Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 111–122

A CHARACTERIZATION OF CALORIC MORPHISMS BETWEEN MANIFOLDS

Masaharu Nishio and Katsunori Shimomura

Osaka City University, Department of Mathematics Osaka 558-8585, Japan; nishio@sci.osaka-cu.ac.jp Ibaraki University, Department of Mathematical Sciences Mito 310-8512, Japan; shimomur@mito.ipc.ibaraki.ac.jp

Abstract. In this paper, we give a characterization of mappings which preserve caloric functions between semi-riemannian manifolds.

1. Introduction

The Appell transformation plays important roles in the study of the heat equation—especially in the study of positive solutions of the heat equation, because it preserves the solution of the heat equation, and also its positivity. In this note, we shall give a characterization of such transformations, called caloric morphisms between manifolds. We treat not only riemannian manifolds but also semi-riemannian manifolds.

H. Leitwiler [4] and the author [7] studied the characterization of caloric morphism on Euclidean domains. For riemannian manifolds, the characterization can be obtained in almost parallel form. But it does not go similarly for semi-riemannian manifolds. The biggest difference is that f^{0} can depend on the space variables, although f^{0} depends only on the time variable for riemannian manifolds.

We organize this paper as follows: In Section 2, we define the caloric morphism, and state the main theorem and related results. In Section 3, we shall give some lemmas, and prove the main theorem in Section 4. Examples are given in the final Section 5.

2. Notation and results

In this paper we always consider manifolds to be connected and infinitely differentiable. Let (M, g) be a semi-riemannian manifold, that is, M is a manifold endowed with non-degenerate and symmetric metric g which is not necessarily positive definite. If g is positive definite, then (M, g) is called a riemannian manifold. For details on semi-riemannian manifolds, we refer to [6] and for our purpose see also [1].

2000 Mathematics Subject Classification: Primary 35K05; Secondary 31B35.

When M is the euclidean space with a translation invariant metric g, we call (M, g) a semi-euclidean space. In [5], the authors determined caloric morphisms between semi-euclidean spaces of same dimension, which is a generalization of the result by H. Leutwiler [4].

We denote by Δ_{g} the Laplace–Beltrami operator on (M, g) which is given in local coordinates (x^{i})_{i=1}^{m} by

Δ_{g}u = ∑_{i,j=1}^{m} \frac{1}{\sqrt{|g|}} \frac{∂}{∂x^{i}} (\sqrt{|g|} g^{ij} \frac{∂u}{∂x^{j}} ),

where

g_{ij} = g(\frac{∂}{∂x^{i}} , \frac{∂}{∂x^{j}} ), g = det(g_{ij})

and (g^{ij}) denotes the inverse matrix of (g_{ij}).

A C^{2} function u(t, x) defined on an open set in R × M is said to be caloric if u satisfies the heat equation

H_{g}u := \frac{∂u}{∂t} − Δ_{g}u = 0.

Definition 1. Let M and N be semi-riemannian manifolds and D be a domain in R × M. A pair (f, φ) of a C^{2} mapping f: D → R × N and a C^{2} function φ > 0 on D is said to be a caloric morphism if:

(1) f(D) is a domain in R × N,

(2) for any caloric function u defined on a domain E ⊂ R × N, the function

φ(t, x) · (u ∘ f)(t, x)

is caloric on f^{−1}(E).

Evidently, the composition of two caloric morphisms is also a caloric morphism. To be precise, let M, N and L be semi-riemannian manifolds and D, E be domains in R × M, R × N, respectively. If (f, φ): E → R × L and (g, ψ): D → R × N are caloric morphisms such that g(D) ⊂ E, then we can make a caloric morphism (F, Φ): D → R × L from (f, φ) and (g, ψ) by the composition (F, Φ) = (f ∘ g, (φ ∘ g)ψ).

Now we shall state our main theorems, first in the case that M is a riemannian manifold and next in the general case. The author gave the characterization theorem of caloric morphisms on Euclidean spaces in [7] (cf. [4]). It can be generalized to the riemannian case in a very natural form as follows.

