Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

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Ann. I. H. Poincaré – AN 30 (2013) 251–274

www.elsevier.com/locate/anihpc

Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

Brian Clarke

^∗

, Yanir A. Rubinstein

Department of Mathematics, Stanford University, Stanford, CA 94305, USA Received 26 April 2011; received in revised form 3 July 2012; accepted 18 July 2012

Available online 21 July 2012

Abstract

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L² metric.

Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi’s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.

1. Introduction

LetMbe ann-dimensional compact closed manifold, and consider the infinite-dimensional spaceMof all smooth Riemannian metrics on M. The spaceMis endowed with a naturalL²-type Riemannian structure, the Ebin metric[14],

gE(h, k)|g:=(h, k)E:=

M

tr

g⁻¹hg⁻¹k

dVg, (1)

whereg∈M,h, k∈TgM,TgMmay be identified with the spaceΓ (S²T^∗M)of smooth symmetric(0,2)-tensor fields on M, andg⁻¹h represents the(1,1)-tensor dual to hwith respect to g. This metric has received much at- tention since being introduced in the 1960s [14], see, e.g., [17,18,7,11], and has found various applications, for example in the Weil–Petersson geometry of moduli spaces of Riemann surfaces[16,25]and in the study of the moduli spaceM/Diff(M)of Riemannian structures (e.g.,[14,15,3]). A related pseudo-Riemannian metric, the DeWitt metric [13,23], has been used in the Hamiltonian formulation of general relativity.

Recently, the metric completion of ME of (M, gE) has been determined [9], and it was shown by means of examples that convergence inMEis too weak to control any geometric quantities or to imply geometric convergence of any sort (e.g., Gromov–Hausdorff convergence)[10]. Therefore, it seems natural to look for other metrics onM

* Corresponding author.

E-mail addresses:bfclarke@stanford.edu(B. Clarke),yanir@member.ams.org(Y.A. Rubinstein).

http://dx.doi.org/10.1016/j.anihpc.2012.07.003

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with the property that their metric completions are strictly contained inME. In other words, metrics for which certain types of degenerations are excluded along convergent sequences. One purpose of this article is to take a first step in this direction by studying conformal deformations of the Ebin metric in the search for metrics with this and other distinguished properties.

Our first observation (Proposition3.1) is that there is a distinguished metric in the conformal class characterized by the property that the tautological vector fieldX|g=gonMis parallel. This metric, which we call thegeneralized Calabi metric(or sometimes thenormalized Ebin metric), is given by

gN:= 1 Vg

gE, g∈M,

whereV_g :=Vol(M, g)is the volume function on M. We then restrict attention to conformal factors that depend on the volume, i.e., metrics on M of the form e^{2f (V}^g⁾gE, with f a smooth function onR>0, and mostly to the metricsgp:=gE/V^p, which serve as the basic models within this family, as they capture the possible degenerations of manifolds in terms of either volume collapse or blow-up. By studying this family of metrics, we then show thatgN

has the smallest metric completion (Theorem5.3), and in particular one that is smaller than that of the Ebin metric.

This provides the first example of anL²-type metric onMwhose metric completion is strictly smaller than that of the Ebin metric.

An additional motivation for introducting g_N comes from the study of the subspace of Kähler metrics H⊂M in a fixed Kähler class (whenM admits a Kähler structure). In our previous work[12], we studied the intrinsic and extrinsic geometry ofHinM. We observed that the Ebin metric induces the so-called Calabi geometry onH, and that this embedding is essentially as far from being totally geodesic as possible. It then seems natural to ask whether there exists a metric onMthat still induces the Calabi geometry onHbut with the property thatHis totally geodesic.

As before, it is natural to restrict to conformal deformations depending on the volume, this time since the volume is an invariant of the Kähler class, and so any such metric will induce the Calabi geometry onH. We then show that to the extent possible,gN is the unique metric with the aforementioned property. In particular,His totally geodesic in the case thatMis a Riemann surface. In generalHis not totally geodesic, but by the Calabi–Yau Theorem it is isometric to the “Riemannian Kähler spaces”Pg∩Mv, consisting of metrics of fixed volume in a fixed conformal class, which are totally geodesic in(M, g_N)(Corollary4.4).

