Objective Rates as Covariant Derivatives on the Manifold of Riemannian Metrics

Abstract

The subject of so-called objective derivatives in Continuum Mechanics has a long history and has generated varying views concerning their true mathematical interpretation. Several attempts have been made to provide a mathematical definition that would at least partially unify the existing notions. In this paper, we demonstrate that, under natural assumptions, all objective derivatives correspond to covariant derivatives on the infinite-dimensional manifold $\textrm{Met}(\mathcal {B})$ of Riemannian metrics on the body. Furthermore, a natural Leibniz rule enables canonical extensions from covariant to contravariant tensor fields and vice versa. This makes the sometimes-used distinction between objective derivatives of “Lie type” and “co-rotational type” unnecessary. For an exhaustive list of objective derivatives found in the literature, we exhibit the corresponding covariant derivative on $\textrm{Met}(\mathcal {B})$.

Appendices

Appendix A. Second-Order Tensors and Their Interpretations

Given a real (finite dimensional) vector space E and denoting its dual by $E^{\star }$, we can define four types of second-order tensors on E, which we interpret as bilinear map**s.

• (1) $\textbf{b}: E \times E \rightarrow \mathbb {R}$, $(\textbf{x},\textbf{y}) \mapsto \textbf{b}(\textbf{x},\textbf{y})$,

• (2) $\textbf{b}: E \times E^{\star } \rightarrow \mathbb {R}$, $(\textbf{x},\beta ) \mapsto \textbf{b}(\textbf{x},\beta )$,

• (3) $\textbf{b}: E^{\star } \times E \rightarrow \mathbb {R}$, $(\alpha ,\textbf{y}) \mapsto \textbf{b}(\alpha ,\textbf{y})$,

• (4) $\textbf{b}: E^{\star } \times E^{\star } \rightarrow \mathbb {R}$, $(\alpha ,\beta ) \mapsto \textbf{b}(\alpha ,\beta )$.

To each of these tensors, we associate, using the convention, of that we call the second argument, a linear map**

• (1) $\tilde{\textbf{b}}: E \rightarrow E^{\star }$, $\textbf{y}\mapsto \textbf{b}(\cdot ,\textbf{y})$,

• (2) $\tilde{\textbf{b}}: E^{\star } \rightarrow E^{\star }$, $\beta \mapsto \textbf{b}(\cdot ,\beta )$,

• (3) $\tilde{\textbf{b}}: E \rightarrow (E^{\star })^{\star } = E$, $\textbf{y}\mapsto \textbf{b}(\cdot ,\textbf{y})$,

• (4) $\tilde{\textbf{b}}: E^{\star } \rightarrow (E^{\star })^{\star } = E$, $\beta \mapsto \textbf{b}(\cdot ,\beta )$,

and if there is no ambiguity, we will not distinguish between $\textbf{b}$ and $\tilde{\textbf{b}}$ and only use the notation $\textbf{b}$. Given a basis $(\varvec{e}_{i})$ of E, and denoting its dual basis by $(\varvec{e}^{i})$, their respective components write , , and .

Given two vector spaces E and F, and a linear map** $L: E \rightarrow F$, its adjoint (or dual linear map**) is defined as

$$\begin{aligned} L^{\star }: F^{\star } \rightarrow E^{\star }, \qquad \alpha \mapsto \alpha \circ L. \end{aligned}$$

Note that $(L^{-1})^{\star } = (L^{\star })^{-1}$ and $(L_{1} \, L_{2})^{\star } = L_{2}^{\star }\, L_{1}^{\star }$.

Remark A.1

A second-order covariant, or contravariant, tensor $\textbf{b}$ is symmetric if and only if $\textbf{b}^{\star } = \textbf{b}$.