Theorem 2.1. Let (M, g) be a riemannian manifold and (N, h) be a semiriemannian manifold. For a C^{2} mapping f on a domain D ⊂ R × M to R × N such that f(D) is a domain and for a C^{2} function φ > 0 on D, the following four statements are equivalent:

(1) (f, φ) is a caloric morphism;

(2) We write f = (f^{0}, f^{1}, …, f^{n}) for a local coordinate (y^{1}, …, y^{n}) of N. Then f^{0}, f^{1}, …, f^{n} and φ satisfy the following equations (E-1)–(E-4): (E-1) H_{g}φ = 0, (E-2) H_{g}f^{α} = 2g(∇_{g} log φ, ∇_{g}f^{α}) − ∑_{β,γ=1}^{n} g(∇_{g}f^{β}, ∇_{g}f^{γ}) ·^{h}Γ_{βγ}^{α} ∘ f for α =

1, …, n, (E-3) ∇_{g}f^{0} = 0, (E-4) g(∇_{g}f^{α}, ∇_{g}f^{β}) = (h^{αβ} ∘ f) · (df^{0}/dt) for α, β = 1, …, n,

where ∇_{g} denotes the gradient operator of (M, g) and ^{h}Γ_{βγ}^{α} denotes the Christoffel symbol of (N, h);

(3) There exists a continuous function λ on D, depending only on t, such that

H_{g}(φ · u ∘ f)(t, x) = λ(t) · φ(t, x) · H_{h}u ∘ f(t, x)

for any C^{2} function u on R × N;

(4) There exists a continuous function λ on D such that

H_{g}(φ · u ∘ f)(t, x) = λ(t, x) · φ(t, x) · H_{h}u ∘ f(t, x)

for any C^{2} function u on R × N.

In the case of semi-riemannian manifolds, we have the following characterization.

Theorem 2.2. Let (M, g) and (N, h) be semi-riemannian manifolds. For a C^{2} mapping f on a domain D ⊂ R × M to a domain f(D) in R × N and for a C^{2} function φ > 0 on D, the following three statements are equivalent:

(1) (f, φ) is a caloric morphism;

(2) f = (f^{0}, f^{1}, …, f^{n}) and φ satisfy the following equations (E-5)–(E-8) for a local coordinate of N: (E-5) H_{g}φ = 0, (E-6) H_{g}f^{α} = 2g(∇_{g} log φ, ∇_{g}f^{α}) − ∑_{β,γ=1}^{n} g(∇_{g}f^{β}, ∇_{g}f^{γ}) ·^{h}Γ_{βγ}^{α} ∘ f for α =

1, …, n, (E-7) g(∇_{g}f^{0}, ∇_{g}f^{α}) = 0 for α = 0, 1, …, n, (E-8) g(∇_{g}f^{α}, ∇_{g}f^{β}) = (h^{αβ} ∘ f) · λ for α, β = 1, …, n,

where λ = H_{g}f^{0} − 2g(∇_{g} log φ, ∇_{g}f^{0});

(3) There exists a continuous function λ on D such that

H_{g}(φ · u ∘ f)(t, x) = λ(t, x) · φ(t, x) · H_{h}u ∘ f(t, x)

for any C^{2} function u on R × N.

Remark 1. If dim M = dim N or (M, g) is riemannian, then it follows from (E-7) that f^{0} depends only on t and then λ = df^{0}/dt, which shows that (E-7) and (E-8) are equivalent to (E-3) and (E-4), respectively. Thus Theorem 2.1 follows from Theorem 2.2. Moreover, since f(D) is a domain in our definition, λ does not vanish. In the case that M is not riemannian, it happens that f^{0} depends both on t and on x (see Example 4). Also, it happens that λ changes its sign in the case that M is not riemannian (see Example 5).

Corollary 1. Let (M, g), (N, h) be semi-riemannian manifolds and let the signature of the metric g (respectively h) be (p, q) (respectively (r, s)). If there exists a caloric morphism from D ⊂ R × M to R × N such that λ(t, x) ≠ 0 at some point (t, x) ∈ D, then

p ≧ r, q ≧ s  or  p ≧ s, q ≧ r

holds. Especially, if M and N have the same dimension, then

p = r, q = s  or  p = s, q = r.