One further possible application of the metricg_Nis to the Ricci flow. Recently, we showed that in the Kähler setting there is a connection between the existence of Einstein metrics, the smooth convergence of the normalized Ricci flow, and the metric geometry of(M, gE). Namely, a Kähler–Einstein metric exists on a Fano manifold if and only if the Kähler–Ricci flow converges in the metric completion of(M, gE), and in particular if and only if the flow path has finite length[12, Theorem 6.3, Corollary 6.9]. It would be very interesting to find analogous results for other classes of Riemannian manifolds, perhaps ones for which the singularities of the Ricci flow can be understood fairly well. In studying this problem, it might prove useful to use the metricgN, for which the submanifoldMv⊂Mof metrics of fixed volumev—which is preserved by the normalized Ricci flow—is also totally geodesic (Corollary3.3).

Motivated by these and possible other applications of the metricg_Nto geometric problems, we thus study the geometry of(M, g_N)in detail. Under the conformal change, the geodesic equation becomes substantially more difficult since it contains non-local terms involving integration over the whole manifold. The solution of the geodesic equation is obtained in several steps, building upon the work of Freed and Groisser forgE[17]. A key extra ingredient here is an invariant of thegN-geodesic flow, or a ‘constant of motion’ (Corollary3.2). The solution of the geodesic equation (Theorem 4.1) gives a precise sense to how geodesics in (M, gN)generalize those discovered by Calabi [4,5]for the subspace of Kähler metrics, which in turn bear several similarities with constrained geodesics of the Wasserstein metric in optimal transportation[6] (cf.[12]). We also compute the curvature of gN and compare it to that of the metricsg_p(Section3.2).

Finally, it should be noted that “weighted”L²type metrics were also studied by several authors on the space of simple closed curves inR²(see[20,22,24]and references therein), and this can also be seen as another motivation for our study. Moreover, very recently, while the present article was being prepared, Bauer, Harms and Michor[2]have written down the geodesic equation for metrics conformal togEonM, as well as much more general Sobolev-type metrics and metrics weighted by the scalar curvature function. Their main result is that for some of these metrics the exponential mapping is a local diffeomorphism. In this article, we go into greater depth for a smaller class of

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metrics by solving the geodesic equation, computing the curvature, estimating the distance function, and determining the metric completion.

The article is organized as follows. In Section2, we briefly review the relevant preliminaries aboutM. In Section3, we discuss general conformal changes, mostly focusing on those involving functions of the volume. For the model metrics g_p we compute the curvature as well as find an invariant (or ‘constant of motion’) of the geodesic flow.

Section 4 contains the solution of the initial value problem for gN-geodesics, making use of the invariant of the geodesic flow. In Section5, we study the distance functions ofg_pand determine their metric completions. Some of the technical facts needed in this analysis are proven in Appendix A. Section6concludes with some further remarks and a few open questions.

2. Preliminaries

Since the preliminaries relevant to our results are covered in detail in[14,17,18], we will simply briefly summarize what we need in this section.

The manifold of metrics,M, is easily seen to be an open cone in the Fréchet spaceΓ (S²T^∗M)of smooth, symmetric (0,2)-tensors on the finite-dimensional, compact manifoldM. As such, it is endowed with the structure of a Fréchet manifold (cf.[19]for background on Fréchet manifolds), and its tangent space atg∈Mis canonically identified withΓ (S²T^∗M).

The Ebin metric, defined in (1), is a smooth Riemannian metric. It is, however, a weak metric, meaning that the tangent spaces ofMare incomplete with respect to the scalar product induced on them by the Ebin metric. For weak Riemannian metrics, the existence of the Levi-Civita connection is not guaranteed by general results. Nevertheless, the Ebin metric has a Levi-Civita connection which can be directly computed. Geodesics and curvature may also be directly computed. The Riemannian curvature of(M, gE)is nonpositive, and the exponential mapping at any point g∈Mis a real-analytic diffeomorphism from an open neighborhood of zero inT_gMto an open neighborhood ofg inM. (Both of these neighborhoods are taken in theC^∞topology.)

With respect togE, we may orthogonally decompose the tangent spaceT_gMinto the subspaces of traceless (sat- isfying tr(g⁻¹h)=0) and pure-trace (satisfyingh=ρg for someρ∈C^∞(M)) tensor fields. Corresponding to this decomposition is a product manifold structure forM. Denote byV the space of all smooth, positive volume forms onM; it is an open cone inΩⁿ(M), the space of smooth top-degree forms. For anyg∈Mwe denote bydVg its induced volume form. Then for anyμ∈V, withMμ:= {g∈M: dVg=μ} ⊂M, there is a diffeomorphism

i_μ:V×Mμ→M, i_μ(ν, h)=(ν/μ)^2/nh. (2)

That is,i_μmaps(ν, h)to the unique metric conformal tohwith volume formν. ThusM∼=V×Mμ, and one sees that (i_μ)_∗(TV)is the subbundle ofTMconsisting of pure-trace tensor fields, while a tangent space to the submanifold Mμis identified with the subspace of traceless tensor fields.

