use Ḡ to denote the metric in both (r, ϕ) and (s, ϕ) coordinates. Let W = lim r + W (r) and W + = lim r W (r). Then W : R > (W, W + ) is a diffeomorphism. This completes the first row in Fig.. The geodesic equation of G in the various representations will be derived in Sect. 5. The formulas for the geodesic equation and later for curvature are infinite-dimensional analoga of the corresponding formulas for warped products; see [, p. 4ff] or [5, Chap. 7]. The four representations are summarized in the following diagram. Dens + (M) R C (M, R > ) Φ R > (S C >) W Id (W, W + ) (S C >).

3 GEOMETRY OF THE FISHER RAO METRIC 3 Dens + (M) R C (M, R > ) Φ R > S C > W Id (W, W + ) S C > Γ C (Vol(M))\{} R C (M, R)\{} Φ R > S C W Id R S C Γ L (Vol(M))\{} R L (M, R)\{} Φ R > S W Id R S Figure. Representations of Dens + (M) and its completions. In the second and third rows we assume that (W, W + ) = (, + ) and we note that R is a diffeomorphism only in the first row. Since Ḡ induces the canonical metric on (W, W + ), a necessary condition for Ḡ to be geodesically complete is (W, W + ) = (, + ). Rewritten in terms of the functions C and C this becomes ( W + = r / ) C (r) dr = or C (r) dr =, and similarly for W =, with the limits of integration being and. If Ḡ is incomplete, i.e., W > or W + <, there are sometimes geodesic completions. See Sect. 8 for details. We now assume that (W, W + ) = (, + ). The metrics Ḡ and G can be extended to the spaces R S C and C (M, R) \ {} and the last two maps in the diagram Γ C (Vol(M))\{} R C (M, R)\{} Φ R > S C W Id R S C are bijections. The extension of R is given by R (f) = f f µ ; it does not map into smooth densities any more, but only into C -sections of the volume bundle; however, R is not surjective into C -sections, because the loss of regularity for R (f) occurs only at point where f is. The last two maps, Φ and W Id, are diffeomorphisms. The following will be shown in Sect. 7: (W, W + ) = (, + ) implies that (R S C, Ḡ) is geodesically complete and hence so are (R > S C, Ḡ) and (C (M, R) \ {}, G). Finally we consider the metric completions, still assuming that (W, W + ) = (, + ). For Ḡ this is R S or R > S in (s, ϕ) or (r, ϕ)-coordinates, respectively, as shown in Sect. 7. The metrics and maps can be extended to Γ L (Vol(M)) \ {} R L (M, R) \ {} Φ R > S W Id R S. Here Γ L denotes the space of L -sections. The extension of R is given by R(µ) = sgn(µ) µ /µ and its inverse is R (f) = f f µ as before. The last two maps are diffeomorphisms and hence (L (M, R) \ {}, G) is metrically complete. The extension of R is bijective, but not a diffeomorphism. It is continuous, but not C, and its inverse is C, but not C ; furthermore DR (f) is not surjective if f = on a set of positive measure. However we can use R to pull back the geodesic distance function from L (M, R) \ {} to Γ L (Vol(M)) \ {} to obtain a complete metric on the latter space, that is compatible with the standard topology.

