Incompatible Deformations in Relativistic Elasticity

Abstract

The work develops differential-geometric methods for modeling incompatible deformations and stresses caused by them in hyperelastic supermassive bodies within the framework of special and general relativity. Particular attention is paid to a unified geometric language used both for modeling distributed defects and gravitational interaction. It is shown that finite incompatible deformations that do not evolve in time can be formalized in terms of the curvature or torsion of the connection on the 3D material manifold obtained as the image of the projection of the body world tube, while the gravitational interaction is formalized in terms of the curvature and torsion of 4D space-time manifold. To describe the evolution of the material composition of a body, which occurs, for example, as a result of accretion, instead of a 3D material manifold, one should consider a body-tube in a 4D material space. In this case, from a formal mathematical point of view, the description of distributed defects and gravity becomes equivalent, and both, material manifold and space-time manifold are derived from a single fundamental object, which is a four-dimensional topological vector space. The developed approach can be used to model the processes of evolution of the stress-strain state of neutron star crusts and their local spontaneous destruction.

Appendix

1.1 REVIEW OF GENERAL AFFINE CONNECTIONS

1.2 1. Covariant Derivative and its Coordinate Representation

There are two approaches to the construction of geometric structures on manifolds. The first approach uses the ‘‘outer’’ geometry methodology, when a manifold is embedded in a Euclidean space of possibly large dimension¹⁷ and the Euclidean parallel transport rule is induced on this manifold, similar to how it is done in classical theory of surfaces. At the same time, despite the seeming simplicity, the implementation of this approach has at least two problems: 1) due to the high dimension, the formulas are cumbersome, and 2) the physical interpretation of the surrounding space is unknown.¹⁸ In this regard, as a rule, the second approach is used, based on the ‘‘internal’’ geometry methodology. Namely, the parallel transfer procedure is defined axiomatically, using only the structure of the considered manifold.¹⁹

Let $M$ be a smooth manifold of dimension $n$. By $\textrm{Vec}(M)$ we denote the $C^{\infty}(M)$-module of smooth vector fields on $M$.

Definition 1. Affine connection on manifold $M$ is a mapping $\nabla:\>\textrm{Vec}(M)\times\textrm{Vec}(M)\rightarrow\textrm{Vec}(M)$, $(\mathsf{u},\>\mathsf{v})\mapsto\nabla_{\mathsf{u}}\mathsf{v}$, that satisfies the following axioms [15, 97]:

• 1. $\forall\mathsf{u},\>\mathsf{v},\>\mathsf{w}\in\textrm{Vec}(M):\;\nabla_{\mathsf{u}+\mathsf{v}}\mathsf{w}=\nabla_{\mathsf{u}}\mathsf{w}+\nabla_{\mathsf{v}}\mathsf{w}$;

• 2. $\forall\mathsf{u},\>\mathsf{v}\in\textrm{Vec}(M)\>\forall f\in C^{\infty}(M):\;\nabla_{f\mathsf{u}}\mathsf{v}=f\nabla_{\mathsf{u}}\mathsf{v}$;

• 3. $\forall\mathsf{u},\>\mathsf{v},\>\mathsf{w}\in\textrm{Vec}(M):\;\nabla_{u}(\mathsf{v}+\mathsf{w})=\nabla_{\mathsf{u}}\mathsf{v}+\nabla_{\mathsf{u}}\mathsf{w}$;

• 4. $\forall\mathsf{u},\>\mathsf{v}\in\textrm{Vec}(M)\>\forall f\in C^{\infty}(M):\;\nabla_{\mathsf{u}}(f\mathsf{v})=f\nabla_{\mathsf{u}}\mathsf{v}+(\mathsf{u}f)\mathsf{v}$.

In Definition 1 the symbol $\mathsf{u}f$ denotes the action of vector field $\mathsf{u}$, interpreted as derivation, onto scalar field $f$. In coordinate neighborhood $U$ one thus has relation $\mathsf{u}f=u^{i}\,\frac{\partial f}{\partial x^{i}}$. The vector field $\nabla_{\mathsf{u}}\mathsf{v}$ is usually referred to as covariant derivative of $\mathsf{v}$ along vector field $\mathsf{u}$.

In arbitrary chart $(U,\,\varphi)=(U,\,x^{1},\,\ldots,\,x^{n})$ on manifold $M$ the restriction of connection $\nabla$ onto $U$ can be expressed in coordinate form. Indeed, let $(\partial_{i})_{i=1}^{n}$ be coordinate frame on $U$, and let $(dx^{i})_{i=1}^{n}$ be the corresponding coordinate coframe, i.e., $\left\langle dx^{i},\>\partial_{j}\right\rangle=\delta^{i}_{j}$, $i,\,j=1,\,\ldots,\,n$. Then, taking $\Gamma^{i}_{jk}:=\left\langle dx^{i},\>\nabla_{\partial_{j}}\partial_{k}\right\rangle$, one obtains relation $\nabla_{\partial_{j}}\partial_{k}=\Gamma^{i}_{jk}\partial_{i}$ on $U$. Scalar functions $\Gamma^{i}_{jk}\in C^{\infty}(U)$, $i,\>j,\>k=1,\>\ldots,\>n$, would be referred to as connection coefficients in chart $(U,\,\varphi)$. Furthermore, utilizing the axioms of affine connection, one obtains that for arbitrary vector fields $\mathsf{u},\,\mathsf{v}\in\textrm{Vec}(U)$, the following formula holds

$$\nabla_{\mathsf{u}}\mathsf{v}=u^{i}\left\{\partial_{i}v^{k}+v^{j}\Gamma^{k}_{ij}\right\}\partial_{k}.$$

