Resume of Papers of H. Shima II

November, 18, 1999

Abstract. The purpose of this talk is to introduce the works of H. Shima, mainly concentrated on affinely flat structures.

Definition. A locally flat linear connection D on a connected smooth manifold M if its torsion and curvature tensors vanish identically. Then for each point p ∈ M, there exists a local coordinate system {x^{1}, ⋯, x^{n}} in a neighborhood of p such that D_{\frac{∂}{∂x^{i}}} \frac{∂}{∂x^{j}} = 0, which we call an affine local coordinate system. A Riemannian metric g on M is said to be locally Hessian with respect to D, if for each point p ∈ M, there exists a real valued C^{∞} function ϕ on a neighborhood of p such that

g = D^{2}ϕ,

that is,

g = ∑_{ij} \frac{∂^{2}ϕ}{∂x^{i}∂x^{j}} dx^{i}dx^{j}.

If ϕ is defined over M, the metric g is called Hessian metric on M. A locally flat manifold with a (locally) Hessian metric is called a (locally) Hessian manifold.

Let M be an affine homogeneous manifold of a connected Lie group G. Assume that M admits a locally flat linear connection D and a volume element ν which are invariant under G. If ν = Kdx^{1} ∧ ⋯ ∧ dx^{n}, then let

α = ∑_{i} \frac{∂ log K}{∂x^{i}} dx^{i},

Dα = ∑_{ij} \frac{∂^{2} log K}{∂x^{i}∂x^{j}} dx^{i}dx^{j}.

Let Ω be an affine homogeneous proper convex domain in (ℝ^{n}, D). Then there exists an invariant volume element ν = Kdx^{1} ∧ ⋯ ∧ dx^{n}, and letting α = ∑_{i} \frac{∂ log K}{∂x^{i}} dx^{i}, we obtain the canonical bilinear form

Dα = ∑_{ij} \frac{∂^{2} log K}{∂x^{i}∂x^{j}} dx^{i}dx^{j},

which defines an invariant Riemannian metric on Ω.

M flat affine manifold, {x^{1}, ⋯, x^{n}} local affine coordinate. g Riemannian metric on M is Hessian if and only if for all p ∈ M, ∃ϕ ∈ C^{∞}(U_{p}) such that g_{ij} = \frac{∂^{2}ϕ}{∂x^{i}∂x^{j}} , where ϕ is called a primitive of g on M.

[S1] H. Shima, On locally symmetric homogeneous domains of completely reducible linear Lie groups, Math. Ann. 217 (1975), 93 – 95.

Theorem. Let Ω = G/H be a locally symmetric homogeneous domain in ℝ^{n}, acted by a completely reducible linear Lie group G. Then

(i) Ω is a ω-domain and the canonical bilinear form Dα of Ω is nondegenerate.

(ii) Let (p, q) be the signature of Dα. Then there exists a p-dimensional subspace W of ℝ^{n} such that Ω^{+} = Ω ∩ W is a homogeneous self-dual cone, and q is equal to the difference between the dimension of a maximal compact subgroup of G and the dimension of a maximal compact subgroup of the identity component H_{0} of H.

(iii) If H_{0} is compact, then there exists a maximal compact subgroup N of G containing H_{0} such that Ω = NΩ^{+}. In particular, H is a maximal compact subgroup of G if and only if Ω is a homogeneous self-dual cone.

[S2]H. Shima, Symmetric spaces with invariant locally Hessian structures, J.

Math. Soc. Japan 29, No. 3 (1977), 581 – 589.

Let G be a connected Lie group, H be a closed Lie subgroup of G. (G, H) is called a symmetric pair if and only if there exists an involutive automor-

phism σ of G such that (H_{σ})_{0} ⊂ H ⊂ H_{σ}, where H_{σ} = {g ∈ G | σ(g) = g}.

If, in addition, H contains no non-trivial normal subgroup of G, then (G, H) is said to be an effective symmetric pair.

Theorem. Let (G, H) be an effective symmetric pair. If M = G/H admits a locally Hessian structure (D, g) such that D and g are invariant under G, then M is affinely diffeomorphic and isometric with respect to D and g respetively to a direct product

M_{0} × M_{1} × ⋯ × M_{r},

where M_{0} is a flat Riemannian manifold and the universal covering manifold of M_{i} (1 ≥ i ≤ r) is an irreducible homogeneous self-dual convex cone with a canonical locally Hessian structure.

Note a canonical locally Hessian structure if Dα = D^{2} log K.

Proposition. Let G be a connected Lie group, H be a closed subgroup of G (not necessarily symmetric pair). If G/H admits an invariant locally Hessian structure, then G is not semisimple.

Proposition. Under the same assumption as in Theorem, assume further that f is a faithful representation of 𝔤 in T_{o}(G/H) = ℝ^{n}. Then the universal covering manifold of G/H is an irreducible homogeneous self-dual convex cone.

[H1] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie

groups, Osaka J. Math. 13 (1976), 213 – 229.

Proposition. Let M be a connected C^{∞}− manifold with locally flat linear connection D, and a Riemannian metric g. Let γ be a cotangent bundle valued 1-form on M defined by

γ(X)(Y) = g(X, Y), for X, Y ∈ 𝔛(M).

The cotangent bundle being locally flat, we may consider the exterior differential \underline{d} for cotangent bundle valued forms on M. Then the following conditions are equivalent:

(i) g is locally Hessian with respect to D.

(ii) For each affine coordinate system {x^{1}, ⋯, x^{n}}, the components g_{ij} of g satisfy the relations

\frac{∂g_{ij}}{∂x^{k}} = \frac{∂g_{ik}}{∂x^{j}} (1 ≥ i, j, k ≤ n).

(iii) (D_{Z}g)(X, Y) = (D_{Y} g)(X, Z) for all vector fields X, Y, Z on M.

(iv) \underline{d}γ = 0.

In addition to these equivalent conditions, assume further H^{1}(M, ℝ) = {0} and that D is flat. Then g is a Hessian metric.