An identity that will be repeatedly used below is that the differential of the mapg →dVgish → ¹₂tr(g⁻¹h) dVg. Therefore, if we denote byV =Vg:=

MdVg the volume function onM, then the differential ofg →Vg ish →

1 2(g, h)E.

For the remainder of the paper, all differentiability properties we refer to are implicitly meant to hold in the category of Fréchet manifolds; we again refer to[19]for background.

3. Conformal deformations of the Ebin metric

Letf:M→Rbe a twice continuously differentiable function, and consider the metric onM, gf(h, k)|g:=e^{2f (g)}gE(h, k)|g=e^{2f (g)}

M

tr

g⁻¹hg⁻¹k

dVg, h, k∈TgM, (3)

conformal to the Ebin metric. The purpose of this section is to characterize two metrics in the conformal class ofgE. One metric, the generalized Calabi metricgN=gE/V, is characterized by its Levi-Civita connection (Proposition3.1), and the other, the second Ebin metricg₂=gE/V², by its curvature tensor (Proposition3.6). We then restrict to the model metrics

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g_p:= 1 Vg^p

gE, g∈M, (4)

for some integerp. We find invariants for their geodesic flows that will be important later in integrating the geodesic equation (Corollary3.2and Lemma3.4), compute their curvature (Section3.2), and describe a natural duality map on Mthat is a conformal isometry betweengp andg2−p (Proposition3.8) and that also conformally relates their curvature tensors.

3.1. Conformal deformations and the Levi-Civita connection

Our first observation is a characterization of thegeneralized Calabi metric gN:=g1=gE/Vg, g∈M.

Proposition 3.1.Letf :M→Rbe a differentiable function. Let∇^f denote the Levi-Civita connection ofe^2fg_E, and suppose that∇^fg=0, wheregdenotes the tautological vector fieldg →gonM. Thenf (g)= −¹₂logV_g+C for some constantC.

Proof. First, note that∇^g^Eg=ⁿ₄δ, whereδis the Kronecker tensor. This follows from the formula for the Levi-Civita connection ofg_E[14, (4.1)],

∇_h^g^Ek|g=D_hk−1 2

hg⁻¹k+kg⁻¹h +1

4 tr

g⁻¹k h+tr

g⁻¹h k−tr

g⁻¹hg⁻¹k g

,

wherehandkare any vector fields onMandD_hk|g=_dt^d|t=0k(g+t h).

Next, recall that[1, p. 58],

∇_h^fk= ∇_h^g^Ek+(∇hf )k+(∇kf )h−(h, k)E∇^g^Ef, (5) so∇g=0 is equivalent to

0=n

4h+df (h)g+df (g)h−(h, g)E∇^g^Ef, for allh.

Plugging in h=g shows that ∇^g^Ef is proportional to g; and by inspecting the equation again then necessarily

−ⁿ₄ =df (g)=D_gf = ∇g^gÊf anddF (h)g=(h, g)_E∇^gÊf. Combining these two equations yields∇^gÊf = −_4V¹g and substituting this back into the second equation yieldsdf (h)= −¹₄(g, h)_N. Now consider a path{g(t)}. Then

d dtf

g(t)

= − 1 4V

M

tr

g(t)⁻¹gt

dVg(t)= −1 2

d

dtlogVg(t), hencef (g)= −¹₂logV_g+C(asMis path connected), as desired. 2

Since, by the proof above, g is the gradient vector field of 2 logV with respect tog_N, we have the following corollary, which will prove crucial in integrating the geodesic equation forg_Nin Section4.

Corollary 3.2.The Hessian oflogV satisfies

∇^g^NdlogV=0.

In particular,logV is linear andV is either strictly monotone or constant alongg_N-geodesics.

By the above corollary, ifg(t)is agN-geodesic and(V_g(t))_t(0)=0, thenV_g(t)is constant. This gives the following fact.

Corollary 3.3.For anyv∈R₊, the submanifoldMv:= {g∈M: V_g=v}is totally geodesic in(M, gN).

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As Corollary3.5below will imply, the above statement is true forg_ponly whenp=1. In particular, it is false for the Ebin metric.