Suppose that $(V,\,\psi)=(V,\,\widetilde{x}^{1},\,\ldots,\,\widetilde{x}^{n})$ is another chart on $M$, that overlaps with the chart $(U,\,\varphi)$, i.e., $U\cap V\neq\varnothing$. Denoting connection coefficients that correspond to the chart $(V,\>\psi)$ by $\widetilde{\Gamma}^{i}_{jk}$, one can show directly that on $U\cap V$ the following transformation law holds

$$\widetilde{\Gamma}^{i}_{jk}=\dfrac{\partial\widetilde{x}^{i}}{\partial x^{p}}\dfrac{\partial x^{l}}{\partial\widetilde{x}^{j}}\dfrac{\partial x^{m}}{\partial\widetilde{x}^{k}}\Gamma^{p}_{lm}+\dfrac{\partial\widetilde{x}^{i}}{\partial x^{p}}\dfrac{\partial^{2}x^{p}}{\partial\widetilde{x}^{j}\partial\widetilde{x}^{k}}.$$

(A1)

Formula (A1) shows, in particular, that quantities $\Gamma^{i}_{jk}$ do not form tensor field on $M$.

1.3 2. Parallel Transport

Let $\gamma:\>\mathbb{I}\rightarrow M$ be some smooth curve on $M$.

Definition 2. A vector field $\mathsf{u}\in\textrm{Vec}(M)$ is called parallel along $\gamma$, if for every value $t\in\mathbb{I}$ the following relation holds

$$\nabla_{\dot{\gamma}(t)}\mathsf{u}=\mathsf{0},$$

(A2)

where $\dot{\gamma}(t)$ is velocity vector of $\gamma$.

Choosing chart $(U,\,x^{1},\,\ldots,\,x^{n})$ on $M$, and supposing that the image of the curve $\gamma$ completely lies in $U$, one can express equation (A2) in components:

$$\dot{u}^{k}+\dot{x}^{i}u^{j}\Gamma^{k}_{ij}=0,\quad k=1,\>\ldots,\>n.$$

Here $\mathsf{u}=u^{i}\partial_{i}$, $\dot{\gamma}=\dot{x}^{i}\partial_{i}$, and the dot above denotes the operation of taking derivative $d/dt$.

If $p=\gamma(t_{0})$, then one can formulate initial condition $\mathsf{u}_{p}=\mathsf{u}_{0}$. Thus, in coordinate neighborhood $U$ there is a unique solution of the following Cauchy problem

$$\dot{u}^{k}+\dot{x}^{i}u^{j}\Gamma^{k}_{ij}=0,\quad k=1,\,\ldots,\,n,$$

$$u_{p}^{k}=u_{0}^{k},\quad k=1,\,\ldots,\,n.$$

(A3)

By this reason, one can define a mapping $P_{t_{0}}^{t}:\>T_{\gamma(t_{0})}M\rightarrow T_{\gamma(t)}M$, that with each vector $\mathsf{u}_{0}\in T_{\gamma(t_{0})}M$ associates value $\mathsf{u}(t)$ of the solution $\mathsf{u}$ of the Cauchy problem (A3) $P_{t_{0}}^{t}:\>\mathsf{u}_{0}\mapsto\mathsf{u}(t)$. This mapping is linear and characterizes parallel transport [98].

If $\mathsf{u},\>\mathsf{v}\in\textrm{Vec}(M)$ are two vector fields and if $\gamma:\>\mathbb{I}\rightarrow M$ is integral curve of $\mathsf{u}$, passing through point $p=\gamma(t_{0})$, then the value of covariant derivative $\nabla_{\mathsf{u}}\mathsf{v}|_{p}$ at $p$ can be calculated via the following limit

$$\nabla_{\mathsf{u}}\mathsf{v}|_{p}=\lim\limits_{h\to 0}\dfrac{P_{t_{0}+h}^{t_{0}}\mathsf{v}_{\gamma(t_{0}+h)}-\mathsf{v}_{\gamma(t_{0})}}{h}=\left.\dfrac{d}{dt}P_{t}^{t_{0}}\mathsf{v}_{\gamma(t)}\right|_{t=t_{0}}.$$

Note that vector $P_{t_{0}+h}^{t_{0}}\mathsf{v}_{\gamma(t_{0}+h)}$ belongs to $T_{\gamma(t_{0})}M$, so, for sufficiently small $h$ the difference $P_{t_{0}+h}^{t_{0}}\mathsf{v}_{\gamma(t_{0}+h)}-\mathsf{v}_{\gamma(t_{0})}$ is well-defined and belongs to $T_{\gamma(t_{0})}M$.

1.4 3. Torsion, Curvature and Nonmetricity

If manifold $M$ is endowed with some affine connection $\nabla$, then we will refer to the structure $(M,\,\nabla)$ as space of affine connection²⁰ [48]. In such the space one can define the notion of parallel transport, so, in particular, there is an opportunity to differentiate tensor fields, that is necessary for formulating of balance equations. If, moreover, the manifold $M$ is endowed with metric tensor $\mathsf{g}$, then one can consider a structure $(M,\,\mathsf{g},\,\nabla)$. We will refer to it as affine-metric manifold.

The parallel transport induced by the connection defines some geometry on the manifold, which, in the general case, is different from the Euclidean one. The connection coefficients are not suitable for describing this geometry: already in the Euclidean space, the connection coefficients are equal to zero in Cartesian coordinates, and are non-zero in curvilinear coordinates.²¹ In this regard, additional tensor fields characterizing the degree of non-Euclideness are introduced.