Corollary 2. Let (M, g), (N, h) be semi-riemannian manifolds of same dimensions. If (f, φ) is a caloric morphism from D ⊂ R × M to R × N such that f has an inverse mapping f^{−1} on a open set E ⊂ f(D), then (f^{−1}, 1/φ ∘ f^{−1}) is a caloric morphism from E into R × M.

Proof. Let v be any C^{2} function on R × M. Then u = (1/φ ∘ f^{−1}) · v ∘ f^{−1} is a C^{2} function on E. By (3), there exists a continuous function λ on D such that H_{g}(φ · u ∘ f) = λ · φ · (H_{h}u) ∘ f. Since dim M = dim N, λ does not vanish (see Remark 1). Hence we have

\frac{1}{λ ∘ f^{−1}} \frac{1}{φ ∘ f^{−1}} (H_{g}v) ∘ f^{−1} = H_{h}(\frac{1}{φ ∘ f^{−1}} · v ∘ f^{−1}).

Therefore (f^{−1}, 1/φ ∘ f^{−1}) is a caloric morphism from E into D by (3).

As an application of Theorem 2.2, we have the following propositions, which enable us to construct new caloric morphisms. These are proved by a calculation similar to the one in [7, Proposition 5].

Proposition 2.1 (direct product). Let I be an open interval of R and M_{j} be a semi-riemannian manifold (j = 1, 2). For two caloric morphisms of form ((f_{0}(t), f_{j}(t, x_{j})), φ_{j}(t, x_{j})) from I ×M_{j} to R×N_{j}, we consider a map (f_{0}, f_{1}, f_{2}) from I × M_{1} × M_{2} to R × N_{1} × N_{2} :

(t, x_{1}, x_{2}) ↦ (f_{0}(t), f_{1}(t, x_{1}), f_{2}(t, x_{2}))

and a function φ_{1}φ_{2} on I × M_{1} × M_{2} :

(t, x_{1}, x_{2}) ↦ φ_{1}(t, x_{1})φ_{2}(t, x_{2}).

Then a pair ((f_{0}, f_{1}, f_{2}), φ_{1}φ_{2}) is a caloric morphism.

Proposition 2.2 (direct sum). Let E be a semi-Euclidean space, I an open interval and M_{j} a semi-riemannian manifold (j = 1, 2). For two caloric morphisms (f_{j}, φ_{j}) from I × M_{j} to R × E, we put

\begin{aligned}f(t, x_{1}, x_{2}) & = f_{1}(t, x_{1}) + f_{2}(t, x_{2}), \\ φ(t, x_{1}, x_{2}) & = φ_{1}(t, x_{1})φ_{2}(t, x_{2}) \end{aligned}

for (t, x_{1}, x_{2}) ∈ I ×M_{1} ×M_{2} . Then (f, φ) is a caloric morphism from I ×M_{1} ×M_{2} to R × E.

Remark 2. Let g be a harmonic morphism with constant dilatation from a domain Ω ⊂ M to N (see [1] for the definition of harmonic morphism). Putting

f(t, x) = (t, g(x)), φ(t, x) = 1,

we obtain a caloric morphism (f, φ).

Finally, we make a remark on the relation between harmonic maps and caloric morphisms. Let (M, g) and (N, h) be semi-riemannian manifolds. For a mapping f: M → N, the tension field τ(f) is defined as

τ^{α}(f) = Δ_{g}f^{α} + ∑_{γ,η} g(∇_{g}f^{γ}, ∇_{g}f^{η}) · (^{h}Γ_{γη}^{α} ∘ f), α = 1, …, n,

in the local coordinates. A harmonic map is the solution of

(1)  τ(f) = 0.

Then (E-2) or (E-6) can be written as

\frac{∂f^{α}}{∂t} = τ^{α}(f) + 2g(∇_{g} log φ, ∇_{g}f^{α}), α = 1, …, n.

Now we introduce a new field τ_{φ}(f) by

τ_{φ}^{α}(f) := τ^{α}(f) + 2g(∇_{g} log φ, ∇_{g}f^{α}), α = 1, …, n.