Koszul :

Let M be a homogeneous manifold with an invariant flat linear connection D and an invariant volume element ν. Then the bilinear form Dα is positive definite if and only if M is an affine homogeneous proper convex domain.

Theorem. Let G be a connected solvable Lie group and M an orientable smooth manifold on which G acts simply transitively. Suppose that M admits a locally flat linear connection D and a locally Hessian metric g with respect ot D, which are invariant under G. Let ν be the volume element defined by g. If the bilinear form Dα determined by ν is non-degenerate, then Dα is positive definite.

Combined with the Koszul’s result,

Corollary. Under the same assumptions as in Theorem, assume further that D is flat. Then M is an affine homogeneous proper convex domain.

Definition. A left-symmetric algebra A is called elementary if A satisfies the following conditions: (E.1) A = {u} + P (direct sum of vector spaces,

(E.2) u · u = u, u ≠ 0, (E.3) u · P ⊂ P, P · u = {0},

(E.4) p · q = Φ(p, q)u for p, q ∈ P, where Φ is a symmetric bilinear form on P.

Proposition. Let Ω be a homogeneous domain in ℝ^{n} containing 0, on which an affine Lie group G acts simply transitively. Suppose that the left-symmetric algebra A of Ω at 0 is elementary, i.e., A = {u} + P satisfies the above conditions from (E.1) to (E.4). Then we have

Ω = \{tu + p | t − \frac{1}{2}Φ(p, p) > −1 for t ∈ ℝ, p ∈ P \}.

In particular, if P hi is a positive definite symmetric bilinear form on P, then Ω is the interior of a paraboloid.

[H2]H. Shima, Homogeneous convex domains of negative sectional curvature,

J. Differential Geometry 12 (1977), 327 – 332.

Theorem. An affine homogeneous proper convex domain Ω has negative sectional curvature with respect to Dα if and only if Ω is the interior of a parabloid :

y^{0} − \frac{1}{2}\{(y^{1})^{2} + ⋯ + (y^{n−1})^{2}\} > −1,

where {y^{0}, y^{1}, ⋯, y^{n−1}} is an affine coordinate system of V.

Definition. A clan A is called elementary if A satisfies the following conditions: (E.1) A = {u} + P (direct sum of vector spaces,

(E.2) u · u = u, u ≠ 0, (E.3) u · p = \frac{1}{2}p, p · u = \{0\} for p ∈ P,

(E.4) p · q = Φ(p, q)u for p, q ∈ P, where Φ is a positive definite symmetric bilinear form on P.

Theorem. Let A be a clan. Then the following conditions are equivalent: (i) the setional curvature K < 0. (ii) A is an elementary clan.

Let Ω be an affine homogeneous proper convex domain in (ℝ^{n}, D). Let G be a connected triangular affine Lie group which acts simply transitively on Ω, Then (ℝ^{n}, ·) becomes a left-symmetric algebra and let L_{x}(y) = x · y.

Define α_{0}(x) = trL_{x} for x ∈ ℝ^{n} = T_{0}Ω and let ⟨x, y⟩ = α_{0}(x · y). Since A is a

LSA, we have

⟨xy, z⟩ − ⟨yx, z⟩ = ⟨x, yz⟩ − ⟨y, xz⟩.

The algebra A together with the linear function α_{0} is said to be a clan corresponding to Ω.

Proposition. The Riemannian connection ∇ for Dα is given by

∇_{x}y = \frac{1}{2}(L_{x} −^{t}L_{x})y,

i.e., ∇_{x} is the skew symmetric part of L_{x}.

Proposition. Let S_{x} be the symmetric part of L_{x}, i.e., S_{x}y = \frac{1}{2}(L_{x} +^{t}L_{x})y. Then we have

(i) S_{x}y = S_{y}x, i.e., ∇_{x} is the skew symmetric part of L_{x}.

(ii) the curvaturetensor R is given by

R(x, y) = −[S_{x}, S_{y}],

(iii) the sectional curvature K is given by

K(x, y) = \frac{|S_{x}y|^{2} − ⟨S_{x}x, S_{y}y⟩}{|x|^{2}|y|^{2} − ⟨x, y⟩^{2}}.

[H3] H. Shima, Compact locally Hessian manifolds, Osaka J. Math. 15 (1978),

509 – 513.

Let M be an oriented manifold with a locally flat connection D and with a Riemannian metric g such that in a neighborhood of every point, g is of the form ∂^{2}ϕ/∂x^{i}∂x^{j} where ϕ is a C^{∞} function and where x^{i} are local coordinate system satisfying D(dx^{i}) = 0.

The connection D and the Riemannain volume ν associated to g determine on M a differentiable form β of degree 2 ; locally, if ν = Kdx^{1} ∧ ... ∧ dx^{n}, β = ∑ β_{ij}dx^{i}dx^{j}, where β_{ij} = ∂^{2} log(K)/∂x^{i}∂x^{j}.

The author prove that if M is compact, then

(i) ∫_{M}(∑ β_{ii})ν ≥ 0, (ii) if this integral is zero, then g is locally flat.

[H4] H. Shima, Homogeneous (locally) Hessian manifolds, Ann. Inst. Fourier, Grenoble 30, No. 3 (1980), 91 – 128.

A differentiable manifold M provided with a locally flat linear connection D and a Riemannian metric g is said to be Hessian(in fact, locally Hessian) (and g is called a Hessian metric) if g has the form g = D^{2}ϕ, where ϕ is a function, in a neighbourhood of any point of M.

The tangent bundle T M of a differentiable manifold M provided with a locally flat linear connection and a Riemannian metric g admits in a natural way a Hermitian metric g^{T}, and g is a Hessian metric on M if and only if g^{T} is a Kahlerian metric on T M.

Thus the geometry of Hessian manifolds is related with that of certain Kahlerian manifolds.

A diffeomorphism of a Hessian manifold onto itself is called an automorphism if it preserves both the locally flat linear connection and the Hessian metric.

A Hessian manifold M is said to be homogeneous if the group of its automorphisms Aut(M) acts transitively on M.