When the spaces E and F are respectively equipped with scalar products, noted $\textbf{q}_{E}$ and $\textbf{q}_{F}$ respectively, the transpose $L^{t}: F \rightarrow E$ of a linear map** $L: E \rightarrow F$ is defined implicitly by the relation

$$\begin{aligned} \langle L\textbf{x}, \textbf{y}\rangle _{F} = \langle \textbf{x}, L^{t}\textbf{y}\rangle _{E}. \end{aligned}$$

The following diagram makes clear the relation between $L^{\star }$ and $L^{t}$

and leads to

$$\begin{aligned} \textbf{q}_{E}\,L^{t} = L^{\star }\,\textbf{q}_{F}. \end{aligned}$$

Appendix B. Pull-Back and Push-Forward

The fundamental concept of differential geometry that allows to pass from material variables to spatial variables (and vice versa) are the operations of pull-back and push-forward. For functions, these operations are defined by

$$\begin{aligned} p^{*}f = f \circ p\quad \text {(pull-back)}, \qquad p_{*}F = F \circ p^{-1} \quad \text {(push-forward)}, \end{aligned}$$

where $f \in \textrm{C}^{\infty }(\Omega ,\mathbb {R})$ and $F \in \textrm{C}^{\infty }(\mathcal {B},\mathbb {R})$. For vector fields, the following diagram

leads immediately to the natural definitions

$$\begin{aligned} p^{*}\varvec{u}= Tp^{-1} \circ \varvec{u}\circ p\quad \text {(pull-back)}, \qquad p_{*}\varvec{U}= Tp\circ \varvec{U}\circ p^{-1} \quad \text {(push-forward)}. \end{aligned}$$

For covector fields, the following diagram

leads to the following definitions

$$\begin{aligned} p^{*}\beta = Tp^{\star } \circ \beta \circ p\quad \text {(pull-back)}, \qquad p_{*}\alpha = (Tp^{\star })^{-1} \circ \alpha \circ p^{-1} \quad \text {(push-forward)}. \end{aligned}$$

The pull-back and push-forward operations are inverse to each other, meaning that $p^{*} = (p_{*})^{-1} = (p^{-1})_{*}$. They are easily extended to higher-order contravariant, covariant or mixed tensor fields.

In local coordinate systems, $(X^{I})$ on $\mathcal {B}$ and $(x^{i})$ on $\mathcal {E}$, where we have set $p(\textbf{X}) = \textbf{x}$, the linear tangent map $Tp: T\mathcal {B}\rightarrow T\mathcal {E}$ is represented by the square matrix $\textbf{F}$ defined by

and its dual tangent linear map $Tp^{\star }: T^{\star }_{\textbf{x}}\mathcal {E}\rightarrow T^{\star }_{\textbf{X}}\mathcal {B}$, by

We will write $\textbf{F}^{-\star }:= (\textbf{F}^{\star })^{-1}=\left( \textbf{F}^{-1}\right) ^{\star }$.

Proposition B.1

For tensor fields of order one or two, we have the following expressions.

• (1) For contravariant vector fields $\varvec{W}=(W^{I})$, $\varvec{w}=(w^{i})$, we have

  $$\begin{aligned} p_{*} \varvec{W}= \textbf{F}\left( \varvec{W}\circ p^{-1}\right) , \qquad p^{*} \varvec{w}= \textbf{F}^{-1} \left( \varvec{w}\circ p\right) . \end{aligned}$$

• (2) For covariant vector fields $\alpha =(\alpha _{I})$, $\beta =(\beta _{i})$, we have

  $$\begin{aligned} p_{*} \alpha = \textbf{F}^{-\star }\left( \alpha \circ p^{-1}\right) , \qquad p^{*} \beta = \textbf{F}^{\star }\left( \beta \circ p\right) . \end{aligned}$$

• (3) For second-order covariant tensor fields ${\varvec{\varepsilon }}= (\varepsilon _{IJ})$, $\textbf{k}= (k_{ij})$, we have

  $$\begin{aligned} p_{*} {\varvec{\varepsilon }}= \textbf{F}^{-\star } \left( {\varvec{\varepsilon }}\circ p^{-1}\right) \textbf{F}^{-1}, \qquad p^{*} \textbf{k}= \textbf{F}^{\star } (\textbf{k}\circ p) \textbf{F}. \end{aligned}$$