Then (E-2) or (E-6) can be simplified as

\frac{∂f}{∂t} = τ_{φ}(f).

The equation

τ_{φ}(f) = 0

is the Euler–Lagrange equation of the weighted energy functional

E_{Ω,φ}(f) = ∫_{Ω} e(f)φ^{2} dμ_{g},

while (1) is the Euler–Lagrange equation of the energy functional

E_{Ω}(f) = ∫_{Ω} e(f) dμ_{g},

where

e(f) = \frac{1}{2}|df|^{2} := \frac{1}{2} ∑_{α,β} g(∇_{g}f^{α}, ∇_{g}f^{β}) · (h_{αβ} ∘ f), dμ_{g}(x) = \sqrt{|g|} dx

and Ω is a relatively compact subdomain of M.

3. Preliminaries

First, we quote a theorem by L. Hörmander from [2], in order to construct local solutions of the heat equation with prescribed derivatives at a given point.

Let P be a second order differential operator of C^{∞} coefficients on R^{n} of the form

P = ∑_{i,j=1}^{n} a^{ij}(x)\frac{∂^{2}}{∂x^{i}∂x^{j}} + ∑_{k=1}^{n} b^{k}(x)\frac{∂}{∂x^{k}} + c(x),

where the matrix (a^{ij}(x)) is symmetric and non-degenerate for all x.

Theorem A. Let m ≧ 2 be an integer. If u ∈ C^{m+1}(R^{n}) and P u(x) = O(|x|^{m−1}) as x → 0, then for any s > m there exists U ∈ C^{s}(R^{n}) such that U(x) − u(x) = O(|x|^{m+1}) as x → 0 and P U = 0 on a neighborhood of 0.

In the proof, he uses the following theorem, which is also useful for our purpose.

Theorem B ([2, Theorem 7.1]). For any positive integer s, there exists a bounded linear operator G_{s} from the Sobolev space H^{s}(R^{n}) to H^{s+1}(R^{n}) such that for every f ∈ H^{s}(R^{n}), P G_{s}f = f on a neighborhood of the origin.

Next we prepare lemmas for the proof of the main theorem. Combining the Sobolev imbedding theorem with Theorem B, we have the following lemma.

Lemma 3.1. For any integers s ≧ 2 and s′ > \frac{1}{2}n+s−1 and any C^{s′} function f defined on a neighborhood of the origin in R^{n}, there exists a C^{s} function u such that P u = f on a neighborhood of the origin.

By Theorem A, we have the following lemma.

Lemma 3.2. Let ((η_{ij})_{i,j=1}^{n}, (η_{k})_{k=1}^{n}, η) ∈ R^{n^{2}+n+1} satisfy η_{ij} = η_{ji} and

∑_{i,j=1}^{n} a^{ij}(0)η_{ij} + ∑_{k=1}^{n} b^{k}(0)η_{k} + c(0)η = 0.

Then for any s > 2 there exists a C^{s} function U such that

\frac{∂^{2}U}{∂x^{i}∂x^{j}} (0) = η_{ij}, \frac{∂U}{∂x^{k}} (0) = η_{k}, U(0) = η, for i, j, k = 1, …, n,

and P U = 0 on a neighborhood of 0.

Proof. Put m = 2 and apply Theorem A to the function

u(x) = \frac{1}{2} ∑_{i,j=1}^{n} η_{ij}x^{i}x^{j} + ∑_{k=1}^{n} η_{k}x^{k} + η.

Using these lemmas, we have the following existence theorem for the solution of the heat equation with prescribed derivatives.

Lemma 3.3. For any ((η_{ij})_{i,j=0}^{n}, (η_{k})_{k=0}^{n}, η) ∈ R^{(n+1)^{2}+(n+1)+1} such that η_{ij} = η_{ji} and

η_{0} = ∑_{i,j=1}^{n} a^{ij}(0)η_{ij} + ∑_{k=1}^{n} b^{k}(0)η_{k} + c(0)η,

there exists a C^{2} function u on a neighborhood of the origin of R × R^{n} such that

(2) \frac{∂^{2}u}{∂x^{i}∂x^{j}} (0, 0) = η_{ij}, \frac{∂u}{∂x^{k}} (0, 0) = η_{k}, u(0, 0) = η, for i, j, k = 0, 1, …, n,

and (∂u/∂t) − P u = 0. Here we use a convention (∂/∂x^{0}) = (∂/∂t).

