Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 27, 2007, 41–46

© J. Kurek and W.M. Mikulski

Abstract. We describe all $\mathcal{ℳ}{f}_m$-natural operators $A:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ transforming Riemannian structures $g$ on $m$-dimensional manifolds $M$ into Riemannian structures $A\left(g\right)$ on the $r$-th order frame bundle $P^{r}M=inv{J}_0^r\left(R^m,M\right)$ over $M$.

Manifolds and maps are assumed to be of class $C^{\infty }$. Manifolds assumed to be finite dimensional and without boundaries.

Let $\mathcal{ℳ}{f}_m$ denote the category of $m$-dimensional manifolds and their embeddings (i.e. diffeomorphisms onto open subsets) and $\mathcal{ℱ}\mathcal{ℳ}$ denote the category of fibred manifolds and their fibred maps.

For any $m$-manifold $M$ we have the $r$-th order frame bundle $P^{r}M=inv{J}_0^r\left(R^m,M\right)$ of $M$. This is a principal bundle with the corresponding Lie group $G_m^r=J_0^r{\left(R^m,R^m\right)}_0$ acting on the right on $P^{r}M$ via compositions of jets. Every $\mathcal{ℳ}{f}_m$-map $\psi :M_1\to M_2$ induces $P^r\psi :P^r{M}_1\to P^r{M}_2$ by $P^r\psi \left(j_0^r\varphi \right)=j_0^r\left(\psi \circ \varphi \right)$, where $\varphi :R^m\to M_1$ is a $\mathcal{ℳ}{f}_m$-map. The correspondence $P^r:\mathcal{ℳ}{f}_m\to \mathcal{ℱ}\mathcal{ℳ}$ is a bundle functor in the sense of [2].

For any $n$-manifold $N$ we have the Riemannian bundle $\mathcal{ℛ}iem\left(N\right)={\bigcup }_{y\in N}Met\left(T_{y}N\right)$ over $N$, where given a vector space $V$ we denote the set of scalar multiplications $G:V\times V\to R$ on $V$ by $Met\left(V\right)$. (We recall that $G:V\times V\to R$ is a scalar multiplication if it is symmetric bilinear and positive define.) Clearly, $\mathcal{ℛ}iem\left(N\right)$ is an open subbundle in the vector bundle $T^{\ast }N\odot T^{\ast }N$ of symmetric tensors of type $\left(0,2\right)$ over $N$. Sections $g:N\to \mathcal{ℛ}iem\left(N\right)$ are the so called Riemannian structures on $N$. Every embedding $\psi :N_1\to N_2$ induces $\mathcal{ℛ}iem\left(\psi \right):\mathcal{ℛ}iem\left(N_1\right)\to \mathcal{ℛ}iem\left(N_2\right)$ being the restriction of $T^{\ast }\psi \odot T^{\ast }\psi :T^{\ast }{N}_1\odot T^{\ast }{N}_1\to T^{\ast }{N}_2\odot T^{\ast }{N}_2$. The correspondence $\mathcal{ℛ}iem:\mathcal{ℳ}{f}_n\to \mathcal{ℱ}\mathcal{ℳ}$ is also a bundle functor in the sense of [2].

In the present short note we study the problem how a Riemannian structure $g$ on an $m$-dimensional manifold $M$ can induce (canonically) a Riemannian structure $A\left(g\right)$ on $P^{r}M$. This problem is reflected in the concept of $\mathcal{ℳ}{f}_m$-natural operators $A:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$. In the note we describe explicitly all $\mathcal{ℳ}{f}_m$-natural operators $A$ in question.

A general concept of natural operators can be found in the fundamental monograph [2]. We need only the following partial case of the definition of natural operators.