• (4) For second-order contravariant tensor fields ${\varvec{\theta }}= ({\varvec{\theta }}^{IJ})$ and ${\varvec{\tau }}= ({\varvec{\tau }}^{ij})$, we have

  $$\begin{aligned} p_{*} {\varvec{\theta }}= \textbf{F}\left( {\varvec{\theta }}\circ p^{-1}\right) \textbf{F}^{\star }, \qquad p^{*} {\varvec{\tau }}= \textbf{F}^{-1} \left( {\varvec{\tau }}\circ p\right) \textbf{F}^{-\star }. \end{aligned}$$

• (5) For second-order mixed tensor fields $\hat{\textbf{T}} = ({\hat{T}^{I}}_{\; J})$ and $\hat{\textbf{t}} = ({\hat{t}^{i}}_{\, j})$, we have

  $$\begin{aligned} p_{*} \hat{\textbf{T}} = \textbf{F}\left( \hat{\textbf{T}} \circ p^{-1}\right) \textbf{F}^{-1}, \qquad p^{*} \hat{\textbf{t}} = \textbf{F}^{-1} \left( \hat{\textbf{t}} \circ p\right) \textbf{F}. \end{aligned}$$

• (6) For second-order mixed tensor fields $\check{\textbf{T}} = ({\check{T}_{I}}^{\,J})$ and $\check{\textbf{t}} = ({\check{t}_{i}}^{\; j})$, we have

  $$\begin{aligned} p_{*} \check{\textbf{T}} = \textbf{F}^{-\star } (\check{\textbf{T}} \circ p^{-1}) \textbf{F}^{\star }, \qquad p^{*} \check{\textbf{t}} = \textbf{F}^{\star } (\check{\textbf{t}} \circ p)\textbf{F}^{-\star }. \end{aligned}$$

Remark B.2

The push-forward and pull-back operations commute with the contraction between covariant and contravariant tensors. So in particular, we get

$$\begin{aligned} (p_{*}\alpha ) \cdot (p_{*}\varvec{W}) = p_{*}(\alpha \cdot \varvec{W}), \qquad (p_{*} {\varvec{\theta }}): (p_{*} {\varvec{\varepsilon }}) = p_{*} ({\varvec{\theta }}: {\varvec{\varepsilon }}), \qquad {{\,\textrm{tr}\,}}(p_{*} \hat{\textbf{T}}) = p_{*}({{\,\textrm{tr}\,}}\hat{\textbf{T}}). \end{aligned}$$

Appendix C. Lie Derivative

Given a manifold M, the Lie derivative of a (time-independent) tensor field $\textbf{t}\in \mathbb {T}(M)$ corresponds to the infinitesimal version of the pull-back operation. More precisely, let $\varvec{u}$ be a vector field on M, $\phi (t)$ its flow and $\textbf{t}$ be a tensor field on M. The Lie derivative ${{\,\textrm{L}\,}}_{\varvec{u}} \textbf{t}$ of the tensor field $\textbf{t}$ with respect to $\varvec{u}$ is defined by

$$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} \textbf{t}:= \left. \frac{\partial }{\partial t}\right| _{t=0} \phi (t)^{*} \textbf{t}. \end{aligned}$$

The Lie derivative has the following properties.

• (1) When $\textbf{t}= \varvec{v}$ is a vector field, ${{\,\textrm{L}\,}}_{\varvec{u}} \varvec{v}= [\varvec{u},\varvec{v}]$, where $[\varvec{u},\varvec{v}]$ is the Lie bracket of the two vector fields $\varvec{u}$ and $\varvec{v}$.