Proof. We shall construct a solution of form

(3)  u(t, x) = u_{0}(x) + u_{1}(x)t + \frac{1}{2}u_{2}(x)t^{2}.

Substitute (3) into the heat equation (∂u/∂t) − P u = 0 and compare the coefficients as the function of t. Then we obtain the equations

P u_{2} = 0,  P u_{1} = u_{2},  P u_{0} = u_{1}.

From the initial value condition (2), u_{0} , u_{1} , u_{2} must satisfy

u_{2}(0) = \frac{∂^{2}u}{∂t^{2}}(0, 0) = η_{00},
u_{1}(0) = \frac{∂u}{∂t}(0, 0) = η_{0},
\frac{∂u_{1}}{∂x^{j}} (0) = \frac{∂^{2}u}{∂t∂x^{j}} (0, 0) = η_{0j} for j = 1, …, n,
u_{0}(0) = u(0, 0) = η,
\frac{∂u_{0}}{∂x^{k}} (0) = \frac{∂u}{∂x^{k}} (0, 0) = η_{k} for k = 1, …, n,
\frac{∂^{2}u_{0}}{∂x^{i}∂x^{j}} (0) = \frac{∂^{2}u}{∂x^{i}∂x^{j}} (0, 0) = η_{ij} for i, j = 1, …, n.

First we shall solve the equation P u_{2} = 0 under the condition u_{2}(0) = η_{00} . Put η̃ = η_{00} and η̃_{k} = 0 for k = 1, …, n. Since the matrix (a^{ij}(0))_{i,j=1}^{n} is non-degenerate, there exists a symmetric matrix (η̃_{ij})_{i,j=1}^{n} satisfying that

(4)  ∑_{i,j=1}^{n} a^{ij}(0)η̃_{ij} + ∑_{k=1}^{n} b^{k}(0)η̃_{k} + c(0)η̃ = 0.

Then we can solve the equation P u_{2} = 0 by Lemma 3.2.

Next we shall solve the equation P u_{1} = u_{2} under the conditions u_{1}(0) = η_{0} and (∂u_{1}/∂x^{j})(0) = η_{0j} for j = 1, …, n. We can take w_{1} with P w_{1} = u_{2} by Lemma 3.1. Putting η̃ = η_{0}−w_{1}(0) and η̃_{k} = η_{0k}−(∂w_{1}/∂x^{k})(0) for k = 1, …, n, we can take η̃_{ij} for i, j = 1, …, n with (4). Then we can solve the equation P v_{1} = 0 under the conditions v_{1}(0) = η_{0}−w_{1}(0) and (∂v_{1}/∂x^{k})(0) = η_{0k}−(∂W_{1}/∂x^{k})(0) for k = 1, …, n by Lemma 3.2. Then u_{1} = v_{1} + w_{1} is a desired solution.

Finally, we shall solve the equation P u_{0} = u_{1} under the conditions u_{0}(0) = η, (∂u_{0}/∂x^{k})(0) = η_{k} and (∂^{2}u_{0}/∂x^{i}∂x^{j})(0) = η_{ij} for i, j, k = 1, …, n. We take w_{0} with P w_{0} = u_{1} by Lemma 3.1 and solve the equation P v_{0} = 0 under the conditions v_{0}(0) = η − w_{0}(0), (∂v_{0}/∂x^{k})(0) = η_{k} − (∂w_{0}/∂x^{k})(0) and (∂^{2}v_{0}/∂x^{i}∂x^{j})(0) = η_{ij} − (∂^{2}w_{0}/∂x^{i}∂x^{j})(0) for i, j, k = 1, …, n, because the condition (4) holds for η̃ = η − w_{0}(0), η̃_{k} = η_{k} − (∂w_{0}/∂x^{k})(0) and η̃_{ij} = η_{ij} − (∂^{2}w_{0}/∂x^{i}∂x^{j})(0). Putting u_{0} = v_{0} + w_{0} , we have the lemma.