4 4 MARTINS BRUVERIS, PETER W. MICHOR 4. The inverse R and geodesic completeness. There is more than one choice for the extension of R (f) = f µ from C (M, R > ) to C (M, R). The choice R (f) = f f µ remains injective and can be further extended to a bijection on the metric completion L (M, R) \ {}. We can consider the equally natural extension Q and its factorizarion given by C (M, R) Q Γ (Vol(M)) Q { f : f C (M, R)} Q(f) = f µ, Q (f) = f, Q ( f ) = f µ. into the space of smooth, nonnegative sections. The map Q is not surjective; see [9] for a discussion of smooth non-negative functions admitting smooth square roots. The image { f : f C (M, R)} of Q looks somewhat like the orbit space of a discrete reflection group: An example of a codimension wall of the image could be { f : f C (M, R), f(x) = } for one fixed point x M. Since this is dense in the L -completion of T f C (M, R) with respect to G f, we do not have a reflection at this wall. Fixing ϕ S C and considering {(r, ϕ) R > S C : ϕ, ϕ = } we can write the orthogonal reflection (r, t ϕ +t ϕ) (r, t ϕ +t ϕ). Geodesics in (C (M, R), G) are mapped by Q to curves that are geodesics in the interior C (M, R > ), and that are reflected following Snell s law at any hyperplanes in the boundary for which the angle makes sense. The mapping Q then smoothes out the reflection to a quadratic glancing of the boundary if one can describe the smooth structure of the boundary. It is tempting to paraphrase this as: The image of Q is geodesically complete. But note that: () The metric G becomes ill-defined on the boundary. () The boundary is very complicated; each closed subset of M is the zeroset of a smooth non-negative function and thus corresponds to a boundary component. Some of them look like reflection walls. One could try to set up a theory of infinite dimensional stratified Riemannian manifolds and geodesics on them to capture this notion of geodesic completeness, similarly to []. But the situation is quite clear geometrically, and we prefer to consider the geodesic completion described by the inverse R used in this paper, which is perhaps more natural. Q 5. Geodesics of the Fisher-Rao metric on Dens + (M). In [7] it was shown that Prob(M) has constant sectional curvature for the Fisher-Rao metric. For fixed µ Prob(M) we consider the mapping µ R : Dens + (M) C (M, R > ), R(µ) =. µ The inverse R : C (M, R > ) Dens + (M) is given by R (f) = f µ ; its tangent mapping is T f R.h = fhµ. Remark. In [8] it was shown that for C and C the rescaled map R(µ) = µ µ is an isometric diffeomorphism from Prob(M) onto the open subset C (M, R > ) {f : f µ = 4} of the L -sphere of radius in the pre-hilbert space (C (M, R),, L (µ )). For a general function C the same holds for R(µ) = λ µ µ and the L -sphere of radius λ, where λ > is a solution of the equation λ = 4C (λ ).

5 GEOMETRY OF THE FISHER RAO METRIC 5 The Fisher Rao metric induces the following metric on the open convex cone C (M, R > ) C (M, R): (a) ( (R ) G ) f (h, k) = G R (f)(t f R.h, T f R.k) = G f µ (fhµ, fkµ ) = C ( f L (µ ) fhµ fkµ ) f µ f f µ + C ( f L µ (µ ) ) fhµ fkµ = 4C ( f ) hkµ + 4C ( f ) fhµ fkµ = 4C ( f ) h, k + 4C ( f ) f, h f, k = 4C ( f f, h f, k ) h f, k f f f ( C ( f ). f + C ( f ) ) f f f, h f, k, where in the last expression we split h and k into the parts perpendicular to f and multiples of f. We now switch to polar coordinates on the pre-hilbert space: Let S = {ϕ L (M, R) : ϕ µ = } denote the sphere, and let S C> be the intersection with the positive cone. Then C (M, R) \ {} = R > S C via Φ : C (M, R) \ {} R > S, Φ(f) = (r, ϕ) = ( f, ) f. f Note that Φ(C (M, R > )) = R > S C >. We have f = Φ (r, ϕ) = r.ϕ thus df = r dϕ + ϕ dr, where r dϕ(h) = h ϕ, h ϕ is the orthogonal projection onto the tangent space of S at ϕ and dr(h) = ϕ, h. The Euclidean (pre-hilbert) metric in polar coordinates is given by df, df = ϕ.dr + r.dϕ, ϕ.dr + r.dϕ = ϕ, ϕ dr + r. ϕ, dϕ.dr + r dϕ, dϕ = dr + r dϕ, dϕ. The pullback metric is then ( ) (b) Ḡ = (Φ ) G = 4C (r )r dϕ, dϕ + 4 ( C (r )r + C (r ) ) dr where we introduced the functions = g (r) dϕ, dϕ + g (r)dr = a(s) dϕ, dϕ + ds, g (r) = 4C (r )r and g (r) = 4 ( C (r )r + C (r ) ), and where in the last expression we changed the coordinate r diffeomorphically to r s(r) = C (ρ )ρ + C (ρ ) dρ and let a(s) = 4C (r(s) )r(s). The resulting metric is a radius dependent scaling of the metric on the sphere times a different radius dependent scaling of the metric on R >. Note that the metric (b) (as well as the metric in the last expression of (a)) is actually welldefined on C (M, R) \ {} = R > S C ; this leads to a (partial) geodesic completion of (Dens + (M), G). Geodesics for the metric (b) follow great circles on the sphere with some time dependent stretching, since reflection at any hyperplane containing this great circle is an isometry.