By using the Koszul formula (or else by using (5) and the known expression for∇^g^E), one can directly compute the Levi-Civita connection ofg_Nfor constant vector fieldsh,kto be

∇_h^g^Nk|g=1

4gN(k, h)g−1 4tr

g⁻¹hg⁻¹k g−1

2hg⁻¹k−1 2kg⁻¹h +1

4tr g⁻¹h

k+1 4tr

g⁻¹k h−1

4gN(h, g)k−1

4gN(k, g)h. (6)

It is torsion free (symmetric inhandk), and one checks directly that it is metric compatible, hence it is the Levi-Civita connection.

It is well known that alonggE-geodesics the volume is quadratic[17]. This is explained by the following lemma and corollary, which are in a similar vein to Corollary3.2.

Lemma 3.4.We have

∇^g^pdV¹⁻^p=n

8(1−p)²g_p.

Proof. By (5) and (6) for constant vector fieldsh,k,

∇_h^g^pk|g=p

4(k, h)_Ng−1 4tr

g⁻¹hg⁻¹k g−1

2hg⁻¹k−1 2kg⁻¹h +1

4tr g⁻¹h

k+1 4tr

g⁻¹k h−p

4(h, g)_Nk−p²

4 (k, g)_Nh.

Thus,

∇^g^pdV¹⁻^p(h, k)=1

2∇_h^g^p(1−p)V⁻^p(g, k)_E−1

2(1−p)V⁻^p

g,∇_h^g^pk

E

= −p(1−p)

4V^p⁺¹ (g, h)_E(g, k)_E−1−p

2V^p (h, k)_E+1−p 4V^p

g,tr g⁻¹h

k

E

−np(1−p)

8V^p (h, k)_E+n(1−p)

8V^p (h, k)_E+1−p 2V^p (h, k)_E

−1−p 4V^p

g,tr g⁻¹h

k

E+p(1−p)

4V^p⁺¹ (g, h)E(g, k)E

=n

8(1−p)²g_p(h, k). 2

Corollary 3.5.Along unit-speedgp-geodesicsg(t),V¹⁻^pgrows quadratically(p=1), V¹⁻^p(t)= n

16(1−p)²t²+1−p

2 a₀t+V¹⁻^p(0), (7)

where

a₀:= 1 V_g^p₀

M

f (0) dVg₀, f (t ):=tr g⁻¹g_t

.

In particular,Vg(t)converges to0 (ifp <1)or to∞(ifp >1)in finite time precisely along constant conformal directions, i.e., ifgt(0)=λg(0)forλ∈Randλnegative(ifp <1)or positive(ifp >1). Also, along a unit-speed g_p-geodesic(for anyp∈R),

d dt

1 V^p

M

f dV_g

=n

4(1−p). (8)

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Proof. Letg(t)be a unit-speedg_p-geodesic. By the previous lemma, we have that d²

dt²V¹⁻^p(t)= d

dtdV¹⁻^p g_t(t)

= ∇gt(t)dV¹⁻^p g_t(t)

=n

8(1−p)².

This proves thatV¹⁻^pgrows quadratically, and (7) follows from the observation that _dt^dV¹⁻^p(0)=¹⁻₂^pa₀.

By the quadratic formula applied to (7), the equation V¹⁻^p(t0)=0 has a real solution t0 if and only if a₀²− nV¹⁻^p(0)0. Furthermore, by the Cauchy–Schwarz inequality for the Hilbert–Schmidt scalar product on symmetric matrices, f (t )²=tr(g⁻¹g_t)²tr((g⁻¹g_t)²)tr(I )=ntr((g⁻¹g_t)²), with equality if and only if g_t =λg for some λ∈R. Therefore, again using Cauchy–Schwarz,

a₀²= 1

V_g^p₀

M

f (0) dVg0

2

V_g¹₀⁻^p 1

V_g^p₀

M

f (0)²dV_g₀

V_g¹₀⁻^p n

V_g^p₀

M

tr

g(0)⁻¹g_t(0)2 dV_g₀

=nV_g¹₀⁻^p,

where the last equality holds becauseg_p(g_t(0), gt(0))=1 by assumption. Note that equality holds in the above if and only ifg_t(0)is a constant scalar multiple ofg(0). Thus this is the only condition under whicha₀²−nV¹⁻^p(0)0, and the statement aboutV_g(t)converging to 0 or∞in finite time follows.