4. Proof of Theorem 2.2

In this section we shall prove our main theorem. Before the proof, we prepare a small lemma from linear algebra.

Lemma 4.1. Let a, b ∈ R^{l} and consider linear forms ϱ(x) = (x, a) and μ(x) = (x, b), which are the usual inner product in R^{l}. If ϱ(x) = 0 implies μ(x) = 0 for all x ∈ R^{l}, then there exists ν ∈ R such that b = νa.

Proof of Theorem 2.2. (1) ⇒ (2): For any point (t, P) ∈ D, we put (τ, Q) = f(t, P) and take a local coordinate y of N near Q such that Q corresponds to the origin. Then since

Δ_{h} = ∑_{α,β=1}^{n} h^{αβ}(y)\frac{∂^{2}}{∂y^{α}∂y^{β}} + ∑_{γ=1}^{n}( ∑_{α,β=1}^{n} h^{αβ}(y)^{h}Γ_{α β}^{γ}(y))\frac{∂}{∂y^{γ}} ,

we put a^{αβ}(y) = h^{αβ}(y), b^{γ}(y) = ∑_{α,β=1}^{n} h^{αβ}(y)^{h}Γ_{α β}^{γ}(y) and c(y) = 0 for α, β, γ = 1, …, n. Hence for every ((η_{αβ})_{α,β=0}^{n}, (η_{γ})_{γ=0}^{n}, η) ∈ R^{(n+1)^{2}+(n+1)+1} such that

η_{0} = ∑_{α,β=1}^{n} a^{αβ}(0)η_{αβ} + ∑_{γ=1}^{n} b^{γ}(0)η_{γ} + c(0)η,

it follows from Lemma 3.3 that there exists a caloric function u on a neighborhood of (τ, Q) such that

\frac{∂^{2}u}{∂y^{α}∂y^{β}} (τ, Q) = η_{αβ},  \frac{∂u}{∂y^{γ}} (τ, Q) = η_{γ},  u(τ, Q) = η

for α, β, γ = 0, 1, …, n. Since (f, φ) is assumed to be a caloric morphism, φ·(u∘f) must be caloric, that is,

0 = H_{g}(φ · (u ∘ f))

= (H_{g}φ) · (u ∘ f) + φ · H_{g}(u ∘ f) − 2g(∇_{g}φ, ∇_{g}(u ∘ f))

= (H_{g}φ) · (u ∘ f) + ∑_{γ=0}^{n}(φ · H_{g}f^{γ} − 2g(∇_{g}φ, ∇_{g}f^{γ})) · (\frac{∂u}{∂y^{γ}} ) ∘ f

− φ ∑_{α,β=0}^{n} g(∇_{g}f^{α}, ∇_{g}f^{β}) · (\frac{∂^{2}u}{∂y^{α}∂y^{β}} ) ∘ f.

Substituting (t, P), we have

\begin{aligned}0 & = (H_{g}φ)(t, P) · η + ∑_{γ=0}^{n}(φ · H_{g}f^{γ} − 2g(∇_{g}φ, ∇_{g}f^{γ}))(t, P) · η_{γ} \\ & − ∑_{α,β=0}^{n}(φ · g(∇_{g}f^{α}, ∇_{g}f^{β}))(t, P) · η_{αβ}, \end{aligned}

which implies that there exists ν(t, P) ∈ R such that

H_{g}φ(t, P) = 0,
(φ · H_{g}f^{0} − 2g(∇_{g}φ, ∇_{g}f^{0}))(t, P) = ν(t, P),
(A) (φ · H_{g}f^{γ} − 2g(∇_{g}φ, ∇_{g}f^{γ}))(t, P) = −ν(t, P) · b^{γ}(0) for γ = 1, …, n,
(φ · g(∇_{g}f^{0}, ∇_{g}f^{β}))(t, P) = 0 for β = 0, 1, …, n,
(B)  (φ · g(∇_{g}f^{α}, ∇_{g}f^{β}))(t, P) = ν(t, P) · a^{αβ}(0) for α, β = 1, …, n,

because of Lemma 4.1. Putting λ(t, P) = ν(t, P)/φ(t, P) and substituting (B) into (A), we obtain (2).