6 6 MARTINS BRUVERIS, PETER W. MICHOR We derive the geodesic equation. Let [, ] ( ε, ε) (t, s) (r(t, s), ϕ(t, s)) be a smooth variation with fixed ends of a curve (r(t, ), ϕ(t, )). The energy of the curve and its derivative with respect to the variation parameter s are as follows, where S is the covariant derivative on the sphere S. ( E(r, ϕ) = g (r) ϕ t, ϕ t + ) g (r).rt dt ( s E(r, ϕ) = g (r).r s ϕ t, ϕ t + g (r) S ϕ s t, ϕ t + = = Thus the geodesic equation is (c) Using the first equation we get: + g (r).r s.r t + g (r).r t.r ts )dt ( g (r).r s ϕ t, ϕ t g (r).r t ϕ s, ϕ t g (r) ϕ s, S t ϕ t + + g (r).r s.r t g (r).r t.r s g (r).r tt.r s )dt ( g (r) ϕ t, ϕ t g (r).r t g (r).r tt ) ( ) g (r).r t ϕ s, ϕ t + g (r) ϕ s, S t ϕ t dt. S t ϕ t = t (log g (r)) ϕ t r tt = g (r) g (r) ϕ t, ϕ t t (log g (r)) r t t ϕ t, ϕ t = t ϕ t, ϕ t = t (log g (r)) ϕ t, ϕ t t (log ϕ t, ϕ t ) = t (log g (r)) log( ϕ t ) = log g (r) + log A with A = g (r) ϕ t, which describes the speed of ϕ(t) along the great circle in terms of r(t); note that the quantity g (r) ϕ t is constant in t. The geodesic equation (c) simplifies to r s (d) S t ϕ t = t (log g (r)) ϕ t r tt = A g (r) g (r) g (r) t (log g (r)) r t with g (r) = 4C (r )r and g (r) = 4 ( C (r )r + C (r ) ). We can solve equation (d) for ϕ explicitely. Given initial conditions ϕ, ψ, the geodesic ϕ(t) on the sphere with radius satisfying ϕ() = ϕ, ϕ t () = ψ is ϕ(t) = cos( ψ t)ϕ + sin( ψ t) ψ ψ. We are looking for a reparametrization ϕ(t) = ϕ(α(t)). Inserting this into the geodesic equation we obtain t ( ϕ(α)) t ϕ(α) ( ϕ(α)), ϕ(α) = t (log g (r)) t ( ϕ(α)) ϕ(α) ( ) S t ϕ t (α)α ϕ(α) t + ϕ t (α)α tt ϕ t (α)α tt, ϕ(α) = t (log g (r)) ϕ t (α)α t ϕ(α) α tt = t (log g (r)) α t.