The main result of the paper under review is the following theorem. If M is a connected homogeneous Hessian manifold then the domain of definition E_{x} for the exponential mapping exp_{x} at a point x ∈ M given by the locally flat linear connection is an affine homogeneous convex domain in the tangent space T_{x}M of M at x; moreover, E_{x} is the universal covering manifold of M with covering projection exp_{x} : E_{x} → M.

Several corollaries are deduced from the theorem. Here are some of them:

(1) If the connected Lie subgroup G of Aut(M) acts transitively on a Hessian manifold M and if the isotropy subgroup of G at a point in M is discrete, then G is a solvable Lie group.

(2) If a connected homogeneous Hessian manifold M admits a transitive reductive Lie subgroup of Aut(M), then the universal covering manifold of M is a direct product of a Euclidean space and an affine homogeneous convex self-dual cone not containing any full straight line.

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(3) A compact connected homogeneous Hessian manifold is a Euclidean torus.

[H5]H. Shima, Hessian manifolds and convexity, in manifolds, and Lie groups, papers in honor of Y. Matsushima, Progress in mathematics 14 (1981), 385 – 392.

Define a cotangent valued 1-form g^{0} by g^{0}(X)(Y) = g(X, Y) for X, Y ∈ 𝔛(M). \underline{d} : exterior differential for tensor bundle valued forms on M.

Theorem. Let M be a connected Hessian manifold with Hessian metric g. suppose g^{0} is \underline{d}-exact, that is, M admits a closed 1-form α such that g^{0} = \underline{d}α. If there exists a subgroup G of affine transformations of M such that M/G is quasi-compact and that G leaves α invariant, then the universal covering manifold of M is a proper convex domain in a real affine space.

Let M be a Hessian manifold with flat affine connection D and a Hessian metric g. Define f ∈ Aut(M) if f ∈ Diff(M), f^{∗}D = D, f^{∗}g = g.

Theprem. Let M be a connected Hessian manifold. If there exists a subgroup G of Aut(M) such that M/G is quasi-compact, then the universal covering manifold of M is a convex domain in a real affine space.

Corollary. Let M be a connected Hessian manifold. If M admits a transitive automorphism group or is compcat then the universal covering manifold of M is a convex domain in a real affine space.

Remark ([Yagi, 1981]) showed in ”On Hessian structures on an affine manifold” :

Flat affine manifold whose universal covering manifold is a convex domain in a real affine space does not necessarily admit a Hessian metric.

Let Ω be a convex domain in a real affine space ℝ^{n} and Γ a properly discontinuous subgroup of affine transformation group of ℝ^{n} acting freely on Ω. Then the quotient space Ω/Γ admits in a natural way a flat affine structure induced from that of ℝ^{n}.

Let A denote the vector space of all affine functions on ℝ^{n}. Then Γ acts on A by g · f = f ∘ g^{−1}, where g ∈ Γ, f ∈ A and this makes A into a Γ-module.

Let H^{r}(Γ, A) be the r-th cohomology group of Γ with coefficients in A.

Theorem. Suppose that Ω/Γ admits a Hessian metric g.

(i) If H^{1}(Γ, A) = 0, there exists a primitive of g globally defined on Ω/Γ. (ii) If Ω/Γ is compact, then we have H^{1}(Γ, A) ≠ 0.

Let P denote the vector space of all parallel (with respect to the natural flat affine connection on ℝ^{n}) 1-forms on ℝ^{n}. Then Γ acts on P by g · ω = (g^{−1})^{∗}ω, where g ∈ Γ, ω ∈ P and this action makes P into a Γ-module.

Let H^{r}(Γ, P) be the r-th cohomology group of Γ with coefficients in P.

Theorem. Suppose that Ω/Γ admits a Hessian metric g. If H^{1}(Γ, P) = 0, the cotangent bundle valued 1-form g^{0} corresponding to g is \underline{d}-exact. Assuming further that Ω/Γ is compact, Ω is a proper convex domain.

[H6]H. Shima, A differential geometric characterization of homogeneous self-

dual cones, Tsukuba J. Math. 6, No. 1 (1982), 79 – 88.

Let Ω be an affine homogeneous convex domain in an n- dimensional real vector space V^{n} not containing any full straight line. Then, by using the characteristic function ϕ of Ω, we can define an invariant Riemannian metric g on Ω in the following way:

g = ∑_{i,j=1}^{n} \frac{∂^{2} log ϕ}{∂x^{i}∂x^{j}}dx^{i}dx^{j},

where (x^{1}, ..., x^{n}) denotes an affine coordinate system on V^{n}.

Let ∇ and R be the Riemannian connection and the Riemannian curvature tensor for g, respectively. Denoting by Γ_{jk}^{i} the components of ∇ with respect to (x^{1}, ..., x^{n}), we can see that Γ_{jk}^{i} defines a tensor field on Ω. We denote this tensor field by the same letter ∇.

In this article, the author gives characterizations of self-dual cones among affine homogeneous convex domains as follows:

(1) A homogeneous convex cone Ω not containing any full straight line is a self-dual cone if and only if R is parallel with respect to ∇. (2) A homogeneous convex domain Ω not containing any full straight line is a self-dual cone if and only if ∇ is parallel with respect to ∇.

The result (1) above gives an affirmative answer to a problem posed by Y. Matsushima in 1965.

The same result as in (1) was also obtained by T. Tsuji independently [Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 185 - 187].

[H7]H. Shima, Vanishing theorems for compact Hessian manifolds, Ann. Inst. Fourier, Grenoble 36, 3 (1986), 183 – 205.

([Serre, 1955] duality Theorem)

Theorem. Let M be a compact oriented flat affine manifold of dimension n. Then we have

H^{p,q}(F) ≅ H^{n−p,n−q}((K ⊗ F)^{∗}),

where K is the canonical line bundle over M and (K ⊗ F)^{∗} denotes the dual bundle of K ⊗ F.

(KodairaNakano’s vanishing Theorem)

Theorem. Let M be a compact oriented Hessian manifold. Denote by K the canonical line bundle over M. Let F be a locally constant line bundle over M.