Finally, to prove (8), note that_dt^dV¹⁻^p(t)=¹⁻₂^p·_Vp¹(t)

Mf (t ) dV_g(t). The result then follows from differentiating this equation and applying the fact, shown above, that _dt^d²2V¹⁻^p(t)=ⁿ₈(1−p)². 2

Eq. (8) in the corollary above provides an integral for the geodesic flow ofgp which allows solving the geodesic equation explicitly, in the spirit of the work of Freed–Groisser. This will be carried out forg_N using Corollary3.2in Section4.

3.2. Conformal deformations and curvature

Our main purpose in this subsection is to study which conformal deformations ofgEstill have nonpositive curvature. The next result shows that there is precisely one metric of the formg_E/V^p, besidesg_Eitself, whose curvature is nonpositive—it is also the unique such metric with curvature conformal to that ofg_E—and this characterizes the second Ebin metric

g2=gE/V²,

among all conformal deformations that depend on the volume.

Proposition 3.6.The curvature ofgpis nonpositive if and only ifp=0or2. Moreover, the curvature ofg2=gE/V² is conformal to the curvature ofg_E,

R^g²=R^g^E/V²,

and this property characterizesg₂, up to scaling, among all conformal deformationse^2fg_Ewithf :M→Ra smooth function depending only onV_g.

In the proof we make use of the following computation:

Proposition 3.7.The curvature tensor ofg_pis given by R^g^p= 1

V^pR^g^E+2p−p² 16 g_p∧

V^2p⁻²g^p⊗g^p−n

2V^p⁻¹g_p

, (9)

whereg^pis the1-form dual to the tautological vector fieldgwith respect tog_pand∧ denotes the Kulkarni–Nomizu product.

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Leth,kbe tangent vectors that are orthonormal with respect tog_p. The sectional curvature of the planeR{h, k}is sec^g^p(h, k)= 1

V^psec^g^E(h, k)−2p−p² 16

V^2p⁻²(g, k)²_p+V^2p⁻²(g, h)²_p−nV^p⁻¹

. (10)

Proof. The formula for the curvature under the conformal changegE →e^2fgEis[1, p. 58], R= 1

V

R^g^E+gE∧

∇^g^Edf−df⊗df+1

2|df|²EgE

, (11)

and this applies in infinite dimensions as can be verified from its proof. (Note that our convention isR(X, Y )Z= (∇X∇Y − ∇Y∇X− ∇_[X,Y])Z, the opposite of Besse’s.) Let f (g):= −¹₂logV_g andf_p(g):=pf (g)= −^p₂logV_g. Assume first thatp=1. We claim that

∇^g^Edf=1

8(g,·)N(g,·)N− n

16gN=2df⊗df− n

16gN. (12)

To see this, compute using the formula∇_h^g^Eg=ⁿ₄h(cf. the proof of Proposition3.1) and the metric property of∇^g^E to deduce

∇^g^Edf (h, k)=

∇_h^g^Edf

(k)= ∇_h^g^E df (k)

−df

∇_h^g^Ek

= −1 4∇_h^g^E

1 V(g, k)_E

+1

4

g,∇_h^g^Ek

N

=1

8(g, h)N(g, k)N− n

16(h, k)N−1 4

g,∇_h^g^Ek

N+1 4

g,∇_h^g^Ek

N

=1

8(g, h)N(g, k)N− n

16(h, k)N. (13)

Second, note thatdf = −¹₄g¹,|df|²N= |∇f|²N=₁₆¹|g|²N=₁₆ⁿ, and|df|²Ng_N= |df|²Eg_E. Thus, R^g^N= 1

VR^g^E+ 1 16gN∧

g¹⊗g¹−n 2gN

. (14)

To conclude the proof, note now that∇^g^Edf_p=p∇^g^Edf =2p df⊗df −^pn₁₆gN,df_p⊗df_p=p²df⊗df, and

1

2|df_p|²Eg_E=^p₂²|df|²Eg_E=^p₁₆²ⁿg_N. From (11) we thus obtain R^g^p= 1

V^pR^g^E+

2p−p² gp∧

df⊗df− n

32V^p⁻¹gp

.

Sincedf = −¹₄(g,·)N= −¹₄g¹= −¹₄V^p⁻¹g^p, (9) follows.

Next, recall that

G∧H (a, b, c, d)=G(a, c)H (b, d)+G(b, d)H (a, c)−G(a, d)H (b, c)−G(b, c)H (a, d).