7 GEOMETRY OF THE FISHER RAO METRIC 7 With intial conditions α() = and α t () = this equation has the solution α(t) = g (r ) t g (r(τ)) dτ, where r = r() is the initial condition for the r-component of the geodesic. If the metric is written in the form Ḡ = ds +a(s) dϕ, dϕ, equation (d) becomes s tt = A a (s) a(s), for A = a(s) ϕ t, where ϕ(t) is given explicitly as above. This can be integrated into the form (e) s t = A a(s) + A, A a constant. 6. Relation to hypersurfaces of revolution. We consider the metric Ḡ on (W, W + ) S C where Ḡs,ϕ = a(s) dϕ, dϕ + ds and a(s) = 4C (r(s) )r(s). Then the map Ψ is an isometric embedding (remember ϕ, dϕ = on S C ), Ψ : ((W, W + ) S C, Ḡ) ( R C (M, R), du + df, df ), ( s Ψ(s, ϕ) = a (σ) 4a(σ) dσ, ) a(s)ϕ, In fact it is defined and smooth only on the open subset { (s, ϕ) (W, W + ) S C : a (s) < 4a(s) }. We will see in Sect. 9 that the condition a (s) < 4a(s) is equivalent to a sign condition on the sectional curvature; to be precise a (s) < 4a(s) Sec (s,ϕ) (span(x, Y )) >, where X, Y T ϕ S is any Ḡ-orthonormal pair of tangent vectors. Fix some ϕ S C and consider the generating curve ( s γ(s) = a (σ) 4a(σ) dσ, ) a(s)ϕ R C (M, R) ; then γ(s) is already arc-length parametrized! Any arc-length parameterized curve I s (c (s), c (s)) in R generates a hypersurface of revolution {(c (s), c (s)ϕ) : s I, ϕ S C } R C (M, R), and the induced metric in the (s, ϕ)-parameterization is c (s) dϕ, dϕ + ds. This suggests that the moduli space of hypersurfaces of revolution is naturally embedded in the moduli space of all metrics of the form a(s) dϕ, dϕ + ds. Let us make this more precise in an example: In the case of S = S and the tractrix (c, c ), the surface of revolution is the pseudosphere (curvature ) whose universal cover is only part of the hyperbolic plane. But in polar coordinates we get a space whose universal cover is the whole hyperbolic plane. In detail: the arc-length parametrization of the tractrix and the induced metric are s c (s) = e σ dσ = Arcosh ( e s) e s, c (s) = e s, s > a(s) dϕ + ds = e s dϕ + ds, s R.

8 8 MARTINS BRUVERIS, PETER W. MICHOR 7. Completeness. In this section we assume that (W, W + ) = (, + ), which is a necessary and sufficient condition for completeness. First we have the following estimate for the geodesic distance dist of the metric Ḡ, which is valid on bounded metric balls. Let dist S denote the geodesic distance on S with respect to the standard metric. Lemma. Let (W, W + ) = (, + ), (s, ϕ ) R S and R >. Then there exists C >, such that C (dist S (ϕ, ϕ ) + s s ) dist ((s, ϕ ), (s, ϕ )) holds for all (s i, ϕ i ) with dist ((s, ϕ ), (s i, ϕ i )) < R, i =,. Proof. First we observe that s s s t (t) dt and hence by taking the infimum over all paths, C (dist S (ϕ, ϕ ) + s s ), a(s) ϕ t + s t dt = Len(s, ϕ), s s dist ((s, ϕ ), (s, ϕ )) < R. Thus s is bounded on bounded geodesic balls. Now let (s i, ϕ i ) be chosen according to the assumptions and let (s(t), ϕ(t)) be a path connecting (s, ϕ ) and (s, ϕ ) with Len(s, ϕ) < dist ((s, ϕ ), (s, ϕ )). Then for t [, ], dist ((s, ϕ ), (s(t), ϕ(t))) dist ((s, ϕ ), (s, ϕ )) + dist ((s, ϕ ), (s, ϕ )) 5R. In particular the path remains in a bounded geodesic