For affine-metric manifold $(M,\,\mathsf{g},\,\nabla)$ the connection $\nabla$ defines tensor fields of torsion $\mathfrak{T}:\>\textrm{Vec}(M)\times\textrm{Vec}(M)\rightarrow\textrm{Vec}(M)$, curvature $\mathfrak{R}:\>\textrm{Vec}(M)\times\textrm{Vec}(M)\times\textrm{Vec}(M)\rightarrow\textrm{Vec}(M)$, and nonmetricity $\mathfrak{Q}:\>\textrm{Vec}(M)\times\textrm{Vec}(M)\times\textrm{Vec}(M)\rightarrow C^{\infty}(M)$ upon formulae [15, 99]

$$\mathfrak{T}(\mathsf{u},\>\mathsf{v}):=\nabla_{\mathsf{u}}\mathsf{v}-\nabla_{\mathsf{v}}\mathsf{u}-[\mathsf{u},\>\mathsf{v}];$$

(A4)

$$\mathfrak{R}(\mathsf{u},\>\mathsf{v},\>\mathsf{w}):=[\nabla_{\mathsf{u}},\>\nabla_{\mathsf{v}}]\mathsf{w}-\nabla_{[\mathsf{u},\>\mathsf{v}]}\mathsf{w};$$

(A5)

$$\mathfrak{Q}(\mathsf{u},\>\mathsf{v},\>\mathsf{w}):=\mathsf{g}(\nabla_{\mathsf{u}}\mathsf{v},\>\mathsf{w})+\mathsf{g}(\mathsf{v},\>\nabla_{\mathsf{u}}\mathsf{w})-\mathsf{u}[\mathsf{g}(\mathsf{v},\>\mathsf{w})].$$

(A6)

Here²²$[\mathsf{u},\>\mathsf{v}]$ and $[\nabla_{\mathsf{u}},\>\nabla_{\mathsf{v}}]$ are commutators: $[\mathsf{u},\,\mathsf{v}]=\mathsf{u}\circ\mathsf{v}-\mathsf{v}\circ\mathsf{u}$, $[\nabla_{\mathsf{u}},\,\nabla_{\mathsf{v}}]=\nabla_{\mathsf{u}}\circ\nabla_{\mathsf{v}}-\nabla_{\mathsf{v}}\circ\nabla_{\mathsf{u}}$, while $\mathsf{u}[\mathsf{g}(\mathsf{v},\,\mathsf{w})]$ is action of vector field $\mathsf{u}$ onto scalar field $\mathsf{g}(\mathsf{v},\,\mathsf{w})$.

In chart $(U,\,\varphi)$ the tensor fields $\mathfrak{T}$, $\mathfrak{R}$ and $\mathfrak{Q}$ (rather, their restrictions on $U$) have the following components

$$\mathfrak{T}^{k}_{ij}=\Gamma^{k}_{ij}-\Gamma^{k}_{ji};$$

$$\mathfrak{R}^{t}_{ijk}=\partial_{i}\Gamma^{t}_{jk}-\partial_{j}\Gamma^{t}_{ik}+\Gamma^{l}_{jk}\Gamma^{t}_{il}-\Gamma^{l}_{ik}\Gamma^{t}_{jl};$$

$$-\mathfrak{Q}_{ijk}=\partial_{i}g_{jk}-\Gamma^{m}_{ij}g_{mk}-\Gamma^{m}_{ik}g_{mj},$$

where $i,\,j,\,k,\,t=1,\,\ldots,\,n$.

1.5 4. Connection in Non-holonomic Frame

Suppose now, that in the coordinate neighborhood $U$ of a chart $(U,\,\varphi)=(U,\,x^{1},\,\ldots,\,x^{n})$ there is a field of non-degenerate $n\times n$-matrices $\Omega=[\Omega^{i}_{j}]:\>U\rightarrow\textrm{GL}(n;\>\mathbb{R})$. With respect to coordinate frame $(\partial_{i})_{i=1}^{n}$ this field defines a new frame $(\mathsf{z}_{i})_{i=1}^{n}$ on the same set $U$ as $\mathsf{z}_{i}=\Omega^{j}_{i}\partial_{j}$. In contrast to the coordinate frame, the latter frame may be non-holonomic, i.e., $[\mathsf{z}_{i},\>\mathsf{z}_{j}]\neq\mathsf{0}$. From physical viewpoint the frame $(\mathsf{z}_{i})_{i=1}^{n}$ may represent the collection of distorted crystal axes, while field $\Omega$ is the distortion field.

The coframe $(\vartheta^{i})_{i=1}^{n}$, dual to $(\mathsf{z}_{i})_{i=1}^{n}$, is determined upon the system of $n^{2}$ equalities: $\left\langle\vartheta^{i},\>\mathsf{z}_{j}\right\rangle=\delta^{i}_{j}$, $i,\,j=1,\,\ldots,\,n$. Explicitly, $\vartheta^{i}=\mho^{i}_{j}dx^{j}$. Here $(dx^{i})_{i=1}^{n}$ is the coordinate coframe, and $\mho=\Omega^{-1}$ is the field of inverse matrices, i.e., $\mho^{i}_{k}\,\Omega^{k}_{j}=\Omega^{i}_{k}\,\mho^{k}_{j}=\delta^{i}_{j}$ on $U$.

Since for all $i,\,j=1,\,\ldots,\,n$, the commutator $[\mathsf{z}_{i},\,\mathsf{z}_{j}]$ is a vector field on $U$, one can decompose it with respect to the frame $(\mathsf{z}_{i})_{i=1}^{n}$: $[\mathsf{z}_{i},\,\mathsf{z}_{j}]=-c^{k}_{ij}\mathsf{z}_{k}$. The coefficients $c^{k}_{ij}\in C^{\infty}(U)$ of this expansion are referred to as the objects of anholonomity and can be expressed explicitly in terms of matrix fields $\Omega$ and $\mho$:

Theorem 1. The fields $c^{k}_{ij}$ can be determined upon equality $c^{k}_{ij}=\mho^{k}_{m}\left(\Omega^{s}_{j}\,\partial_{s}\Omega^{m}_{i}-\Omega^{s}_{i}\,\partial_{s}\Omega^{m}_{j}\right),$ or, equivalently, from $c^{k}_{ij}=\Omega^{m}_{i}\Omega^{q}_{j}\left(\partial_{m}\mho^{k}_{q}-\partial_{q}\mho^{k}_{m}\right).$

The proof of this claim can be found in ([12], p. 222).