• (2) Given a diffeomorphism $\varphi$ of M, we have

  $$\begin{aligned} \varphi ^{*} {{\,\textrm{L}\,}}_{\varvec{u}} \textbf{t}= {{\,\textrm{L}\,}}_{\varphi ^{*}\varvec{u}} \varphi ^{*}\textbf{t}. \end{aligned}$$

  (C.1)

• (3) At any time t where $\phi (t)$ is defined, we have (note that $\phi (t)^{*} \varvec{u}=\varvec{u}$)

  $$\begin{aligned} \frac{\partial }{\partial t}\left( \phi (t)^{*} \textbf{t}\right) = \phi (t)^{*} {{\,\textrm{L}\,}}_{\varvec{u}} \textbf{t}= {{\,\textrm{L}\,}}_{\varvec{u}} \phi (t)^{*} \textbf{t}. \end{aligned}$$

  (C.2)

• (4) Given two vector fields $\varvec{u},\varvec{v}$ on M, we have

  $$\begin{aligned} {{\,\textrm{L}\,}}_{[\varvec{u},\varvec{v}]} \textbf{t}= {{\,\textrm{L}\,}}_{\varvec{u}} {{\,\textrm{L}\,}}_{\varvec{v}} \textbf{t}- {{\,\textrm{L}\,}}_{\varvec{v}} {{\,\textrm{L}\,}}_{\varvec{u}} \textbf{t}. \end{aligned}$$

Consider now a time dependent vector field $\varvec{u}(t)$ on M. Its flow $\phi (t,s)$ is defined as the solution a time t of the initial value problem

$$\begin{aligned} \dot{c}(t) = \varvec{u}(t,c(t)), \qquad c(s)=\textbf{x}. \end{aligned}$$

(C.3)

Then, $\phi (t,s)$ is a local diffeomorphism with inverse $\phi (s,t)$ and we have

$$\begin{aligned} \phi (t,s) = \phi (t,\tau ) \circ \phi (\tau ,s), \end{aligned}$$

as soon as the three map**s are defined. The Lie derivative can be extended to time dependent vector fields as follows.

$$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}(t)} \textbf{t}:= \left. \frac{\partial }{\partial \tau }\right| _{\tau =t} \phi (\tau ,t)^{*} \textbf{t}. \end{aligned}$$

Remark C.1

Let $\widetilde{\varphi } = (\varphi (t))$ be a path of diffeomorphism. Its Eulerian velocity is defined as the time dependent vector field $\varvec{u}(t):= \partial _{t}\varphi \circ \varphi (t)^{-1}$. Given a (time-independent) tensor field $\textbf{t}$, we have

$$\begin{aligned} \frac{\partial }{\partial t} \left( \varphi (t)^{*} \textbf{t}\right) =\varphi (t)^{*} ({{\,\textrm{L}\,}}_{\varvec{u}(t)} \textbf{t}) \qquad \text {and} \qquad \frac{\partial }{\partial t} \left( \varphi (t)_{*} \textbf{t}\right) =-{{\,\textrm{L}\,}}_{\varvec{u}(t)} (\varphi (t)_{*} \textbf{t}). \end{aligned}$$

The preceding results extend to paths of embeddings between two manifolds $\mathcal {B}$ and M and will be summarized by the following lemma.

Lemma C.2

Let $\widetilde{p} = (p(t))$ be a path of embeddings, $\varvec{u}(t):= (\partial _{t}p)\circ p(t)^{-1}$ be its (right) Eulerian velocity and $\textbf{t}(t)$ be a tensor field defined along $p(t)$ (i.e. on $\Omega _{p(t)} = p(t)(\mathcal {B})$ and possibly time-dependent). Then

$$\begin{aligned} \partial _{t}(p(t)^{*}\textbf{t}(t)) = p(t)^{*} \left( \partial _{t}\textbf{t}+ {{\,\textrm{L}\,}}_{\varvec{u}(t)}\textbf{t}(t) \right) . \end{aligned}$$

Proof

Let $\phi (t,s)$ be the flow of $\varvec{u}(t)$. Then we get

$$\begin{aligned} p(s) = \phi (s,t) \circ p(t). \end{aligned}$$

We have thus

$$\begin{aligned} \partial _{t}(p(t)^{*}\textbf{t}(t))&= \left. \frac{\partial }{\partial s}\right| _{s=t} (\phi (s,t) \circ p(t))^{*} \textbf{t}(s)\\&= p(t)^{*} \left. \frac{\partial }{\partial s} \right| _{s=t} \phi (s,t)^{*} \textbf{t}(s)\\&= p(t)^{*}\left( \partial _{t} \textbf{t}(t) + {{\,\textrm{L}\,}}_{\varvec{u}(t)}\textbf{t}(t)\right) . \end{aligned}$$