ball. Thus there exists a constant C >, such that C a(s) C holds along (s(t), ϕ(t)). From there we obtain C ϕ t + s t dt a(s) ϕ t + s t dt C ϕ t + s t dt, and by taking the infimum over paths connecting (s, ϕ ) and (s, ϕ ) the desired result follows. Proposition. If (W, W + ) = (, + ), the space (R S, Ḡ) is metrically and geodesically complete. The subspace (R S C, Ḡ) is geodesically complete. Proof. Given a Cauchy sequence (s n, ϕ n ) n N in R S with respect to the geodesic distance, the lemma shows that (s n ) n N and (ϕ n ) n N are Cauchy sequences in R and S respectively. Hence they have limits s and ϕ and by the lemma the sequence (s n, ϕ n ) n N converges to (s, ϕ) in the geodesic distance as well. It is shown in [, Prop. 6.5] that a metrically complete, strong Riemannian manifold is geodesically complete. Since the ϕ-part of a geodesic in R S is a reparametrization of a great circle, if the initial conditions lie in R S C, so will the whole geodesic. Hence R S C is geodesically complete. The map W Id Φ : L (M, R) \ {} R S is a diffeomorphism and an isometry with respect to the metrics G and Ḡ. Corollary. If (W, W + ) = (, + ), the space (L (M, R) \ {}, G) is metrically and geodesically complete. The subset (C (M, R)\{}, G) is geodesically complete.

9 GEOMETRY OF THE FISHER RAO METRIC 9 It remains to consider the existence of minimal geodesics. Theorem. If (W, W + ) = (, + ), then any two points (s, ϕ ) and (s, ϕ ) in R S can be joined by a minimal geodesic. If ϕ and ϕ lie in S C, then the minimal geodesic also lies in R S C. Proof. If ϕ and ϕ are linearly independent, we consider the -space V = V (ϕ, ϕ ) spanned by ϕ and ϕ in L. Then R V S is totally geodesic since it is the fixed point set of the isometry (s, ϕ) (s, s V (ϕ)) where s V is the orthogonal reflection at V. Thus there is exists a minimizing geodesic between (s, ϕ ) and (s, ϕ ) in the complete 3-dimensional Riemannian submanifold R V S. This geodesic is also length-minimizing in the strong Hilbert manifold R S by the following argument: Given any smooth curve c = (s, ϕ) : [, ] R S between these two points, there is a subdivision = t < t < < t N = such that the piecewise geodesic c which first runs along a geodesic from c(t ) to c(t ), then to c(t ),..., and finally to c(t N ), has length Len(c ) Len(c). This piecewise geodesic now lies in the totally geodesic (N + )-dimensional submanifold R V (ϕ(t ),..., ϕ(t N )) S. Thus there exists a geodesic c between the two points (s, ϕ ) and (s, ϕ ) which is length-minimizing in this (N +)-dimensional submanifold. Therefore Len(c ) Len(c ) Len(c). Moreover, c = (s c, ϕ c ) lies in R V (ϕ, (ϕ c ) ()) S which also contains ϕ, thus c lies in R V (ϕ, ϕ ) S. If ϕ = ϕ, then R {ϕ } is a minimal geodesic. If ϕ = ϕ we choose a great circle between them which lies in a -space V and proceed as above. When ϕ, ϕ C, then the 3-dimensional submanifold R V S lies in R S C and hence so does the minimal geodesic. 