(2) ⇒ (3): By a direct calculation, we have

H_{g}(φ · (u ∘ f)) = (H_{g}φ) · (u ∘ f) + ∑_{γ=0}^{n}(φ · H_{g}f^{γ} − 2g(∇_{g}φ, ∇_{g}f^{γ})) · (\frac{∂u}{∂y^{γ}} ) ∘ f

− φ ∑_{α,β=0}^{n} g(∇_{g}f^{α}, ∇_{g}f^{β}) · (\frac{∂^{2}u}{∂y^{α}∂y^{β}} ) ∘ f

= (λ · φ) · (\frac{∂u}{∂y^{0}} + ∑_{γ=1}^{n}( ∑_{α,β=1}^{n} h^{αβ}^{h}Γ_{αβ}^{γ})\frac{∂u}{∂y^{γ}} ) ∘ f

− (λ · φ) · ( ∑_{α,β=1}^{n} h^{αβ} \frac{∂^{2}u}{∂y^{α}∂y^{β}} ) ∘ f

= (λ · φ) · (H_{h}u) ∘ f.

(3) ⇒ (1) is trivial, which completes the proof.

5. Examples

In the following Examples 1–3, we consider the case that M = R^{n} \ {0} and g has the form g_{ij} = ϱ(|x|)δ_{ij}, where ϱ is a C^{∞} function on (0, ∞).

Example 1. Let n = 2, ϱ(r) = 1/r^{2} . Then

f(t, x) = (t + b, \frac{e^{t}}{|x|^{2}} R(t)x), φ(t, r, θ) = r^{−1/2} exp(\frac{1}{2}θ + \frac{1}{2}t),

is a caloric morphism, where

R(t) = ( \begin{matrix}cos t & − sin t \\ sin t & cos t \end{matrix} )

and (r, θ) is the polar coordinate of R^{2} .

Example 2. Let n ≧ 3, ϱ(r) = 1/r^{2} . Then

f(t, x) = (t, e^{t} \frac{x}{|x|^{2}} ), φ(t, x) = |x|^{−1/2} exp(\frac{1}{4}t)

is a caloric morphism.

Example 3 (Appell transformation). Let ϱ(r) = r^{k}, k ∈ R, k ≠ −2. Then

\begin{matrix}f(t, x) = & (\frac{ct + d}{at + b}, \frac{x}{|at + b|^{2/(k+2)}} ), \\ φ(t, x) = & \frac{1}{|at + b|^{n/2}} exp(−\frac{a|x|^{k+2}}{(k + 2)^{2}(at + b)}) \end{matrix}

is a caloric morphism, where a, b, c, d ∈ R, bc − ad = 1. In particular, in the case of k = 0, this example includes the usual Appell transformation:

f(t, x) = (−\frac{1}{t}, \frac{x}{t}), φ(t, x) = \frac{1}{(4π|t|)^{n/2}} exp(−\frac{|x|^{2}}{4t} ).

Next we consider the non-riemannian case.

Example 4. Let M = R^{3} be a semi-Euclidean space whose metric is g = (dx^{1})^{2} + (dx^{2})^{2} − (dx^{3})^{2} and N = R^{1} be the Euclidean space. Then

f(t, x) = (t + x^{2} + x^{3}, x^{1}), φ(t, x) = 1

satisfy (E-5)–(E-8). In this case λ(t, x) = 1. Composing this to the Appell transformation on R × R

(−\frac{1}{τ} , \frac{y}{τ} ), \frac{1}{\sqrt{|τ|}} exp(−\frac{y^{2}}{4τ} ),

we have another example

\begin{aligned}f(t, x) & = (−\frac{1}{t + x^{2} + x^{3}} , \frac{x^{1}}{t + x^{2} + x^{3}} ), \\ φ(t, x) & = \frac{1}{\sqrt{|t + x^{2} + x^{3}|}} exp(−\frac{(x^{1})^{2}}{4(t + x^{2} + x^{3})}) \end{aligned}

such that λ(t, x) = 1/τ^{2} = 1/(t + x^{2} + x^{3})^{2} depends on both t and x.