The objects of anholonomity are components of exterior differential $d\vartheta^{k}$ with respect to coframe $(\vartheta^{i})_{i=1}^{n}$.

Theorem 2. The following relation holds

$$d\vartheta^{k}=\displaystyle\sum\limits_{p<q}c^{k}_{pq}\vartheta^{p}\wedge\vartheta^{q}.$$

(A7)

The proof can be found in ([12], p. 223).

Now, let $\gamma^{i}_{jk}=\left\langle\vartheta^{i},\>\nabla_{\mathsf{z}_{j}}\mathsf{z}_{k}\right\rangle$ and $\Gamma^{i}_{jk}=\left\langle dx^{i},\>\nabla_{\partial_{j}}\partial_{k}\right\rangle$ be coefficients of connection $\nabla$ with respect to frames $(\mathsf{z}_{i})_{i=1}^{n}$ and $(\partial_{i})_{i=1}^{n}$:

$$\nabla_{\mathsf{z}_{j}}\mathsf{z}_{k}=\gamma^{i}_{jk}\mathsf{z}_{i}\quad\text{and}\quad\nabla_{\partial_{j}}\partial_{k}=\Gamma^{i}_{jk}\partial_{i}.$$

(A8)

The next claim expresses relation between $\gamma^{i}_{jk}$ and $\Gamma^{i}_{jk}$.

Theorem 3. Fields $\gamma^{i}_{jk}$ and $\Gamma^{i}_{jk}$ on $U$, defined by relations (A8), are related as

$$\gamma^{i}_{jk}=\Gamma^{m}_{sq}\mho^{i}_{m}\Omega^{s}_{j}\Omega^{q}_{k}+\mho^{i}_{m}\Omega^{s}_{j}\partial_{x^{s}}\Omega^{m}_{k}.$$

(A9)

For the proof see ([12], p. 213, Section 9.2.1).

From equality (A9), as particular case (when $(\mathsf{z}_{i})_{i=1}^{n}$ is coordinate frame, generated by another chart), the equality (A1) follows.

Define components of torsion, curvature and nonmetricity in non-holonomic frame. Let

$$\mathfrak{T}=\mathfrak{T}^{k}_{ij}\,\mathsf{z}_{k}\otimes\vartheta^{i}\otimes\vartheta^{j},\quad\mathfrak{R}=\mathfrak{R}^{m}_{ijk}\,\mathsf{z}_{m}\otimes\vartheta^{i}\otimes\vartheta^{j}\otimes\vartheta^{k},\quad\mathfrak{Q}=\mathfrak{Q}_{ijk}\,\vartheta^{i}\otimes\vartheta^{j}\otimes\vartheta^{k}$$

be polyadic decompositions of fields $\mathfrak{T}$, $\mathfrak{R}$ and $\mathfrak{Q}$ with respect to the frame $(\mathsf{z}_{i})_{i=1}^{n}$ and coframe $(\vartheta^{i})_{i=1}^{n}$. Expressions of components of these expansions are stated in the following claim.

Theorem 4. With respect to the frame $(\mathsf{z}_{i})_{i=1}^{n}$ , related with the coordinate frame $(\partial_{i})_{i=1}^{n}$ by equalities $\mathsf{z}_{i}=\Omega^{j}_{i}\partial_{j}$ , $i=1,\,\ldots,\,n$ , one has the following formulas

$$\mathfrak{T}^{k}_{ij}=\gamma^{k}_{ij}-\gamma^{k}_{ji}+c^{k}_{ij},$$

(A10)

$$\mathfrak{R}^{m}_{ijk}=\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}-\Omega^{s}_{j}\,\partial_{s}\gamma^{m}_{ik}+\gamma^{s}_{jk}\gamma^{m}_{is}-\gamma^{s}_{ik}\gamma^{m}_{js}+c^{s}_{ij}\gamma^{m}_{sk},$$

(A11)

$$-\mathfrak{Q}_{ijk}=\Omega^{s}_{i}\,\partial_{s}g_{jk}-g_{mk}\gamma^{m}_{ij}-g_{jm}\gamma^{m}_{ik}.$$

(A12)

Here $\gamma^{i}_{jk}$ are coefficients of connection $\nabla$ in frame $(\mathsf{z}_{i})_{i=1}^{n}$, while $g_{ij}$ are metric tensor components; $g_{ij}=\mathsf{g}(\mathsf{z}_{i},\,\mathsf{z}_{j})$.

Proof. Due to formula (A4),

$$\mathfrak{T}(\mathsf{z}_{i},\>\mathsf{z}_{j})=\nabla_{\mathsf{z}_{i}}\mathsf{z}_{j}-\nabla_{\mathsf{z}_{j}}\mathsf{z}_{i}-[\mathsf{z}_{i},\>\mathsf{z}_{j}]=\gamma^{k}_{ij}\mathsf{z}_{k}-\gamma^{k}_{ji}\mathsf{z}_{k}+c^{k}_{ij}\mathsf{z}_{k},$$

that with respect to equality $\mathfrak{T}^{k}_{ij}=\left\langle\vartheta^{k},\>\mathfrak{T}(\mathsf{z}_{i},\>\mathsf{z}_{j})\right\rangle$ leads to (A10).