(i) If 2F + K is positive, then

H^{p,q}(F) = 0 for p + q > n.

(ii) If 2F + K is negative, then

H^{p,q}(F) = 0 for p + q < n.

([Koszul, 1968])

Theorem. Let M be a compact oriented hyperbolic affine manifold. Then we have

H^{p,q}(1) = 0 for p, q > 0,

where 1 is the trivial line bundle over M.

Let F be a locally constant line bundle over M. Choose an open covering {U_{λ}} of M such that the local triviality holds on each U_{λ}. Denote {f_{λμ}}the constant transition functions with respect to {U_{λ}}. A fiber metric a = {a_{λ}} on F is a collection of positive C^{∞}-functions a_{λ} on U_{λ} such that

a_{μ} = f_{λμ}^{2}a_{λ}.

Using this we can define a globally defined closed 1-form −D log a_{λ} and

−D^{2} log a_{λ}.

Definition. A locally constant line bundle F is said to be positive (resp. negative) if the globally defined 2-form (−D^{2} log a_{λ}) is positive definite [resp. negative definite].

[H8] H. Shima, Hessian manifolds of constant Hessian sectional curvature,

J. Math. Soc. Japan 47, No. 4 (1995), 735 – 753.

Let (M, D, g) a Hessian manifold, {x^{1}, ⋯, x^{n}} be an affine coordinate with respect to D. Let ∇ be the Levi-Civita connection for g. Let γ(X, Y) = ∇_{X}Y − D_{X}Y, and S = Dγ. Let ν be the volume element for g, and D_{X}ν =

α(X)ν, β = Dα. Then S_{jkl}^{i} = \frac{∂γ_{jl}^{i}}{∂x^{k}}.

Let S be an endomorphism of the space of covariant symmetric tensor fields of degree 2 defined by

(ξ)^{ik} = S_{j}^{i}_{l}^{k}ξ^{jl}.

Definition. For a non-zero contravariant symmetric tensor ξ_{x} of degree 2 at x we set

\frac{h(ξ_{x}) = ⟨S(ξ_{x}), ξ_{x}⟩}{⟨ξ_{x}, ξ_{x}⟩,}

and call it the Hessian sectional curvature in the direction ξ_{x}, where ⟨, ⟩ is the inner product by g.

Definition. A Hessian manifold (M, D, g) is said to be a space of constant sectional curvature c if h(ξ_{x}) is constant c for all contravariant symmetric tensor ξ_{x} at x and for all point x in M.

Theorem 1. Let (M, D, g) be a Hessian manifold of dim ≥ 2. if the Hessian sectional curvature h(ξ_{x}) depends only on x, then (M, D, g)is of constant Hessian sectional curvature c if and only if

S_{ijkl} = \frac{c}{2}(g_{ij}g_{kl} + g_{il}g_{kj}).

Corollary 2. If a Hessian manifold (M, D, g) is of constant Hessian sectional curvature c, then the Riemannian manifold (M, g) is a space of constant sectional curvature \frac{−c}{4}.

Corollary 3. Let (M, D, g) be a simply connected Hessian manifold of constant Hessian sectional curvature c. If the Riemannian metric g is complete, then c ≥ 0.

Definition. A Hessian manifold (M, D, g) is said to be Hessian-Einstein if

β = λg.

Corollary 4. A Hessian manifold of constant Hessian sectional curvature c, is Hessian-Einstein ;

β = \{\frac{(n + 1)c}{2}\}g.

Define a tensor field W by

W_{jkl}^{i} = S_{jkl}^{i} − \frac{1}{n + 1}(δ_{j}^{i}β_{kl} + δ_{l}^{i}β_{kj}).

Theorem 5. (M, D, g) is of constant Hessian sectional curvature if and only if W = 0.

Theorem 6. We have

trS^{2} ≥ \frac{2}{n(n + 1)}(trβ)^{2},

where trβ = β_{i}^{i}. The equality holds if and only if (M, D, g) is of constant Hessian sectional curvature

2. Constructions of Hessian manifold of constant Hessian sectional curvature.

4. Characterizations of Hessian manifolds of constant Hessian sectional curvature by affine Chern classes.

Theorem 10. Let (M, D, g) be a Hessian-Einstein manifold of dimension n. Then we have

{−nc_{1}^{2} + 2(n + 1)c_{2})c_{2}} ∧ g^{n−2} ≥ 0.

The equality holds if and only if (M, D, g) is of constant Hessian sectional curvature.

Theorem 11. Let (M, D, g) be a Hessian-Einstein manifold of dimension n. Then we have

{−(n − 2)c_{1}^{2} + 2(n − 1)c_{2})c_{2}} ∧ g^{n−2} ≥ 0.

The equality holds if and only if (M, D, g) is of constant Hessian sectional curvature.

[H9]H. Shima, K. Yagi, Geometry of Hessian manifolds, Differential Geom. Appl. 7, No. 3 (1997), 277–290.

Theorem 2.1. If M is a simply connected Hessian manifold whose metric g is complete, then the affine development of M is an affine-diffeomorphism whose image is a convex domain in 𝔼^{n}.

Theorem 4.2. The following conditions are equivalent :

(i) M admits a D-parallel volume element ;

(ii) ∇ = D ; (iii) α = 0 ; (iv) β = 0.

Corollary 4.3. Suppose the Hessian metric is rational. If D is complete, then ∇ = D.

Let M is an oriented compact Hessian manifold with Hessian structure (D, g). Let {x_{1}, ⋯, x^{n}} be a local affine coordinate with respect to D, i.e., Ddx^{i} = 0. Let ∇ be the levi-Civita connection for g. Denote by ν the volume element determined by g. Define a closed 1-form α and a symmetric bilinear form β by

D_{X}ν = α(X)ν,
β = Dα.

For a flat vector bundle F over M, denote Γ(F) the space of all smooth sections of F. Set

Ω^{p,q} = Γ((Λ^{p}T^{∗}M) ⊗ (Λ^{q}T^{∗}M)).