So ifhandkareg_p-orthonormal,g_p∧g_p(h, k, k, h)=2(h, k)²_p−2|h|²p|k|²p= −2, and gp∧

g^p⊗g^p

(h, k, k, h)=2(h, k)p

g^p⊗g^p (h, k)

−(h, h)_p

g^p⊗g^p

(k, k)−(k, k)_p

g^p⊗g^p (h, h)

= −(g, k)²_p−(g, h)²_p.

By definition, sec^g^p(h, k)=g_p(R^g^p(h, k)k, h), and so (10) follows. 2

Proof of Proposition 3.6. Let f =f (Vg) be a smooth function on M. Then df =fdV = ¹₂f(g,·)E, or

∇^gÊf =¹₂fg and ¹₂|∇^gÊf|²EgE=ⁿ₈(f)²V gE, whiledf ⊗df =¹₄(f)²gÊ⊗gÊ. A computation similar to (13) gives∇^gÊdf=¹₄fgÊ⊗gÊ+ⁿ₈fgE. So

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R^g^f =e^2fR^g^E+1

4e^2fgE∧ f−

f2

g^E⊗g^E+n 2

f+V f2

gE

,

and analogously to the proof of (10), we may compute sec^g^f(h, k)=e^2fsec^g^E(h, k)−f−(f)²

4e^4f

(g, k)²_f+(g, h)²_f

− n 4e^2f

f+V f2

.

Suppose now that sec^g^f =e^2fsec^g^E. Then considering directionsh,ktangent to Mμgives that f+V (f)²=0, from which it follows that eitherf= −logV +C, i.e.,gf =e^2CgE/V², or elsef=0, i.e.,gf =e^2CgE. 2 3.3. Conformal transformations and a duality map

By Proposition3.7,g₂₋_p andg_phave the same curvature tensor, up to a conformal factor. Here we observe that there is also a conformal diffeomorphismF :M→Mthat relates these two metrics, so they are in fact isometric, and in this senseg₂does not provide a new geometry compared tog_E.

Consider the mapF:M→Mdefined byF (g):=V^qg. Leth∈TMbe a constant vector field. Then dF (h)= d

dt

t=0

(g+t h)V_g^q₊_{t h}=V^qh+1

2q(g, h)_EV^q⁻¹g.

Hence, a careful computation shows that g_pdF (h), dF (k)

F (g)=g_p₊ⁿ

2q(p−1)(h, k)+ n

4q²+q

V⁽¹⁻^p)(1⁺ⁿ²^q)⁻²(g, h)E(g, k)E.

To summarize, we have:

Proposition 3.8. The diffeomorphism F (g)=V⁻⁴ⁿg of M is an isometry between the spaces (M, g₂₋_p) and (M, g_p), and we haveV_{F (g)}=V_g⁻¹andF⁻¹=F. In particular,(M, g2)and(M, gE)are isometric.

It is interesting to note that using this result one obtains rather effortlessly the solution of the geodesic equation for g₂, building on the much simpler one forgE([18, Thm. 3.2],[17, Thm. 2.3]). In fact, a direct solution of the g₂-geodesic equation using the fact that the inverse of the volume is quadratic (Corollary3.5) is substantially more involved.

Remark 3.9.Ifφis a positive differentiable function, andF (g):=φ(V_g)g, then dF (h), dF (k)

p

F (g)=φ^n/2(V_{F (g)})

V^p (h, k)E|g+V_{F (g)}(g, h)E(g, k)E

φ^n/2φ φ

nφ 4φV +1

.

Hence, the only such mapF that is an isometry betweengEand anyg_pis given by Proposition3.8.

4. Geometry of the generalized Calabi metric

In this section, we study the geometry of the metricgNin more detail. In the first subsection we solve its geodesic equation for any given initial data. In the second subsection, we compute the sectional curvature, and examine the extrinsic geometry of certain submanifolds in the spirit of[12], showing that the Riemannian analogues of the space of Kähler metrics are totally geodesic. These spaces are naturally isometric (via the Calabi–Yau Theorem) to the usual spaces of Kähler metrics. These facts, together with the explicit formula for geodesics, give a precise meaning to the statement thatg_Ngeneralizes Calabi’s geometry on the space of Kähler metrics.

4.1. Geodesics

From (6) we obtain the geodesic equation for(M, g_N), g⁻¹g_t

t=1 4tr

g⁻¹g_tg⁻¹g_t δ−1

2tr g⁻¹g_t

g⁻¹g_t+1

2(g_t, g)_Ng⁻¹g_t−1

4|g_t|²Nδ, (15)

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whereδ denotes the Kronecker tensor corresponding to the identity matrix. The last two terms are the new terms compared to the geodesic equation forgE. Since they are non-local, the solution of the equation becomes substantially more involved and requires making use the ‘constant of motion’ of the geodesic flow found in (8).