Using definition of curvature (formula (A5)), one gets

$$\mathfrak{R}(\mathsf{z}_{i},\>\mathsf{z}_{j},\>\mathsf{z}_{k})=\nabla_{\mathsf{z}_{i}}\nabla_{\mathsf{z}_{j}}\mathsf{z}_{k}-\nabla_{\mathsf{z}_{j}}\nabla_{\mathsf{z}_{i}}\mathsf{z}_{k}-\nabla_{[\mathsf{z}_{i},\>\mathsf{z}_{j}]}\mathsf{z}_{k}$$

$${}=\nabla_{\mathsf{z}_{i}}\left(\gamma^{m}_{jk}\mathsf{z}_{m}\right)-\nabla_{\mathsf{z}_{j}}\left(\gamma^{m}_{ik}\mathsf{z}_{m}\right)-\nabla_{(-c^{s}_{ij}\mathsf{z}_{s})}\mathsf{z}_{k}$$

$${}=\gamma^{m}_{jk}\nabla_{\mathsf{z}_{i}}\mathsf{z}_{m}+\mathsf{z}_{i}[\gamma^{m}_{jk}]\mathsf{z}_{m}-\gamma^{m}_{ik}\nabla_{\mathsf{z}_{j}}\mathsf{z}_{m}-\mathsf{z}_{j}[\gamma^{m}_{ik}]\mathsf{z}_{m}+c^{s}_{ij}\nabla_{\mathsf{z}_{s}}\mathsf{z}_{k}$$

$${}=\gamma^{m}_{jk}\gamma^{s}_{im}\mathsf{z}_{s}+\Omega^{s}_{i}\partial_{s}\gamma^{m}_{jk}\mathsf{z}_{m}-\gamma^{m}_{ik}\gamma^{s}_{jm}\mathsf{z}_{s}-\Omega^{s}_{j}\partial_{s}\gamma^{m}_{ik}\mathsf{z}_{m}+c^{s}_{ij}\gamma^{m}_{sk}\mathsf{z}_{m}.$$

From the latter equality, according to relation $\mathfrak{R}^{m}_{ijk}=\left\langle\vartheta^{m},\>\mathfrak{R}(\mathsf{z}_{i},\>\mathsf{z}_{j},\>\mathsf{z}_{k})\right\rangle$, the formula (A11) follows. Finally, for nonmetricity (formula (A6)) one gets

$$\mathfrak{Q}_{ijk}=\mathfrak{Q}(\mathsf{z}_{i},\>\mathsf{z}_{j},\>\mathsf{z}_{k})=\mathsf{g}(\nabla_{\mathsf{z}_{i}}\mathsf{z}_{j},\>\mathsf{z}_{k})+\mathsf{g}(\mathsf{z}_{j},\>\nabla_{\mathsf{z}_{i}}\mathsf{z}_{k})-\mathsf{z}_{i}[\mathsf{g}(\mathsf{z}_{j},\>\mathsf{z}_{k})]$$

$${}=\mathsf{g}(\gamma^{m}_{ij}\mathsf{z}_{m},\>\mathsf{z}_{k})+\mathsf{g}(\mathsf{z}_{j},\>\gamma^{m}_{ik}\mathsf{z}_{m})-\Omega^{s}_{i}\partial_{s}g_{jk}=\gamma^{m}_{ij}g_{mk}+\gamma^{m}_{ik}g_{jm}-\Omega^{s}_{i}\partial_{s}g_{jk},$$

and then the equality (A12) directly follows. $\Box$

Using formulae (A10) and (A12), one can obtain direct expression for connection coefficients $\gamma^{i}_{jk}$ in terms of metric, torsion, nonmetricity and objects of anholonomity.

Theorem 5. The following relation holds

$$\gamma^{i}_{jk}=\frac{g^{is}}{2}\left[\mathsf{z}_{j}(g_{sk})+\mathsf{z}_{k}(g_{js})-\mathsf{z}_{s}(g_{jk})\right]+\frac{g^{is}}{2}\left(g_{jm}c^{m}_{ks}-g_{ms}c^{m}_{jk}-g_{mk}c^{m}_{sj}\right)$$

$${}+\frac{g^{is}}{2}\left(g_{ms}\mathfrak{T}^{m}_{jk}+g_{mk}\mathfrak{T}^{m}_{sj}-g_{mj}\mathfrak{T}^{m}_{ks}\right)+\frac{g^{is}}{2}\left(\mathfrak{Q}_{jks}+\mathfrak{Q}_{ksj}-\mathfrak{Q}_{sjk}\right).$$

(A13)

Proof. Under cyclic permutation $(i,\,j,\,k)\mapsto(k,\,i,\,j)\mapsto(j,\,k,\,i)$ one gets three counterparts of formula (A12):

$$\mathsf{z}_{i}(g_{jk})-g_{mk}\gamma^{m}_{ij}-g_{jm}\gamma^{m}_{ik}=-\mathfrak{Q}_{ijk},$$

$$\mathsf{z}_{k}(g_{ij})-g_{mj}\gamma^{m}_{ki}-g_{im}\gamma^{m}_{kj}=-\mathfrak{Q}_{kij},\quad\mathsf{z}_{j}(g_{ki})-g_{mi}\gamma^{m}_{jk}-g_{km}\gamma^{m}_{ji}=-\mathfrak{Q}_{jki}$$

(here we take into account that $\mathsf{z}_{i}(g_{jk})=\Omega^{s}_{i}\,\partial_{s}g_{jk}$). Adding two latter formulae with each other and subtracting the first formula from the result, one gets

$$\mathsf{z}_{j}(g_{ki})+\mathsf{z}_{k}(g_{ij})-\mathsf{z}_{i}(g_{jk})-g_{mi}\gamma^{m}_{jk}-g_{im}\gamma^{m}_{kj}+g_{mk}(\gamma^{m}_{ij}-\gamma^{m}_{ji})-g_{jm}(\gamma^{m}_{ki}-\gamma^{m}_{ik})$$

$${}=\mathfrak{Q}_{ijk}-\mathfrak{Q}_{kij}-\mathfrak{Q}_{jki}.$$

Expressing the fourth term in the left hand side as $g_{mi}\gamma^{m}_{jk}=2g_{mi}\gamma^{m}_{jk}-g_{mi}\gamma^{m}_{jk}$, and then replacing the differences $\gamma^{m}_{jk}-\gamma^{m}_{kj}$, $\gamma^{m}_{ij}-\gamma^{m}_{ji}$, $\gamma^{m}_{ki}-\gamma^{m}_{ik}$, according to formula (A10), one gets

$$\mathsf{z}_{j}(g_{ki})+\mathsf{z}_{k}(g_{ij})-\mathsf{z}_{i}(g_{jk})-2g_{mi}\gamma^{m}_{jk}+g_{mi}(\mathfrak{T}^{m}_{jk}-c^{m}_{jk})+g_{mk}(\mathfrak{T}^{m}_{ij}-c^{m}_{ij})$$

$${}-g_{jm}(\mathfrak{T}^{m}_{ki}-c^{m}_{ki})=\mathfrak{Q}_{ijk}-\mathfrak{Q}_{kij}-\mathfrak{Q}_{jki}.$$