8. Some geodesic completions. The relation to hypersurfaces of revolution in Sect. 6 suggests that there are functions C and C such that geodesic incompleteness of the metric G is due to a coordinate singularity at W or at W +. Let us write I = (W, W + ). We work in polar coordinates on the infinite-dimensional manifold I (S C ) with the metric Ḡ = ds + a(s) dϕ, dϕ. Example. For I = (, ) the metric ds + s dϕ, dϕ describes the flat space C (M, R)\{} with the L -metric in polar coordinates. Putting back in geodesically completes the space. Moreover, for β (, π/4] the metric ds + sin (β)s dϕ, dϕ describes the cone with radial opening angle β. Putting in generates a tip; sectional curvature is a delta distribution at the tip of size ( sin(β))π. This is an orbifold with symmetry group Z/kZ at the tip if sin(β) = /k. More generally, ds + K s dϕ, dϕ describes the generalized cone whose angle defect at the tip is π( K); there is negative curvature at the tip if K > in which case we cannot describe it as a surface of revolution. Example. For I = (, π), the metric ds + sin (s) dϕ, dϕ describes the infinitedimensional round sphere of dimension higher with equator S C and with north- and south-pole omitted. This can be seen from the formula for sectional curvature from Sect. 9 below, or by transforming it to the hypersurface of revolution according to Sect. 6. Putting back the two poles gives the geodesic completion. To realize this on the space of densities, we may choose a smooth and positive function

10 MARTINS BRUVERIS, PETER W. MICHOR g (r) freely, and then put g (r) = sin ( r ) g (ρ) / dρ, C (m) = g ( m) 4m, C (m) = 4m g ( m) 4m g ( m). Choosing g (r) = 4r we get g (r) = sin (r ) so that C (m) = 4m sin (m ) and C (m) = 4m sin (m ). The general situation can be summarized in the following result: Theorem. If W > and if C and C have smooth extensions to [, ) and C () >, then the metric Ḡ has a smooth -point geodesic completion at r = (or s = W ). If W + < and if C and C have smooth extensions to (, ] in the coordinate /m, then the metric Ḡ has a smooth -point geodesic completion at r = (in the coordinate /r), or at s = W +. Proof. From the formulas in Sect. 3 we get Φ (g (r) dϕ, dϕ + g (r)dr ) = g ( ( f ) f df, df + g ( f ) f g ) ( f ) f 4 f, df = 4C ( f ) df, df + 4C ( f ) f, df. By a classical theorem of Whitney the even smooth functions h(r) are exactly the smooth functions of r. So the metric extends smoothly at to C (M, R). The proof for the case W + < is similar. 9. Covariant derivative and curvature. In this section we will write I = (W, W + ). In order to calculate the covariant derivative we consider the infinitedimensional manifold I S with the metric Ḡ = ds + a(s) dϕ, dϕ and smooth vector fields f(s, ϕ) s + X(s, ϕ) where X(s, ) X(S) is a smooth vector field on the Hilbert sphere S. We denote by S the covariant derivative on S and get s Ḡ ( g s + Y, h s + Z ) = s ( gh + a Y, Z ) = = g s h + gh s + a s Y, Z + a Y s, Z + a Y, Z s = Ḡ( g s s + a s a Y + Y s, h s + Z ) + Ḡ( g s + Y, h s s + a s a Z + Z ) s XḠ( g s + Y, h s + Z ) = X ( gh + a Y, Z ) = dg(x).h + g.dh(x) + a S XY, Z + a Y, S XZ, = Ḡ( dg(x) s + S XY, h s + Z ) + Ḡ( g s + Y, dh(x) s + S XZ ). Thus the following covariant derivative on I S, which is not the Levi-Civita covariant derivative, f s+x(g s + Y ) = f.g s s + f a s a Y + fy s + dg(x) s + S XY, respects the metric ds + a dϕ, dϕ. But it has torsion which is given by Tor(f s + X, g s + Y ) = = f s+x(g s + Y ) g s+y (f s + X) [f s + X, g s + Y ] = = a s (fy gx). a