An $\mathcal{ℳ}{f}_m$-natural operator $A:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ is a family of $\mathcal{ℳ}{f}_m$-invariant regular operators (functions)

for any $\mathcal{ℳ}{f}_m$-object $M$, where $\underset{¯}{\mathcal{ℛ}iem}\left(N\right)$ is the set of all Riemannian structures on $N$ (sections of $\mathcal{ℛ}iem\left(N\right)\to N$) for any manifold $N$. The invariance means that if $g_1\in \underset{¯}{\mathcal{ℛ}iem}\left(M_1\right)$ and $g_2\in \underset{¯}{\mathcal{ℛ}iem}\left(M_2\right)$ are related by an $\mathcal{ℳ}{f}_m$-map $\psi :M_1\to M_2$ (i.e. $\mathcal{ℛ}iem\left(\psi \right)\circ g_1=g_2\circ \psi$) then $A\left(g_1\right)$ and $A\left(g_2\right)$ are $P^r\psi$-related. The regularity means that $A$ transforms smoothly parametrized families of Riemannian structures into smoothly parametrized ones.

For $r=1$, $P^{1}M$ is equivalent with the bundle of linear frames over $M$. In this case we have the following example basing on a very important classical construction presented in the proof of Theorem 1.5 in [1].

Example 1. Let $g$ be a Riemannian structure on an $m$-manifold $M$. Let $\nabla$ be the Levi-Civita connection of $g$ and let $\omega =\left({\omega }_k^j\right):T{P}^{1}M\to gl\left(m\right)$ be its connection form. Let $\theta =\left({\theta }^i\right):T{P}^{1}M\to R^m$ be the canonical form on $P^{1}M$. We put

Then $\tilde{g}$ is a Riemannian structure on $P^{1}M$, see the proof of Theorem 1.5 in [1]. Clearly, the correspondence $A^1:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^1$ given by $A^1\left(g\right)=\tilde{g}$ for all $g\in \underset{¯}{\mathcal{ℛ}iem}\left(M\right)$ is an $\mathcal{ℳ}{f}_m$-natural operator.

To generalize Example 1 on all $r$ we firstly reformulate it as follows.

Example 2. Let $g$ be a Riemannian structure on $M$. Let Let $\left({\theta }^i,{\omega }_k^j\right)$ be the basis of $1$-forms on $P^{1}M$, where $\theta =\left({\theta }^i\right)$ and $\omega =\left({\omega }_k^j\right)$ is as in Example 1 for $g$. Let $X^1\left(g\right),\dots ,X^{L_1}\left(g\right)$, where $L_1=dim\left(P^1{R}^m\right)=m+m^2$, be the dual (to $\left({\theta }^i,{\omega }_k^j\right)$) basis of vector fields on $P^{1}M$. Then $\tilde{g}={\sum }_{s=1}^{L_1}{\left(X^s\left(g\right)\right)}^{\ast }\odot {\left(X^s\left(g\right)\right)}^{\ast }$ is a Riemannian structure on $P^{1}M$ (the same as in Example 1).

So, to generalize Example 1 (or Example 2) on all $r$ we need to construct an absolute parallelism (basis of global vector fields) on $P^{r}M$ canonically dependent on a given Riemannian structure $g$ on $M$.

From now on let $A_1,\dots ,A_{dim\left(g_m^r\right)}$ be the standard basis in $g_m^r=\mathcal{ℒ}ie\left(G_m^r\right)$ (i.e. the basis $j_0^r\left(x^{\alpha }\frac{\partial }{\partial x^i}\right)\in {\left(J_0^{r}T{R}^m\right)}_0=g_m^r$ for $i=1,\dots ,m$, $1\le \mid \alpha \mid \le r$).