We call φ in Ω^{p,q} a (p, q)-form and express φ locally as follow ;

φ = c ∑_{I,\overline{J}} φ_{I}dx^{I} ⊗ dx\overline{J},

where I = (i_{1}, ⋯, i_{p}), J = (j_{1}, ⋯, j_{q}), dx^{I} = dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{p}}. Define an operator ∂_{D} : Ω^{p,q} → Ω^{p+1,q} by

∂_{D}φ = c ∑_{I,\overline{J}}\{dφ_{I} ∧ dx^{I}\} ⊗ dx\overline{J}.

Then {∑_{p} Ω^{p,q}, ∂_{D}} is a complex which is similar to the Dolbeault complex for a complex manifold.

Definition. We define certain covariant derivatives D′_{X} and \overline{D}′_{X} of Γ(T M ⊗ T M) in the direction of X ∈ Γ(T M) by

D′_{X} = 2γ_{X} ⊗ I + D_{X},
\overline{D}′_{X} = 2I ⊗ γ_{x} + D_{X},

where γ_{X} = ∇_{X} − D_{X} and I is the identity map.

Let 𝔤 be the abelian Lie algebra of all D-parallel vector fields on M By Corollary 4.5 and (H_{6}), the restriction of the Hessian metric g on 𝔤 is an inner product on 𝔤. Let {X_{1}, ⋯, X_{r}} be an orthonomal basis of 𝔤 with respect to g ; g(X_{i}, X_{j}) = δ_{ij}. Again by Corollary 4.5, the dual 1-forms {ω^{1}, ⋯, ω^{r}} defined by

ω^{i}(Y) = g(X_{i}, Y) for Y ∈ Γ^{(}T M)

are D-parallel.

Set 𝔤_{p} = {X_{p} ∈ T_{p}M | X ∈ 𝔤} and denote by D_{p} the orthogonal complement of 𝔤_{p} with respect to g. Since D_{p} = {v ∈ T_{p}M | ω^{i}(v) = 0 for alli} and dω^{i} = 0, the assignment D : p ↦ D_{p} is an integrable distribution.

Theorem 5.1. Let G be the abelian Lie group generated by 𝔤 and N_{p} the maximal integal submanifold of D through p. Then we have

(i) Each G-orbit G · q is totally geodesic with respect to D, D′ and ∇. The restrictions of (D, g) and (D′, g) on G · q are Hessian structures

and D = D′ = ∇ on G · q.

(ii) N_{p} is totally geodesic with respect to D, D′ and ∇ and the restrictions

of (D, g) and (D′, g) on N_{p} are Hessian structures.

(iii) M = G · N_{p}.

Corollary 5.2. Suppose N_{p} is closed in M. Let H be a closed subgroup of G which consists of all elements in G preserving N_{p}. Then G/H is a torus and

M = G ×_{H} N_{p} → G/H

is a fiber bundle.

[I1] J. H. Hao, H. Shima, Level surfaces of non-degenerate functions in ℝ^{n+1},

Geometriae Dedicata 50, No. 2 (1994), 193 – 204.

Thorem1. Let Ẽ denote the affine normal on a level surface M of ϕ with respect to the volume element dx^{1} ∧ ⋯ d ∧ x^{n+1} on Ω. Then we have

Ẽ = Z + μẼ,

w here μ = |\frac{det(g̃_{ij})}{(Ẽϕ)^{n+1}} |^{\frac{1}{n+2}} , and Z ∈ 𝔛(M) is given by

g̃(X, Z) = (d(μẼϕ))(X) for X ∈ 𝔛(M).

Moreover, we have

d log |μẼϕ| = \frac{1}{n + 1}(τ + 2α̃) on 𝔛(M).

Corollary 2. Ẽ is the direction of the affine normal \overline{E} if and only if

α̃(X) = −\frac{1}{2}τ(X) for X ∈ 𝔛(M).

Corollary 3. Suppose τ = 0. Ẽ is the direction of the affine normal \overline{E} if and only if

α̃(X) = fdϕ.

Theorem 4. The following

S = I, τ = 0, λ = 1, Dh = 0

hold for all level surfaces of ϕ if and only if ϕ is a non-degenerate polynomial of degree 2. In this case Ẽ is in the direction of the affine normal \overline{E}.

Theorem 5. The following

S = 0, τ = 0, λ = −1

hold for all level surfaces of ϕ if and only if, with an appropriate choice of affine coordinate systems, ϕ is expressed as

ϕ = k log(x^{n+1} − F (x^{1}, ⋯, x^{n})),

where F is a smooth function on a domain in ℝ^{n} with det(\frac{∂^{2}F}{∂x^{i}∂x^{j}}) ≠ 0 and k is a non-zero constant.

Corollary 6. Let F be a smooth convex function defined on ℝ^{n}, i.e., (\frac{∂^{2}F}{∂x^{i}∂x^{j}}) is positive definite on ℝ^{n}. For the non-degenerate function ϕ = log(x^{n+1} − F (x^{1}, ⋯, x^{n})) defined on x^{n+1} > F (x^{1}, ⋯, x^{n}), Ẽ is in the direction of the affine normal \overline{E} if and only if F is a convex polynomial of degree 2.

Theorem 7. The following

S = −I, τ = 0, λ = −1

hold for all level surfaces of ϕ if and only if dϕ is invariant under a certain 1-parameter group of dilations.

Corollary 8. Let Ω be an open proper convex cone and ψ be the characteristic function of Ω. For a non-degenerate function ϕ = log ψ, we have

S = −I, τ = 0, λ = −1.

Corollary 9. Let Ω be an open proper convex domain. Suppose Ω is homogeneous under the action of a group G of affine automorphism group of Ω. For the non-degenerate function ϕ = log ψ, we have

τ = 0, λ = −1 α̃ = dϕ,

and Ẽ is in the direction of the affine normal \overline{E}. Moreover, Ω is a cone if and only if S = −I.

Let D̃ be the canonical flat affine connection on ℝ^{n+1}, and let {x^{1}, ⋯ x^{n+1}} be an affine coordinate system on ℝ^{n+1}.