Replacing all occurrences of index $i$ by $s$ and performing simple algebraic transformations, one arrives at the formula

$$g_{ms}\gamma^{m}_{jk}=\frac{1}{2}\left[\mathsf{z}_{j}(g_{sk})+\mathsf{z}_{k}(g_{js})-\mathsf{z}_{s}(g_{jk})\right]+\frac{1}{2}\left(g_{jm}c^{m}_{ks}-g_{ms}c^{m}_{jk}-g_{mk}c^{m}_{sj}\right)$$

$${}+\frac{1}{2}\left(g_{ms}\mathfrak{T}^{m}_{jk}+g_{mk}\mathfrak{T}^{m}_{sj}-g_{mj}\mathfrak{T}^{m}_{ks}\right)+\frac{1}{2}\left(\mathfrak{Q}_{jks}+\mathfrak{Q}_{ksj}-\mathfrak{Q}_{sjk}\right).$$

Finally, multiplying both sides of the latter relation by $g^{is}$ and performing summation over $s$, one obtains (A13). $\Box$

In particular case, when the frame $(\mathsf{z}_{i})_{i=1}^{n}$ is $\mathsf{g}$-orthonormal, one gets that $g_{ij}=\delta_{ij}$, $g^{ij}=\delta^{ij}$, and the equality (A13) takes the form

$$\gamma^{i}_{jk}=\frac{1}{2}\left(c^{j}_{ki}-c^{i}_{jk}-c^{k}_{ij}\right)+\frac{1}{2}\left(\mathfrak{T}^{i}_{jk}+\mathfrak{T}^{k}_{ij}-\mathfrak{T}^{j}_{ki}\right)+\frac{1}{2}\left(\mathfrak{Q}_{jki}+\mathfrak{Q}_{kij}-\mathfrak{Q}_{ijk}\right).$$

1.6 5. Cartan Structural Equations and Bianchi Identities

It is convenient to represent affine connection in terms of family of $1$-forms, since within such representation one can obtain relations, that define the pair $(\nabla,\,(\vartheta_{i})_{i=1}^{n})$ as solution of system of equations, which right hand sides are torsion, curvature and nonmetricity, expressed in terms of $2$-forms (for torsion and curvature) and in terms of $1$-forms (for curvature). These relations are referred to as Cartan equations, since Cartan was first who derived them.

Suppose that on some coordinate neighborhood $U\subset M$ the frame $(\mathsf{z}_{i})_{i=1}^{n}$ is defined. This field is related with the original coordinate frame $(\partial_{i})_{i=1}^{n}$ as $\mathsf{z}_{i}=\Omega^{j}_{i}\partial_{j}$, where $\Omega=[\Omega^{i}_{j}]$ is smooth field of invertible matrices on $U$.

Define $1$-forms $\omega^{i}_{k}:=\gamma^{i}_{jk}\vartheta^{j}$, where $\gamma^{i}_{jk}$ are coefficients of connection $\nabla$ with respect to the frame $(\mathsf{z}_{i})_{i=1}^{n}$, while $(\vartheta^{i})_{i=1}^{n}$ is coframe, dual to $(\mathsf{z}_{i})_{i=1}^{n}$. We will refer to each $1$-form $\omega^{i}_{k}$ as connection form.

Introduce families of forms $Q_{jk}$, $T^{i}$ and $R^{i}_{j}$ as

$$Q_{jk}=\mathfrak{Q}_{ijk}\vartheta^{i},\quad T^{i}=\dfrac{1}{2}\mathfrak{T}^{i}_{ms}\vartheta^{m}\wedge\vartheta^{s},\quad R^{i}_{j}=\dfrac{1}{2}\mathfrak{R}^{i}_{klj}\vartheta^{k}\wedge\vartheta^{l}.$$

Here $\mathfrak{Q}_{ijk}$, $\mathfrak{T}^{i}_{ms}$ and $\mathfrak{R}^{i}_{klj}$ are components of the corresponding fields with respect to $(\mathsf{z}_{i})$. The field $Q_{jk}$ is referred to as nonmetricity form, the quantity $T^{i}$ is torsion form, while $R^{i}_{j}$ is curvature form.

The following claim is central for this exposition.

Theorem 6 (Cartan Structural Equations). The following relations take place:

$$g_{mk}\omega^{m}_{j}+g_{mj}\omega^{m}_{k}-dg_{jk}=Q_{jk}$$

(A14)

($0th$ Cartan structural equation; here $g_{jk}=\mathsf{g}(\mathsf{z}_{j},\,\mathsf{z}_{k})$),

$$d\vartheta^{i}+\omega^{i}_{j}\wedge\vartheta^{j}=T^{i}$$

(A15)

($1$ st Cartan structural equation) and

$$d\omega^{i}_{j}+\omega^{i}_{k}\wedge\omega^{k}_{j}=R^{i}_{j}$$

(A16)

( $2$ nd Cartan structural equation).

Proof. Cartan structural equations are nothing but the reformulations of equalities (A10)–(A12) in terms of differential forms.