The solution of the initial value problem for the geodesic equation is given by the following theorem.

Theorem 4.1.Letg(0)∈M, and letμ₀:=dV_g(0). Then the geodesic in(M, gN)emanating fromg(0), with initial tangent vector(α, A)∈T_μ₀Vol(M)×T_g(0)Mμ0, is given by the following.

Defineσ:= |(α, A)|Nand

a₀:= 2 V (0)

M

α, b₀:= nσ²−a²₀

4 , q:= α

μ₀−a₀

2, r:=

n 4tr

g(0)⁻¹A2 .

First, ifb0=0, theng(t, x)=e^{t σ}^√ⁿg(0, x).

Ifb₀=0, then for eachx∈M, g(t, x)=

1 2

1−q²+r² b₀²

cos(b0t )+ q b0

sin(b0t )+1 2

1+q²+r² b²₀

²

n

·e^a⁰^{t /n}g(0)exp 2

rtan⁻¹

rsin(b0t )

b₀+b₀cos(b0t )+qsin(b0t )

g(0)⁻¹A

. (16)

Here, we take the exponential term to be the identity ifA(x)=0.

IfA(x)=0, then in(16), arctangent takes values in[π k−^π₂, π k+^π₂]whenevert∈ [^2π(k⁻_b₀¹⁾⁺^θ,^{2π k}_b⁺^θ

0 ]fork∈Z, where

θ (x):=

⎧⎪

⎨

⎪⎩

2π−cos⁻¹(^q(x)

2−b₀²

q(x)²+b₀²) ifq(x)0, cos⁻¹(^q(x)

2−b²₀

q(x)²+b²₀) ifq(x) <0.

(Here, arccosine takes values in[0, π].)

The domain of definition ofg(t)is[0,∞)ifb₀=0. Ifb₀=0, the domain of definition is[0, t0), wheret₀ is the infimum ofθ (x)at points whereA(x)=0.(We take the infimum to be∞if there are no such points.)In the case where the geodesic exists for only finite time, it approaches a limit point on the boundary ofM⊂Γ (S²T^∗M)ast→t₀; i.e.,μ_g(t)(x)→0for at least one pointx∈M.

Remark 4.2.Theorem4.1gives a precise meaning to the statement thatgN generalizes Calabi’s geometry on the space of Kähler metrics. Indeed, on the level of volume forms, Calabi’s geodesics in the space of Kähler metrics (or, via the Calabi–Yau Theorem, on the space of volume forms with total volumev) are given by

dVg(t)=dVg

G√

vsin 1

2t /√ v

+cos

1 2t /√

v 2

,

where(dV_g(0))_t =G dV_g(0) [12, Remark 5.7]. On the other hand, in proving (16) one shows that the volume forms alonggN-geodesics satisfy an equation of a similar form—see (28)—and the two equations can actually be shown to exactly coincide whenA≡0 anda0=0 by using trigonometric formulas and carefully identifying the integration constants.

Before we give the proof of this theorem, let us point out a contrast to the Ebin metric. Like the case ofgE(cf.[17,

§2]), geodesics in(M, gN)exist for all time ifA(x)=0 for allx∈M. However, the converse of this statement also holds—ifA(x)=0 for somex∈M, then the geodesic only exists for finite time—unlessb0=0. (In the case ofgE, this happens only when there is a point whereA(x)=0 and(α/μ₀)(x) <0.)

Note also that, as in the case of the Ebin metric, any conformal class—a submanifold of the formPgwithg∈M—

is totally geodesic, as can be seen from Theorem4.1by puttingA≡0.

We will solve the geodesic equation in the following subsections, beginning with general considerations and then considering various special cases.

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4.1.1. The general case

We letC:=g⁻¹g_t and decompose into pure trace and traceless parts:C=:E+^f_nI, with trE=0, i.e.,f =trC.

From (15), we obtain the pair of coupled equations Et= −1

2f E+ E 2V

M

f dVg, (17)

and

f_t=n 4tr

E²

−f² 4 −nσ²

4 + f

2V

M

f dV_g, (18)

whereσ= |gt|N, which is constant sincegis a geodesic. The last term in the first equation and the last two terms in the second equation are new compared to the unnormalized metric.