11 GEOMETRY OF THE FISHER RAO METRIC To remove the torsion we consider the endomorphisms Tor f s+x, Tor f s+x : T (I S) T (I S), Tor f s+x(g s + Y ) = Tor(f s + X, g s + Y ), Ḡ ( Tor f s+x(g s + Y ), h s + Z) = Ḡ( g s + Y, Tor f s+x(h s + Z) ) The endomorphism A f s+x(g s + Y ) := ( Tor(f s + X, g s + Y ) Tor f s+x(g s + Y ) Tor g s+y (f s + X) ) = is then Ḡ-skew, so that f s+x(g s + Y ) = f s+x(g s + Y ) A f s+x(g s + Y ) still respects Ḡ and is now torsion free. In detail we get Tor f s+x(g s + Y ) = a s X, Y s + a s a fy A f s+x(g s + Y ) = a s X, Y s a s a gx, so that is the Levi-Civita connection of Ḡ: f s+x(g s + Y ) = ( f.g s + dg(x) a s X, Y ) s + a s a (fy + gx) + fy s + S XY. For the curvature computation we assume from now on that all vector fields of the form f s + X have f constant and X = X(ϕ) so that in this case f s+x(g s + Y ) = a s X, Y s + a s a (fy + gx) + S XY, [f s + X, g s + Y ] = [X, Y ] S, in order to obtain ( a s f s+x g s+y (h s + Z) = f s+x Y, Z s + a s a (gz + hy ) + S Y Z ) ( = f a ss Y, Z a s S XY, Z a s Y, S XZ a s 4a g X, Z a s 4a h X, Y a s X, S Y Z + a s 4a fgz + a s 4a fhy + a s a f S Y Z a s Y, Z X 4a + ( a s a )s fgz + ( a s a )s fhy + a s a g S XZ + a s a h S XY + S X S Y Z [f s+x,g s+y ](h s + Z) = [X,Y ] S (h s + Z) = + a s [X, Y ]S, Z s a s a h[x, Y ]S S [X,Y ] S Z Summing up we obtain the curvature (for general vector fields, since curvature is of tensorial character) R(f s + X, g s + Y )(h s + Z) = = ( a ss a ) s gx fy, Z s + R S (X, Y )Z 4a (( a s a )s + a ) s a ( ) h(gx fy ) + s X, Z Y Y, Z X. 4a 4a ) s

12 MARTINS BRUVERIS, PETER W. MICHOR Figure. Fixing ϕ(), ϕ t () with ϕ t () =, the figure shows geodesics r(t).ϕ(t) starting at r() = for various choices of r t (); the geodesics are shown in the orthonormal basis {ϕ(), ϕ t ()}. A periodic geodesic can be seen on the right. The coefficients in the metric are C (λ) = λ and C (λ) =. and the numerator for sectional curvature Ḡ ( R(f s + X, g s + Y )(g s + Y ), f s + X ) = a R S (X, Y )Y, X ( a ss a ) s a ( gx fy, gx fy + s X, Y Y, Y X, X ). 4a 4 Let us take X, Y T ϕ S with X, Y = and X, X = Y, Y = /a(s), then Sec (s,ϕ) (span(x, Y )) = a a s 4a, Sec (s,ϕ)(span( s, Y )) = a ss a + a s 4a, are all the possible sectional curvatures. Compare this with the formulae for the principal curvatures of a hypersurface of revolution in [6] and with the formulas for rotationally symmetric Riemannian metrics in [3, Sect. 3..3].. Example. The simplest case is the choice C (λ) = λ and C (λ) =. The Riemannian metric is G µ (α, β) = α β µ(m) M µ µ µ. Then g (r) = 4 and g (r) = 4 r. This metric is geodesically complete on C (M, R)\ {}. The geodesic equation (d) simplifies to r tt = r t r. This ODE can be solved explicitely and the solution is given by ( ) rt () r(t) = r() exp r() t. The reparamterization map is α(t) = t and thus the geodesic ϕ(t) = cos ( ϕ t () t) + sin ( ϕ t () t) ϕ t () ϕ t (), describes a great circle on the sphere with the standard parametrization. Note that geodesics with r t () = are closed with period π/ ϕ t (). The spiraling behaviour of the geodesics can be seen in Fig..