Indeed, multiply both sides of equality (A12) on $\vartheta^{i}$ from the right and sum over $i$:

$$-\Omega^{s}_{i}\partial_{s}g_{jk}\vartheta^{i}+g_{mk}\gamma^{m}_{ij}\vartheta^{i}+g_{jm}\gamma^{m}_{ik}\vartheta^{i}=\mathfrak{Q}_{ijk}\vartheta^{i}.$$

The right hand side of the obtained relation is just $Q_{jk}$. As for the left hand side, its second and third terms are equal, respectively, to $g_{mk}\omega^{m}_{j}$ and $g_{jm}\omega^{m}_{k}$. Finally, the first term is $dg_{jk}$:

$$dg_{jk}=\partial_{s}g_{jk}dx^{s}=\Omega^{s}_{i}\partial_{s}g_{jk}\vartheta^{i}.$$

All this implies (A14).

Multiplying both sides of the equality (A10) on $2$-covector $\dfrac{1}{2}\vartheta^{i}\wedge\vartheta^{j}$ from the right and summing over all values $i,\,j$, one has

$$\dfrac{1}{2}\gamma^{k}_{ij}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\gamma^{k}_{ji}\vartheta^{i}\wedge\vartheta^{j}+\dfrac{1}{2}c^{k}_{ij}\vartheta^{i}\wedge\vartheta^{j}=\dfrac{1}{2}\mathfrak{T}^{k}_{ij}\vartheta^{i}\wedge\vartheta^{j}.$$

The right hand side of the obtained equality is torsion form $T^{k}$. The third term of the left hand side, according to (A7), is equal to differential $d\vartheta^{k}$. As for the first two terms on the left hand side, they give

$$\dfrac{1}{2}\gamma^{k}_{ij}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\gamma^{k}_{ji}\vartheta^{i}\wedge\vartheta^{j}=\dfrac{1}{2}\omega^{k}_{j}\wedge\vartheta^{j}+\dfrac{1}{2}\omega^{k}_{i}\wedge\vartheta^{i}=\omega^{k}_{i}\wedge\vartheta^{i}$$

(in the second term, the factors of the $2$-covector are interchanged). Thus, up to index notation, one has (A15).

Finally, multiply both sides of the equality (A11) on $\dfrac{1}{2}\vartheta^{i}\wedge\vartheta^{j}$ from the right and sum over all values $i,\>j$. This results in the following formula

$$\dfrac{1}{2}\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\Omega^{s}_{j}\,\partial_{s}\gamma^{m}_{ik}\vartheta^{i}\wedge\vartheta^{j}+\dfrac{1}{2}\gamma^{s}_{jk}\gamma^{m}_{is}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\gamma^{s}_{ik}\gamma^{m}_{js}\vartheta^{i}\wedge\vartheta^{j}$$

$${}+\dfrac{1}{2}c^{s}_{ij}\gamma^{m}_{sk}\vartheta^{i}\wedge\vartheta^{j}=\dfrac{1}{2}\mathfrak{R}^{m}_{ijk}\vartheta^{i}\wedge\vartheta^{j},$$

the right hand side of which is curvature form $R^{m}_{k}$. The third and fourth terms give

$$\dfrac{1}{2}\gamma^{s}_{jk}\gamma^{m}_{is}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\gamma^{s}_{ik}\gamma^{m}_{js}\vartheta^{i}\wedge\vartheta^{j}=\dfrac{1}{2}\omega^{m}_{s}\wedge\omega^{s}_{k}-\dfrac{1}{2}\omega^{s}_{k}\wedge\omega^{m}_{s}=\omega^{m}_{s}\wedge\omega^{s}_{k}.$$

It remains to consider the sum

$$\dfrac{1}{2}\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\Omega^{s}_{j}\,\partial_{s}\gamma^{m}_{ik}\vartheta^{i}\wedge\vartheta^{j}+\dfrac{1}{2}c^{s}_{ij}\gamma^{m}_{sk}\vartheta^{i}\wedge\vartheta^{j}.$$

(A17)

Interchanging indices $i$ and $j$ in the second term and then interchanging factors of $2$-covector, one gets

$$\dfrac{1}{2}\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{j}\wedge\vartheta^{i}=\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}.$$

Thus, (A17) has the form

$$\dfrac{1}{2}\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\Omega^{s}_{j}\,\partial_{s}\gamma^{m}_{ik}\vartheta^{i}\wedge\vartheta^{j}+\dfrac{1}{2}c^{s}_{ij}\gamma^{m}_{sk}\vartheta^{i}\wedge\vartheta^{j}=\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}+\gamma^{m}_{sk}d\vartheta^{s},$$

where the equality (A7) was utilized as well. The first term of the resulting expression can also be transformed

$$\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}=\partial_{s}\gamma^{m}_{jk}dx^{s}\wedge\vartheta^{j}=d\gamma^{m}_{sk}\wedge\vartheta^{s}.$$

Therefore, in fact, the expression (A17) is equal to

$$\dfrac{1}{2}\Omega^{s}_{i}\,\partial_{s}\gamma^{m}_{jk}\vartheta^{i}\wedge\vartheta^{j}-\dfrac{1}{2}\Omega^{s}_{j}\,\partial_{s}\gamma^{m}_{ik}\vartheta^{i}\wedge\vartheta^{j}+\dfrac{1}{2}c^{s}_{ij}\gamma^{m}_{sk}\vartheta^{i}\wedge\vartheta^{j}$$

$${}=d\gamma^{m}_{sk}\wedge\vartheta^{s}+\gamma^{m}_{sk}d\vartheta^{s}=d(\gamma^{m}_{sk}\wedge\vartheta^{s})=d\omega^{m}_{k}.$$

Collecting the obtained results, one arrives at the relation (A16). $\Box$

Cartan structural equations make it possible to derive relations for the differentials of torsion, curvature, and non-metricity.