A real valued smooth function ϕ defined on a domain Ω in ℝ^{n+1} is said to be non-degenerate if it satisfies the following conditions:

(i) dϕ_{x} ≠ 0 for x ∈ Ω,

(ii) the Hessian D̃dϕ = ∑ \frac{∂^{2}ϕ}{∂x^{i}∂x^{j}} dx^{i}dx^{j} of ϕ is non-degenerate on Ω.

From now on ϕ si always assumed to be a non-degenerate function on Ω. The Hessian D̃dϕ of ϕ gives a pseudo-Riemannian meric on Ω and is denoted by g^{}˜ .

Define a canonical vector field Ẽ on Ω by

g^{}˜ (X̃, Ẽ) = (dϕ)(X̃) for X̃ ∈ 𝔛(Ω).

Assume that dϕ(Ẽ) does not identically vanish in Ω.

Let M be a level surface of ϕ. Denote 𝔛(M) the set of all tangent vector fields on M. A vector field X along M is tangent to M if and only if

(dϕ)(X) = 0.

Whenever Ẽ is transversal to M, we can define the induced affine connection D and the affine fundamental form h by

D̃_{X}Y = D_{X}Y + h(X, Y)Ẽ for X, Y ∈ 𝔛(M).

We also define the shape operator S and the transversal connection form τ on M by

D̃_{X} Ẽ = S(X) + τ(X)Ẽ for X ∈ 𝔛(M).

Lemma 1. The covariant tensor field of degree 3 defined by

(D̃X̃ g̃)(Ỹ , Z̃)) for X̃, Ỹ , Z̃ ∈ 𝔛(Ω),

is symmetric with respect to X̃, Ỹ , Z̃.

Lemma 2. We have

(i) h(X, Y) = −(Ẽϕ)^{−1}g̃(X, Y), (ii) τ(X) = (d log |Ẽϕ|)(X) for X, Y ∈ 𝔛(M).

Lemma 3. The following conditions are equivalent :

(i) τ = 0.

(ii) D̃ẼẼ = λẼ.

In this case, we have

d(Ẽϕ) = (λ + 1)dϕ,

in particular, Ẽϕ is a constant on Ω if and only if λ = −1.

We define a closed 1-formα̃ on Ω by

α̃ = d log | det(g̃_{ij})|^{\frac{1}{2}} .

[I2] H. Shima, Harmonicity of grandient mapping of level surfaces in a real

affine spaces, Geometriae Dedicata 56, No. 2 (1995), 177 – 184.

Theorem 1. Let D̃′ be a flat affine connection on Ω locally defined by

ι_{∗}(D̃′ ; X̃(Ỹ)) = D̃∗ ; X̃(Ỹ) for X̃, Ỹ ∈ 𝔛(Ω),

where the right-hand D̃^{∗} of the above formula is the covariant differentiation along ι induced by the flat affine connection D̃^{∗} on 𝔼_{n+1}^{∗}. Then we have

(i) D̃′ = 2 ˜∇ − D̃,

(ii) X̃g̃(Ỹ , Z̃) = g̃(D̃X̃ Ỹ , Z̃) + g̃(Ỹ , D̃′ ; X̃ Z̃) for X̃, Ỹ , Z̃ ∈ 𝔛(Ω).

The connection D̃′ is called the dual(conjugate) affine connection of D̃ with respect to g^{}˜ .

Let D̃ be the canonical flat affine connection on 𝔼^{n+1}, and let {x^{1}, ⋯ x^{n+1}} be an affine coordinate system on 𝔼^{n+1}. A domain Ω in 𝔼^{n+1} admitting a function ϕ on Ω is said to be a Hessian domain if the Hessian D̃dϕ = ∑ \frac{∂^{2}ϕ}{∂x^{i}∂x^{j}} dx^{i}dx^{j} of ϕ is non-degenerate, and is denoted by (Ω, D̃, g^{}˜ = D̃dϕ).

Let 𝔼_{n+1}^{∗} be the dual affine space of 𝔼^{n+1} with the dual affine coordinate system {x_{1}^{∗}, ⋯, x_{n+1}^{∗}} and let D̃^{∗} be the canonical flat affine connection given

by D̃^{∗}dx_{i}^{∗} = 0.

Consider an immersion ι of a Hessian domain (Ω, D̃, D̃dϕ) into 𝔼_{n+1}^{∗} defined by

x_{i}^{∗} ∘ ι = \frac{∂ϕ}{∂x^{i}} .

Since ι is locally invertible, we define locally the Legendre transformation ϕ^{∗} of ϕ by

ϕ^{∗} = ∑_{i=1}^{n+1}(x^{i} ∘ ι^{−1})x_{i}^{∗} − ϕ ∘ ι^{−1}.

Then we have

\frac{∂ϕ^{∗}}{∂x_{i}^{∗}} = x^{i} ∘ ι^{−1},

\frac{∂^{2}ϕ^{∗}}{∂x_{i}^{∗}∂x_{j}^{∗}} = g^{}^{˜ ij} ∘ ι^{−1},

where (g^{}^{˜ ij}) = (g^{}_{˜ ij})^{−1} and g^{}_{˜ ij} = \frac{∂^{2}ϕ}{∂x^{i}∂x^{j}}.

Suppose ι is invertible on a domain Ω_{0} in Ω. Then ι maps a Hessian domain (Ω_{0}, D̃, D̃dϕ) to a Hessian domain (Ω_{0}^{∗} = ι(Ω_{0}), D̃^{∗}, D̃^{∗}dϕ^{∗}) and ι is locally involutive, i.e., ι^{−1} = ι on Ω_{0}^{∗}.

Let recall the Laplacian Δι of ι relative to (g^{}˜ , D̃^{∗}) :

Δι = ∑_{ij} g^{}^{˜ ij}\{D̃∗_{\frac{∂}{∂x^{i}}} ι_{∗}(\frac{∂}{∂x^{j}}) − ι_{∗}( ˜∇_{\frac{∂}{∂x^{i}}} \frac{∂}{∂x^{j}})\}.