13 GEOMETRY OF THE FISHER RAO METRIC Figure 3. Fixing ϕ(), ϕ t () with ϕ t () =, the figure shows geodesics r(t).ϕ(t) for various choices of r t (); on the left the extended Fisher Rao metric with C = C = with geodesics starting from r() = ; on the right the metric with C = r with geodesics starting from r() =... Example. By setting C (λ) = and C (λ) = we obtain the Fisher Rao metric on the space of all densities. The Riemannian metric is α β G µ (α, β) = µ µ µ. In this case g (r) = 4r and g (r) = 4. The metric is incomplete towards on C (M, R) \ {}. The pullback metric (b) is M G = 4r dϕ, dϕ + 4dr, and hence geodesics are straight lines in C (M, R) \ {}. In terms of the variables (r, ϕ), the geodesic equation (d) for r is with A = 4r() ϕ t (). r tt = A 6 r 3,. Example. Setting C (λ) = and C (λ) = we obtain the extended metric α β G µ (α, β) = µ µ µ + α β. M In this case g (r) = 4r and g (r) = 4r + 4. The geodesic equation (d) is M r tt = A 6r 4 r t 6r 3 ( + r ). The metric on C (M, R) \ {} is incomplete towards. Geodesics for the metric can be seen in Fig. 3. Note that only the geodesic going straight into the origin seems to be incomplete. M 3. Example. Setting C (λ) = λ and C (λ) = we obtain the metric α β G µ (α, β) = µ(m) µ µ µ, M

14 4 MARTINS BRUVERIS, PETER W. MICHOR which is complete towards, but incomplete towards infinity on C (M, R) \ {}. We have g (r) = 4/r and g (r) = 4/r. The geodesic equation (d) is r tt = r t A r 6. 6r Examples of geodesics can be seen in Fig. 3. Note that the geodesic ball extends more towards infinity than towards the origin. References [] D. Alekseevsky, A. Kriegl, M. Losik, and P. W. Michor. The Riemannian geometry of orbit spaces the metric, geodesics, and integrable systems. Publ. Math. Debrecen, 6(3-4):47 76, 3. [] S.-I. Amari. Differential-Geometrical Methods in Statistics, volume 8 of Lecture Notes in Statistics. Springer-Verlag, New York, 985. [3] N. Ay, J. Jost, H. V. Lê, and L. Schwachhöfer. Information geometry and sufficient statistics. Probab. Theory Related Fields, 6(-):37 364, 5. [4] M. Bauer, M. Bruveris, and P. W. Michor. Uniqueness of the Fisher Rao metric on the space of smooth densities. Bull. London Math. Soc., 48(3):499 56, 6. [5] B.-Y. Chen. Pseudo-Riemannian Geometry, δ-invariants and Applications. World Scientific, Singapore,. [6] V. Coll and M. Harrison. Hypersurfaces of revolution with proportional principal curvatures. Adv. Geom., 3(3): , 3. [7] T. Friedrich. Die Fisher-Information und symplektische Strukturen. Math. Nachr., 53:73 96, 99. [8] B. Khesin, J. Lenells, G. Misio lek, and S. C. Preston. Geometry of diffeomorphism groups, complete integrability and geometric statistics. Geom. Funct. Anal., 3(): , 3. [9] A. Kriegl, M. Losik, and P. W. Michor. Choosing roots of polynomials smoothly. II. Israel J. Math., 39:83 88, 4. [] S. Lang. Fundamentals of Differential Geometry, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 999. [] P. W. Michor. Topics in Differential Geometry, volume 93 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 8. [] B. O Neill. Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York, 983. [3] P. Petersen. Riemannian geometry, volume 7 of Graduate Texts in Mathematics. Springer, New York, second edition, 6. [4] N. N. Čencov. Statistical Decision Rules and Optimal Inference, volume 53 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 98. Translation from the Russian edited by Lev J. Leifman. Martins Bruveris: Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, United Kingdom Peter W. Michor: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern- Platz, A-9 Wien, Austria. address: martins.bruveris@brunel.ac.uk address: peter.michor@univie.ac.at