Theorem 7 (Bianchi Identities). The following identities hold

$$dQ_{jk}=g_{mk}R^{m}_{j}+g_{mj}R^{m}_{k}-Q_{mk}\wedge\omega^{m}_{j}-Q_{mj}\wedge\omega^{m}_{k},$$

(A18)

$$dT^{i}=R^{i}_{j}\wedge\vartheta^{j}-\omega^{i}_{j}\wedge T^{j},$$

(A19)

$$dR^{i}_{j}=R^{i}_{k}\wedge\omega^{k}_{j}-\omega^{i}_{k}\wedge R^{k}_{j}.$$

(A20)

Proof. Each of the relations (A18)–(A20) can be obtained just with Cartan equations (A14)–(A16) only. Indeed, applying the exterior differential to both sides of $0$th Cartan equation (A14), one gets²³

$$dQ_{jk}=dg_{mk}\wedge\omega^{m}_{j}+g_{mk}d\omega^{m}_{j}+dg_{mj}\wedge\omega^{m}_{k}+g_{mj}d\omega^{m}_{k}.$$

(A21)

From another perspective, $0$th and $2$nd Cartan equations (A14) and (A16) imply

$$dg_{mk}=g_{sk}\omega^{s}_{m}+g_{sm}\omega^{s}_{k}-Q_{mk},\quad dg_{mj}=g_{sj}\omega^{s}_{m}+g_{sm}\omega^{s}_{j}-Q_{mj},$$

$$d\omega^{m}_{j}=R^{m}_{j}-\omega^{m}_{s}\wedge\omega^{s}_{j},\quad d\omega^{m}_{k}=R^{m}_{k}-\omega^{m}_{s}\wedge\omega^{s}_{k}.$$

Substituting the obtained expressions into (A21) and collecting similar terms, one arrives at (A18).

Acting by exterior derivative $d$ onto both sides of the $1$st Cartan equation (A15) and taking into account that $d\circ d=0$, one gets

$$dT^{i}=d\omega^{i}_{j}\wedge\vartheta^{j}-\omega^{i}_{j}\wedge d\vartheta^{j}.$$

(A22)

Further, from the first and second Cartan equations (A15) and (A16) one can obtain expressions for $d\omega^{i}_{j}$ and $d\vartheta^{j}$:

$$d\omega^{i}_{j}=R^{i}_{j}-\omega^{i}_{k}\wedge\omega^{k}_{j},\quad d\vartheta^{j}=T^{j}-\omega^{j}_{k}\wedge\vartheta^{k}.$$

Their substitution into (A22) gives (A19).

Finally, let us act by exterior derivative $d$ onto $2$nd Cartan equation (A16) (and, like in previous cases, take into account that $d\circ d=0$):

$$dR^{i}_{j}=d\omega^{i}_{k}\wedge\omega^{k}_{j}-\omega^{i}_{k}\wedge d\omega^{k}_{j}.$$

(A23)

On the other hand, from the second Cartan equation one can obtain representations for $d\omega^{i}_{k}$ and $d\omega^{k}_{j}$:

$$d\omega^{i}_{k}=R^{i}_{k}-\omega^{i}_{s}\wedge\omega^{s}_{k},\quad d\omega^{k}_{j}=R^{k}_{j}-\omega^{k}_{s}\wedge\omega^{s}_{j}.$$

Substituting them into (A23), one obtains (A20). $\Box$

We will call the equalities (A18)–(A20) as Bianchi identities.

The Cartan equations and the Bianchi identities can be written in a more concise form. This can be achieved if one introduces the generalization of exterior derivative, namely, covariant exterior derivative ([100], p. 139). Consider family of $p$-forms

$$\Pi^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots j_{l}}=\frac{1}{p!}A^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots j_{l}j_{l+1}j_{l+p}}\,\vartheta^{j_{l+1}}\wedge\cdots\wedge\vartheta^{j_{l+p}},\quad i_{1},\,\ldots,\,i_{k},\,j_{1},\,\ldots,\,j_{l}=1,\,\ldots,\,n,$$

where $A^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots j_{l}j_{l+1}j_{l+p}}$ are components of $(k,\,l+p)$-tensor, that are antisymmetric with respect to the latter $p$ lower indices. Then, covariant exterior derivative $D\Pi^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots j_{l}}$ is defined as

$$D\Pi^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots j_{l}}=d\Pi^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots j_{l}}+\omega^{i_{1}}_{s}\wedge\Pi^{s\ldots i_{k}}{}_{j_{1}\ldots j_{l}}+\cdots+\omega^{i_{k}}_{s}\wedge\Pi^{i_{1}\ldots s}{}_{j_{1}\ldots j_{l}}$$

$${}-\omega^{s}_{j_{1}}\wedge\Pi^{i_{1}\ldots i_{k}}{}_{s\ldots j_{l}}-\cdots-\omega^{s}_{j_{l}}\wedge\Pi^{i_{1}\ldots i_{k}}{}_{j_{1}\ldots s},$$

(A24)

where $d$ is exterior differential. In particular,

$$D\Pi^{i}=d\Pi^{i}+\omega^{i}_{k}\wedge\Pi^{k}\quad\text{and}\quad D\Pi^{i}_{j}=d\Pi^{i}_{j}+\omega^{i}_{k}\wedge\Pi^{k}_{j}-\omega^{k}_{j}\wedge\Pi^{i}_{k}.$$

Thus, $D$ increases the rank of the ‘‘outer’’ part by $1$.

Using operation $D$, one can rewrite Cartan equations (A14)–(A16) as

$$Dg_{ij}=-Q_{ij},\quad D\vartheta^{i}=T^{i},\quad D\omega^{i}_{j}=R^{i}_{j}.$$

(A25)

The Bianchi identities (A18)–(A20), in turn, are represented as

$$DQ_{jk}=g_{mk}R^{m}_{j}+g_{mj}R^{m}_{k},\quad DT^{i}=R^{i}_{j}\wedge\vartheta^{j},\quad DR^{i}_{j}=\mathsf{0}.$$

(A26)