By calculation, then we have

Δι = α̃.

Theorem 2. Let ϕ be a convex function on 𝔼^{n+1}. Then the following conditions are equivalent :

(i) ι is harmonic, i.e., Δι = 0;

(ii) α̃ = 0;

(iii) ϕ is a polynomial of degree 2.

Assume that a function ϕ on Ω satisfies

(i) D̃dϕ is non-degenerate ;

(ii) dϕ(x) ≠ 0 for x ∈ Ω ;

(iii) dϕ(Ẽ) does not vanish at any point in Ω, where Ẽ is a canonical vector field Ẽ on Ω defined by

g^{}˜ (X̃, Ẽ) = (dϕ)(X̃) for X̃ ∈ 𝔛(Ω).

Let M be a level surface of ϕ. Denote 𝔛(M) the set of all tangent vector fields on M. A vector field X along M is tangent to M if and only if

(dϕ)(X) = 0.

Whenever Ẽ is transversal to M, we can define the induced affine connection D and the affine fundamental form h by

D̃_{X}Y = D_{X}Y + h(X, Y)Ẽ for X, Y ∈ 𝔛(M).

D̃_{X} Ẽ = S(X) + τ(X)Ẽ for X ∈ 𝔛(M).

Let ι be the restriction of ι̃ (= ι on Ω ain the above) to M and g be the restriction of g̃ to M with ∇ the Levi-Civita connection for g. Let {X_{1}, ⋯, X^{n}} be a local basis on M. Then

Δι = ∑_{ij} g^{ij}{D̃_{X_{i}}^{∗}ι_{∗}(X_{j}) − ι_{∗}(∇_{X_{i}}X_{j})},

where (g^{ij}) = (g(X_{i}, X_{j}))^{−1}.

Theorem 3. We have

(i) Δι(X) = α̃(X̃) + \frac{1}{2}τ(X) for X ∈ 𝔛(M), (ii) Δι(Ẽ) = −trS.

Corollary 4. The following conditions are equivalent :

(i) ι is harmonic, i.e., Δι = 0 ;

(ii) trS = 0 and α̃ = −\frac{1}{2}τ on M ;

(iii) trS = 0 and Ẽ is in the direction of the affine normal.

The affine Bernstein problem conjectured by Chern can be restated : Let F be a smooth convex function defined on 𝔼^{n}. If the conormal

mapping of the graph x^{n+1} = F (x^{1}, ⋯, x^{n}) is harmonic, then the graph is an elliptic paraboloid.

Corollary 5. Let F be a smooth convex function defined on 𝔼^{n}, and let Ω be the domain in 𝔼^{n+1} lying above the graph x^{n+1} = F (x^{1}, ⋯, x^{n}). For a level surface M of the function

ϕ = log(x^{n+1} − F (x^{1}, ⋯, x^{n})) on Ω,

the mapping ι, defined at first, is harmonic if and only if M is elliptic paraboloid.

[P1]H. Shima, Homogeneous spaces with invariant projectively flat affine

connections, Trans. Amer. Math. Soc. 351, No. 12 (1999), 4713 – 4726.

Let D be an affine connection on an n-dimensional manifold. Assume that D is torsion free and Ricci-symmetric. D is said to be projectively flat if D is locally projectively equivalent to a flat affine connection.

T (X, Y) = T^{D}(X, Y) = D_{X}Y − D_{Y} X − [X, Y],

R(X, Y)Z = R^{D}(X, Y)Z = D_{X}D_{Y} Z − D_{Y} D_{X}Z − D_{[X,Y]}Z.

If T = 0 and R is identically 0 on M, we say that ∇ is a flat affine connection. We always assume the torsion tensor is zero.

Definition 1. An affine connection D with zero torsion is said to be (locally) equiaffine if it admits locally a parallel volume element, that is, a nonvanishing n-form ω such that Dω = 0.

Definition 2. The Ricci tensor of an affine connection D is defined by

(1)  Ric(Y, Z) = trace of the linear map X ↦ R(X, Y)Z

for any vector fields X, Y, Z.

Proposition 3. A torsion free affine connection D has symmetric Ricci tensor if and only if it admits locally a parallel volume element if and only if D is equiaffine.

For a proof, see [NP] or [NS, p14].

Definition 4. Two torsion free (locally) equiaffine connections D and \overline{D} are projectively equivalent if there is a 1-form ρ such that

(2)  \overline{D}_{X}Y = D_{X}Y + ρ(X)Y + ρ(Y)X

for all tangent vector fields X and Y. In this case, ρ is necessarily closed, i.e, ρ = d log λ for some positive function λ (see [NS, p38]).

Proposition 5. Let D be a torsion free affine connection on a manifold M. Given a volume element ω on M, there is a unique torsion free affine connection \overline{D} on M such that

(i) \overline{D} is projectively equivalent to D ;

(ii) \overline{D}ω = 0, i.e., ω is parallel relative to \overline{D}.

For a proof, see [NP] or [Ei, p104].

Definition 6. An affine connection D with zero torsion and symmetric Ricci tensor is said to be projectively flat if in a neighborhood of each point, D is projectively equivalent to a flat affine connection.

Proposition 7. A torsion free equiaffine connection D is projectively flat if and only if

(i) the Weyl projective curvature tensor W defined by

W (X, Y)Z = R(X, Y)Z − {γ(Y, Z)X − γ(X, Z)Y }

is identically zero ; and (ii) γ satisfies Codazzi equation :

(D_{X}γ)(Y, Z) = (D_{Y} γ)(X, Z),

where γ = \frac{1}{n-1}Ric for all vector fields X, Y, Z.

Furthermore, if dim ≥ 3 then (ii) follows from (i). For dim = 2, (i) is automatically satisfied.

Remark 8. (i) For proofs see [Ei, p95]. (ii) It is known that the Levi-Civita connection of a nondegenerate metric g is projectively flat if and only if g has constant sectional curvature (see [NP, p411]).