Example 5. Let M = R^{4} be a semi-Euclidean space whose metric is g = (dx^{1})^{2} − (dx^{2})^{2} + (dx^{3})^{2} − (dx^{4})^{2} and N = R^{2} be a semi-Euclidean space whose metric is g = (dx^{1})^{2} − (dx^{2})^{2} . Then

\begin{aligned}f(t, x) & = (\frac{1}{t(t − 1)}, \frac{x^{1}}{t} + \frac{x^{4}}{t − 1}, \frac{x^{2}}{t} + \frac{x^{3}}{t − 1}), \\ φ(t, x) & = \frac{1}{|t(t − 1)|} exp(−\frac{(x^{1})^{2} − (x^{2})^{2}}{4t} − \frac{(x^{3})^{2} − (x^{4})^{2}}{4(t − 1)} ) \end{aligned}

satisfy (E-5)–(E-8) on (0, 1)×M. Then λ(t) = (1/t^{2})−(1/(t − 1)^{2}), which changes the sign at t = \frac{1}{2} . Note that (f, φ) is a direct sum of two Appell transfomations.

Example 6 (Reverse). Put f(t, x) = (−t, x) and φ(t, x) = 1. Then (f, φ) is a caloric morphism from R × (M, g) onto R × (M, −g). Moreover, if M is a 2ndimensional semi-euclidean space such that the signature of g is equal to (n, n), then there exists a (2n × 2n)-matrix R such that R^{2} = I and ^{t}RGR = −G, where I is the identity matrix and G = (g_{ij}). Hence putting f(t, x) = (−t, Rx) and φ(t, x) = 1, we have a caloric morphism on R × (M, g). We remark that (f^{0})′ is negative in these examples.

Example 7 (Appell transformation). For integers p, q ≧ 0 and k ∈ R\{−2}, put

\begin{aligned}M & = \{x ∈ R^{p+q} : (x^{1})^{2} + ⋯ + (x^{p})^{2} − (x^{p+1})^{2} − ⋯ − (x^{p+q})^{2} > 0\}, \\ ⟨x⟩ & = \sqrt{(x^{1})^{2} + ⋯ + (x^{p})^{2} − (x^{p+1})^{2} − ⋯ − (x^{p+q})^{2}} \end{aligned}

and

g_{ij}(x) = \{ \begin{matrix}⟨x⟩^{k}, & i = j = 1, …, p, \\ −⟨x⟩^{k}, & i = j = p + 1, …, p + q, \\ 0, & i ≠ j. \end{matrix}

Then

f(t, x) = (−\frac{1}{t}, \frac{x}{t^{2/(k+2)}} ), φ(t, x) = \frac{1}{t^{(p+q)/2}} exp(−\frac{⟨x⟩^{k+2}}{(k + 2)^{2}t})

is a caloric morphism from (0, ∞) × M to (−∞, 0) × M. This is a generalization of Example 3.

References

[1] Fuglede, B.: Harmonic morphisms between semi-riemannian manifolds. - Ann. Acad. Sci. Fenn. Math. 21, 1996, 31–50.

[2] Hörmander, L.: Differential operators of principal type. - Math. Ann. 140, 1960, 124–146.

[3] Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions. - J. Math. Kyoto Univ. 19, 1979, 215–229.

[4] Leutwiler, H.: On Appell transformation. - In: Potential Theory, edited by Král, J., J. Lukes̆, I. Netuka, and J. Veselý. Plenum, New York, 1988, 215–222.

[5] Nishio, M., and K. Shimomura: Caloric morphisms on semi-euclidean space. - Preprint, 2001.

[6] O’Neill, B.: Semi-Riemannian Geometry. - Academic Press, 1983.

[7] Shimomura, K.: The determination of caloric morphisms on Euclidean domains. - Nagoya Math J. 158, 2000, 133–166.

Received 22 February 2002