Example 3. Construction of an absolute parallelism on $P^{r}M$ from a Riemannian structure on $M$. Let $g$ be a Riemannian structure on an $m$-manifold $M$. Let $\nabla$ be the Levi-Civita connection of $g$. Let $i=1,\dots ,m$. We have a vector field $Y^i\left(g\right)$ on $P^{r}M$ defined as follows. Let $\sigma =j_0^r\varphi \in {\left(P^{r}M\right)}_x$, $x\in M$. Let $v^i=T\varphi \left({\frac{\partial }{\partial x^i}}_{\mid 0}\right)\in T_{x}M$. We extend $v^i$ to the constant vector field ${ṽ}^i$ on $T_{x}M$. Then on some neighborhood of $x$ we have the vector field $V^i\left(g\right)={\left(Ex{p}_x^{\nabla }\right)}_{\ast }{ṽ}^i$, where $Ex{p}_x^{\nabla }:T_{x}M\supset U_{0_x}\to {Ũ}_x\subset M$ is the exponent of $\nabla$. We define $Y^i{\left(g\right)}_{\sigma }:={\mathcal{P}}^r{\left(V^i\left(g\right)\right)}_{\sigma }$, where ${\mathcal{P}}^{r}V$ is the flow lifting of a vector field $V$ on $M$ to $P^{r}M$ (if $\left\{{\varphi }_t\right\}$ is the flow of $V$ then $\left\{P^r{\varphi }_t\right\}$ is the flow of ${\mathcal{P}}^{r}V$). It is easy to see that $Y^i{\left(g\right)}_{\sigma }$ projects onto $v^i$ by the bundle projection $P^{r}M\to M$. So, it is a simple observation that $Y^i\left(g\right),A_j^{\ast }$ for $i=1,\dots ,m$, $j=1,\dots ,dim\left(g_m^r\right)$ is an absolute parallelism on $P^{r}M$ (canonically depending on $g$), where given $A\in \mathcal{ℒ}ie\left(G_m^r\right)$ we denote the fundamental vector field on the principal bundle $P^{r}M$ by $A^{\ast }$.

Now, we are in position to extend Example 1 (or 2) on all $r$.

Example 4. Let $g$ be a Riemannian structure on an $m$-manifold $M$. Let $\left(Y^i\left(g\right),A_j^{\ast }\right)$ be the parallelism from Example 3. Let ${\omega }^s\left(g\right)$ for $s=1,\dots ,dim\left(P^r{R}^m\right)$ be the dual basis of $1$-form on $P^{r}M$. We put

Clearly, ${\tilde{g}}^r$ is a Riemannian structure on $P^{r}M$. Clearly, the correspondence $A^{\left[r\right]}:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ given by $A^{\left[r\right]}\left(g\right)={\tilde{g}}^r$ for all $g$ in question is an $\mathcal{ℳ}{f}_m$-natural operator. One can observe easily that $A^1=A^{\left[1\right]}$.

To present a general example of $\mathcal{ℳ}{f}_m$-natural operators $\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ we need some preparation and notations.

According to the global basis of vector fields $Y^i\left(g\right),A_j^{\ast }$ on $P^{r}M$ from Example 3, given $g\in \underset{¯}{\mathcal{ℛ}iem}\left(M\right)$ we have a canonical (in $g$) fibred diffeomorphism

covering $i{d}_{P^{r}M}$ defined by the condition that the matrix of $I_g\left(\sigma ,G\right)$ in the basis $\left(Y^i\left(g\right)\left(\sigma \right),A_j^{\ast }\left(\sigma \right)\right)$ is the same as the one of $G$ in the usual canonical basis of $R^{L_r}$.

Given $g\in \underset{¯}{Riem}\left(M\right)$ we have the projection

given by the Gramm orthonormalization with respect to $g$ (for $l=\left(l_i\right)\in L_{x}M$, $Ort\left(g\right)\left(l\right)$ is the orthonormalization of $l$ with respect to $g_x$).

From now on we denote

where $g^o$ is the usual flat Riemannian structure on $R^m$ and $l^o$ is the usual canonical basis in $R^m=T_0{R}^m$ and ${\pi }_1^r:P^{r}M\to P^{1}M$ is the jet projection. Of course, $Q^r$ is a submanifold in ${\left(P^r{R}^m\right)}_0$.

For $s=0,1,\dots ,\infty$, let $Z^s=J_0^s\left(\mathcal{ℛ}iem\left(R^m\right)\right)$ be the set of all $s$-jets $j_0^{s}g$ of Riemannian structures $g$ on $R^m$. If $s$ is finite, $Z^s$ is a finite dimensional manifold (as the fibre of the $s$-jet prolongation $J^s\left(\mathcal{ℛ}iem\left(R^m\right)\right)$ of the bundle $\mathcal{ℛ}iem\left(R^m\right)\to R^m$). $Z^{\infty }$ is a topological space with respect to the inverse limit topology given by the inverse system $\dots \to Z^{s+1}\to Z^s\to \dots \to Z^0$ of jet projections.