The following relations hold betweenE,f, and data related to the splittingM∼=V×Mμ₀, whereμ₀:=dV_g₀ (cf. Section2). We writeg=(^μ_μ^g

0)^2/nh, whereμ_g=dV_g andh∈Mμ₀, i.e.,his the unique metric conformal tog withdV_h=μ₀. Theng⁻¹g_t=h⁻¹h_t+²_n^(μ_μ^g_g⁾^tI, implying thatE=h⁻¹h_t andf=2^(μ_μ^g⁾^t

g . We define

φ:=f− 1 V

M

f dV_g.

Note thatV⁻¹

Mf dV_g=2_dt^d(logV_g(t)). Hence, by Corollary3.2, this quantity is constant alongg(t). So defining a0:= 1

V (0)

M

f (0) dVg(0), we haveφ=f −a₀andφ_t=f_t.

Now, note that

−f² 4 + f

2V

M

f= −1 4φ²+1

4 1

V

M

f 2

. (19)

Using this, together with the considerations of the previous paragraph, we can rewrite (17)–(18) in terms ofφ, E_t= −φ

2E, (20)

φ_t=n 4tr

E²

−nσ² 4 −φ²

4 +a₀²

4 . (21)

Note that tr

E²

t =2 tr(EtE)=

V⁻¹

M

f dV_g−f

tr E²

= −φtr E² (so tr(E²)=exp(_t

0(V⁻¹

Mf dV_g−f ) ds)tr(E²(0))). Hence, differentiating (21) yields φ_{t t}= −n

4φtr E²

−1 2φφ_t,

and substituting for tr(E²)using (21) we obtain 4φt t+6φφt+φ³=φ

a₀²−nσ²

. (22)

We now let p:=μg

μ₀e⁻^a⁰^{t /2}. (23)

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It follows that 2pt/p=2(μg)_t/μ₀−a₀=φ. Thus, the left-hand side of (22) equals 8pt t t/p, hencepsatisfies 4pt t t−

a₀²−nσ²

pt=0. (24)

Let

b₀:= nσ²−a₀²

4 .

Note thatb₀is real-valued and positive, sinceσ= |g_t(0)|Nand so σ²= 1

V (0)

M

tr C(0)²

dV_g(0) 1 nV (0)

M

f (0)²dV_g(0)

1

n 1

V (0)

M

f (0) dVg(0)

2

=a²₀

n , (25)

where the second inequality is Cauchy–Schwarz. Note that the first inequality is an equality if and only ifE(0)≡0, and the second inequality is an equality if and only iff is constant. Therefore,b₀=0 if and only ifg_t(0)=λg(0)for someλ∈R.

Now, integrating (24), we have

p_{t t}+b²₀p=C, (26)

for someC∈R. It follows that p(t )=

C₁cos(b0t )+C₂sin(b0t )+C₃, ifb₀=0,

C₁t²+C₂t+C₃, ifb₀=0. (27)

By (23), then, μ_g μ₀ =

(C₁cos(b0t )+C₂sin(b0t )+C₃)e^a⁰^{t /2}, ifb₀=0,

(C₁t²+C₂t+C₃)e^a⁰^{t /2}, ifb₀=0. (28)

We now consider the initial value data needed to determine the constants of integration. Note that (23) implies that p(0)=1 andp_t(0)=α/μ₀−a₀/2.

To determinep_{t t}(0), we first use thatφ=2pt/pto see that on the one hand, φ_t(0)=2

pt

p

t

(0)=2pt t(0)p(0)−(pt(0))²

p(0)² =2pt t(0)−2 α

μ0−a0

2 2

.

On the other hand, we see by (21) and the fact thatf (0)=2α/μ0that p_{t t}(0)=n

4tr E(0)²

−nσ² 4 −φ²

4 +a₀² 4 +2

α μ0−a0

2 2

=n 4tr

g(0)⁻¹A2

−1 4

2 α μ₀ −a₀

2

−b²₀+2 α

μ₀−a0

2 2

=1 2

q²+r²−b²₀ ,

whereq:=α/μ₀−a₀/2 andr:=

n

4tr((g(0)⁻¹A)²).

This gives all the information needed to solve forμ_g/μ₀in the individual cases. To solve forh, we must use (20) and the fact thatφ=2pt/pto see thatE_t= −(logp)_tE, implyingE=E(0)/p=g(0)⁻¹A/p. SinceE=h⁻¹h_t, this gives

h⁻¹h_t(t)=g(0)⁻¹A/p(t ). (29)

We now give the solution of the geodesic equation for each special case.