$\square$

Appendix D. Vector Bundles and Covariant Derivatives

The interested reader may consult [30, 53, 57, 60, 82] for complementary points of view on differential geometry. Let $\mathbb {E}$ be a vector bundle over a manifold M. We denote by $\Gamma (\mathbb {E})$ the space of smooth sections of $\mathbb {E}$ and by $\Omega ^{k}(M,\mathbb {E})$ the space of k-forms with values in $\mathbb {E}$, in other words, the space of sections of the vector bundle $\Lambda ^{k}T^{\star }M \otimes \mathbb {E}$ (in particular, $\Omega ^{0}(M,\mathbb {E}) = \Gamma (\mathbb {E})$).

Definition D.1

(Covariant derivative) A covariant derivative on vector bundle $\mathbb {E}$ over M is a linear operator

$$\begin{aligned} \nabla : \Gamma (\mathbb {E}) \rightarrow \Omega ^{1}(M,\mathbb {E}), \qquad \textbf{s}\mapsto \nabla \textbf{s}, \end{aligned}$$

which satisfies the Leibniz identity.

$$\begin{aligned} \nabla (f\textbf{s}) = df \otimes \textbf{s}+ f \, \nabla \textbf{s}, \end{aligned}$$

for any function $f \in \textrm{C}^{\infty }(M)$ and any section $\textbf{s}\in \Gamma (\mathbb {E})$.

Remark D.2

The set of all covariant derivatives defined on a given vector bundle $\mathbb {E}$ has an affine structure. Indeed, given two covariant derivatives $\nabla ^{1}$ and $\nabla ^{2}$, the difference $\nabla ^{2}-\nabla ^{1}$ is a section of the vector bundle

$$\begin{aligned} (T^{\star }M \otimes \mathbb {E}^{\star }) \otimes \mathbb {E}. \end{aligned}$$

Hence, this set is well an affine space with associated vector space $\Gamma ((T^{\star }M \otimes \mathbb {E}^{\star }) \otimes \mathbb {E})$.

Remark D.3

If a vector bundle $\mathbb {E}$ with base manifold M and fiber type E (a vector space) is trivializable, meaning that there exists a vector bundle isomorphism

$$\begin{aligned} \Psi : \mathbb {E}\rightarrow M \times E, \qquad \varvec{v}_{x} \mapsto (x,\varvec{v}), \end{aligned}$$

then, each section $\textbf{s}$ of $\mathbb {E}$ corresponds bijectively to a vector valued function S defined by

$$\begin{aligned} S: M \rightarrow E, \qquad x \mapsto p_{2} \circ \Psi (\textbf{s}(x)), \end{aligned}$$

where $p_{2}: M \times E$ is the projection onto the second factor. Therefore, there is a canonical covariant derivative associated with this trivialization which is given by

$$\begin{aligned} (\nabla _{X} \textbf{s})(x):= \Psi ^{-1}(x, d_{x}S.X). \end{aligned}$$

Definition D.4

(Curvature) Given a covariant derivative $\nabla$ on a vector bundle $\mathbb {E}$, its curvature is the map**

$$\begin{aligned} R: \Gamma (\mathbb {E}) \rightarrow \Omega ^{2}(M,\mathbb {E}) \end{aligned}$$

defined by

$$\begin{aligned} R(X,Y)\textbf{s}:= \nabla _{X}\nabla _{Y}\textbf{s}- \nabla _{Y}\nabla _{X}\textbf{s}- \nabla _{[X,Y]}\textbf{s}. \end{aligned}$$