Proposition 9. Let g be a metric tensor with Levi-Civita connection D with respect to g. If M has constant sectional curvature k, then

R(X, Y)Z = k{g(Y, Z)X − g(X, Z)Y }.

The converse is also true. For a proof see [O’ne, p80] or [KN, p203].

We say that an affine connection D on M is metric if Dg = 0. That is,

0 = (D_{X}g)(Y, Z) = Xg(Y, Z) − g(D_{X}Y, Z) −g(Y, D_{X}Z). If x, y, z in 𝔤 then with

xy = D_{X}Y, this is equivalent to 0 = g(xy, z) + g(y, xz) and also λ_{x} + λ′_{x} = 0 under the nondegenerate g.

Definition 10. A hypersurface M embedded in ℝ^{n+1} is said to be a centroaffine hypersurface if the position vector x (from 0) for each point x ∈ M is transversal to the tangent plane of M at x.

Let G be a simply connected Lie group and K a connected closed subgroup of G. Assume that G acts effectively on G/K. Denote 𝔤 and 𝔨 the Lie algebras of G and K, resp.

Enlarge 𝔤̃ = 𝔤 ⊕ ℝE, with [𝔤̃, E] = {0}.

Theorem. A simply connected effective homogeneous space G/K admits a G-invariant projectively flat affine connection if and only if 𝔤̃ has an affine representation (f̃, q̃) on a real affine space Ṽ, that is,

(i) f̃: 𝔤̃ → 𝔤𝔩(Ṽ) is a Lie algebra homomorphism,

(ii) q̃ is a linear mapping from 𝔤̃ to Ṽ such that

q̃([X̃, Ỹ]) = f̃(X̃)q̃(Ỹ) − f̃(Ỹ)q̃(X̃),

with the following properties,

(iii) dimṼ = dimG/K + 1,

(iv) q̃ is surjective and the kernel id 𝔨,

(v) f̃(E) is the identity mapping IṼ of Ṽ and q̃(E) ≠ 0.

Corollary. Let G/K be a simply connected homogeneous space. Then the following conditions are equivalent.

(i) G/K admits an invariant projectively flat affine connection.

(ii) G/K has an equivalent centro-affine hypersurface immersion into a real affine space.

Theorem. ([Koszul, Vinberg]) There is a natural one-to-one correspondence between

(i) n-dimensional simply connected Lie groups with left invariant flat affine connections up to affine diffeomorphism ;

(ii) n-dimensional left symmetric algebra over ℝ up to algebraic isomorphism.

Theorem. There is a natural one-to-one correspondence between

(i) n-dimensional simply connected Lie groups with left invariant projectively flat affine connections up to equivariant projective diffeomorphism ;

(ii) (n + 1)-dimensional left symmetric algebra over ℝ with unit up to algebraic isomorphism.

Theorem. Let G/K be an effective symmetric pair where G is semisimple. Suppose that the space G/K admits a G-invarinat projectively flat affine connection. Then there exists a central-simple Jordan algebra Ṽ with unit e such that

(i) Ṽ = V ⊕ ℝe (direct sum as vector spaces).

(ii) Let 𝔪(V) = {L_{u} | u ∈ V } and let 𝔨(V) be the vector space spanned by [L_{u}, L_{v}] where u, v ∈ V. Then 𝔤(V) = 𝔨(V)+𝔪(V) is a Lie algebra and is isomorphic to the Lie algebra 𝔤 = 𝔨+𝔪 of G including decompositions.

Conversely, we have

Theorem. Let Ṽ be a central-simple Jordan algebra with unit e. We set

V = {v ∈ Ṽ | trL_{v} = 0}. Let 𝔪(V) = {L_{v} | v ∈ V } and let 𝔨(V) be the vector

space spanned by [L_{u}, L_{v}] for u, v ∈ V. Then 𝔨(V) and 𝔤(V) = 𝔨(V) + 𝔪(V) are linear Lie algebras. Let G(V) and K(V) are linear Lie groups generated by 𝔤(V) and 𝔨(V), respectively. Then (G(V), K(V)) is a symmetric pair, where G(V) is a semi-simple Lie group, and G(V)/K(V) admits a G(V)-invariant projectively flat affine connection.

Let (G, K) be an effective symmetric pair where G is semi-simple and let 𝔤 = 𝔨 + 𝔪 be the canonical decomposition, that is,

[𝔨, 𝔨] ⊂ 𝔨, [𝔨, 𝔪] ⊂ 𝔪, [𝔪, 𝔪] ⊂ 𝔨.

Suppose that G/k admits a G-invariant projectively flat affine connection. Enlarg 𝔤 so that 𝔤̃ = 𝔤 ⊕ ℝE, with [𝔤̃, E] = {0}, and set 𝔨̃ = 𝔨, 𝔪̃ = 𝔪 + ℝE. Then

[𝔨̃, 𝔪̃] ⊂ 𝔪̃, [𝔪̃, 𝔪̃] ⊂ 𝔨̃.

By above Theorem, there exists an affine representation (f̃, q̃) : 𝔤̃ → 𝔞𝔣𝔣(Ṽ), where V = T_{0}G/K, Ṽ is a real affine space with dimṼ = dimV + 1. The restriction of q̃ to 𝔪̃ being an siomorphism, for each u ∈ Ṽ there exists a unique element X_{u}𝔪̃ such that q̃(X_{u}) = u. We put

L_{u} = f̃(X_{u}),

and define a multiplication law in Ṽ by

u · v = L_{u}(v).

Then the algebra Ṽ is commutative and has unit e = q̃(E).

An algebra Ṽ over ℝ is said to Jordan algebra if for all u, v in Ṽ,

u · v = v · u,
u · (u^{2} · v) = u^{2} · (u · v).

The center Z(Ṽ) of a Jordan algebra Ṽ is by definition([Braun, Koecher])

Z(Ṽ) = {u ∈ Ṽ | (u, v, w) = 0}.

A Jordan algebra Ṽ with unit e is said to be central-simple if Ṽ is simple

and Z(Ṽ) = ℝe.

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