Now, we are in position to present the following general construction.

Example 5. General construction. Let $\mu :Z^{\infty }\times Q^r\to Met\left(R^{L_r}\right)$, where $L_r=dim\left(P^r{R}^m\right)$, be a map satisfying the following local finite determination property ($a_r$):

($a_r$) For any $\rho \in Z^{\infty }$ and $\sigma \in Q^r$ we can find an open neighborhood $U\subset Z^{\infty }$ of $\rho$, an open neighborhood $V\subset Q^r$ of $\sigma$, a natural number $s$ and a smooth map $f:{\pi }_s\left(U\right)\times V\to Met\left(R^{L_r}\right)$ such that $\mu =f\circ \left({\pi }_s\times i{d}_V\right)$ on $U\times V$, where ${\pi }_s:Z^{\infty }\to Z^s$ is the jet projection.

(A simple example of such $\mu$ is $\mu =f\circ \left({\pi }_s\times i{d}_{Q^r}\right)$ for smooth $f:Z^s\times Q^r\to Met\left(R^{L_r}\right)$ for finite $s$.) Given a Riemannian structure $g$ on an $m$-manifold $M$ we define a Riemannian structure $A^{<\mu >}\left(g\right)$ on $P^{r}M$ as follows. Let $\sigma \in {\left(P^{r}M\right)}_x$, $x\in M$. Choose a $g$-normal coordinate system $\psi$ on $M$ with center $x$ such that $P^r\psi \left(\sigma \right)\in Q^r$. Of course, such $\psi$ exists. Then $ger{m}_x\left(\psi \right)$ is uniquely determined. We put

Since $ger{m}_x\left(\psi \right)$ is uniquely determined the definition (**) is correct. The family $A^{<\mu >}:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ is an $\mathcal{ℳ}{f}_m$-natural operator.

The main result of the present note is the following theorem.

Theorem 1. Any $\mathcal{ℳ}{f}_m$-natural operator $A:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ is $A^{<\mu >}$ for some $\mu :Z^{\infty }\times Q^r\to Met\left(R^{L_r}\right)$ satisfying the property ($a_r$).

Proof. Let $A:\mathcal{ℛ}iem⇝\mathcal{ℛ}iem{P}^r$ be an $\mathcal{ℳ}{f}_m$-natural operator. Define $\mu :Z^{\infty }\times Q^r\to Met\left(R^{L_r}\right)$ by

Then by the non-linear Peetre theorem [2], $\mu$ satisfies the property ($a_r$). Then by the definition of $\mu$ and $A^{<\mu >}$ we see that $A\left(g\right)\left(\sigma \right)=A^{<\mu >}\left(g\right)\left(\sigma \right)$ for any Riemannian structure $g$ on $R^m$ such that the identity map $i{d}_{R^m}$ is a $g$-normal coordinate system with center $0$ and any $\sigma \in Q^r$. Then by the invariance of $A$ and $A^{<\mu >}$ with respect to normal coordinates we deduce that $A=A^{<\mu >}$. $\square$

Remark 1. One can observe that $A^{\left[r\right]}=A^{<\mu >}$ for constant $\mu :Z^{\infty }\times Q^r\to Met\left(R^{L_r}\right)$ equal to the standard scalar multiplication, where $A^{\left[r\right]}$ is as in Example 4.

Remark 2. The map $\mu$ from Theorem 1 is not uniquely determined by $A$. One can observe that $A^{<{\mu }_1>}=A^{<{\mu }_2>}$ iff ${\mu }_1\left(j_0^{\infty }g,\sigma \right)={\mu }_2\left(j_0^{\infty }g,\sigma \right)$ for all Riemannian structures $g$ on $R^m$ such that the identity map $i{d}_{R^m}$ is a $g$-normal coordinate system with center $0$ and all $\sigma \in Q^r$.

References

Institute of Mathematics, Maria Curie-Sklodowska University, Pl. Marii Curie Sklodowska 1, Lublin, Poland

E-mail address: kurek@golem.umcs.lublin.pl

Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, Kraków, Poland

E-mail address: mikulski@im.uj.edu.pl

Received August 13, 2007