Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.

By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has $$\omega(M',f^*g)=f^*\omega(M,g).$$

Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:

1. the components $g_{ij}$ of the metric tensor;

2. a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;

3. the inverse of $\det g=\det(g_{ij})$.

Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$ $$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$

Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in Pontryagin forms (thus must have weight 0).

QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.

Below are some examples of such regular $0$-form (function) valued invariants of negative weight.

Example 1. The scalar curvature of the metric. It has weight -2.

Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted up using the metric. For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$: \begin{eqnarray*} H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e}, \end{eqnarray*} where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sign of this permutation is denoted above by $sgn(\alpha,\beta)$.

Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.

By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has $$\omega(M',f^*g)=f^*\omega(M,g).$$

Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:

1. the components $g_{ij}$ of the metric tensor;

2. a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;

3. the inverse of $\det g=\det(g_{ij})$.

Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$ $$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$

Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in the Pontryagin forms (thus must have weight 0).

QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.

Below are some examples of such regular $0$-form (function) valued invariants of negative weight.

Example 1. The scalar curvature of the metric. It has weight -2.

Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted up using the metric. For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$: \begin{eqnarray*} H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e}, \end{eqnarray*} where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sign of this permutation is denoted above by $sgn(\alpha,\beta)$.

Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.

By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has $$\omega(M',f^*g)=f^*\omega(M,g).$$

Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:

1. the components $g_{ij}$ of the metric tensor;

2. a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;

3. the inverse of $\det g=\det(g_{ij})$.

Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$ $$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$

Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in Pontryagin forms (thus must have weight 0).

QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.

Below are some examples of such regular $0$-form (function) valued invariants of negative weight.

Example 1. The scalar curvature of the metric. It has weight -1.

Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted up using the metric. For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$: \begin{eqnarray*} H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e}, \end{eqnarray*} where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sign of this permutation is denoted above by $sgn(\alpha,\beta)$.

Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.

By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has $$\omega(M',f^*g)=f^*\omega(M,g).$$

Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:

1. the components $g_{ij}$ of the metric tensor;

2. a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;

3. the inverse of $\det g=\det(g_{ij})$.

Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$ $$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$

Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in Pontryagin forms (thus must have weight 0).

QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.

Below are some examples of such regular $0$-form (function) valued invariants of negative weight.

Example 1. The scalar curvature of the metric. It has weight -2.

Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted up using the metric. For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$: \begin{eqnarray*} H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e}, \end{eqnarray*} where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sign of this permutation is denoted above by $sgn(\alpha,\beta)$.

Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.

By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has $$\omega(M',f^*g)=f^*\omega(M,g).$$

Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:

1. the components $g_{ij}$ of the metric tensor;

2. a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;

3. the inverse of $\det g=\det(g_{ij})$.

Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$ $$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$

Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in Pontryagin forms (thus must have weight 0).

QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.

Below are some examples of such regular $0$-form (function) valued invariants of negative weight.

Example 1. The scalar curvature of the metric. It has weight -1.

Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted by using the metric. For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$: \begin{eqnarray*} H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e}, \end{eqnarray*} where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sigh of this permutation is denoted above by $sgn(\alpha,\beta)$.

Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.

Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.

By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has $$\omega(M',f^*g)=f^*\omega(M,g).$$

Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:

1. the components $g_{ij}$ of the metric tensor;

2. a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;

3. the inverse of $\det g=\det(g_{ij})$.

Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$ $$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$

Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in Pontryagin forms (thus must have weight 0).

QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.

Below are some examples of such regular $0$-form (function) valued invariants of negative weight.

Example 1. The scalar curvature of the metric. It has weight -1.

Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted up using the metric. For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$: \begin{eqnarray*} H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e}, \end{eqnarray*} where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sign of this permutation is denoted above by $sgn(\alpha,\beta)$.

Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.