When $\mathbb {E}= TM$ is the tangent bundle of a manifold M, we define the torsion of this covariant derivative, by the formula

$$\begin{aligned} T(X,Y):= \nabla _{X} Y - \nabla _{Y} X - [X, Y], \qquad X, Y \in \textrm{Vect}(M), \end{aligned}$$

which is a mixed tensor field of type (1, 2). The curvature tensor of $\nabla$ is a mixed tensor field of type (1, 3), which writes

$$\begin{aligned} R(X,Y)Z = \nabla _{X}\nabla _{Y}Z - \nabla _{Y} \nabla _{X}Z - \nabla _{[X,Y]}Z, \qquad X, Y, Z \in \textrm{Vect}(M). \end{aligned}$$

Definition D.5

A covariant derivative on the tangent bundle TM of a manifold M is symmetric if its torsion is zero, that is if

$$\begin{aligned} \nabla _{\varvec{v}} \varvec{w}- \nabla _{\varvec{w}} \varvec{v}= [\varvec{v}, \varvec{w}], \qquad \forall \varvec{v}, \varvec{w}\in \textrm{Vect}(M), \end{aligned}$$

where $[\varvec{v}, \varvec{w}]:={{\,\textrm{L}\,}}_{\varvec{v}} \varvec{w}$ is the Lie bracket of the vector fields $\varvec{v}$ and $\varvec{w}$.

Remark D.6

One can show the existence on any differential manifold M of a covariant derivative. However, there are an infinite number of such derivatives and none of them play a particular role. On the other hand, if a manifold M has a Riemannian metric g, then there is a unique symmetric covariant derivative $\nabla$ such that $\nabla g = 0$ (see for example [30, Theorem 2.51]), this is the Riemannian covariant derivative.

Any covariant derivative on TM induces by the Leibniz rule a covariant derivative on all tensor bundles of M. The link between the Lie derivative and a symmetric covariant derivative is then recalled in the following theorem.

Proposition D.7

Let M be a differential manifold with a symmetric covariant derivative $\nabla$. Then we have the following relations:

• (1) Lie derivative of a function f:

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} f = \nabla _{\varvec{u}} f = df.\varvec{u}; \end{aligned}$$

• (2) Lie derivative of a vector field $\varvec{w}= (w^{i})$:

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} \varvec{w}= [\varvec{u}, \varvec{w}] = \nabla _{\varvec{u}} \varvec{w}- \nabla _{\varvec{w}} \varvec{u}; \end{aligned}$$

• (3) Lie derivative of a covector field (1-form) $\alpha = (\alpha _{i})$:

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} \alpha = \nabla _{\varvec{u}} \alpha + (\nabla \varvec{u})^{\star } \alpha ; \end{aligned}$$

• (4) Lie derivative of a second-order covariant tensor field, $\textbf{k}= (k_{ij})$:

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} \textbf{k}= \nabla _{\varvec{u}} \textbf{k}+ (\nabla \varvec{u})^{\star } \textbf{k}+ \textbf{k}(\nabla \varvec{u}); \end{aligned}$$

• (5) Lie derivative of a second-order contravariant tensor field, ${\varvec{\tau }}= (\tau ^{ij})$:

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} {\varvec{\tau }}= \nabla _{\varvec{u}} {\varvec{\tau }}- (\nabla \varvec{u}) {\varvec{\tau }}- {\varvec{\tau }}(\nabla \varvec{u})^{\star }; \end{aligned}$$

• (6) Lie derivative of a second-order mixed tensor field, :

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} \hat{\textbf{t}} = \nabla _{\varvec{u}} \hat{\textbf{t}} - (\nabla \varvec{u}) \hat{\textbf{t}} + \hat{\textbf{t}} (\nabla \varvec{u}); \end{aligned}$$

• (7) Lie derivative of a second-order mixed tensor field, :

  $$\begin{aligned} {{\,\textrm{L}\,}}_{\varvec{u}} \check{\textbf{t}} = \nabla _{\varvec{u}} \check{\textbf{t}} + (\nabla \varvec{u})^{\star } \check{\textbf{t}} - \check{\textbf{t}} (\nabla \varvec{u})^{\star }. \end{aligned}$$

In order to intrinsically define the geodesic equation of a Riemannian manifold, but also the covariant derivative of the Lagrangian velocity, it is necessary to extend the notion of covariant derivative to vector fields (and more generally tensor fields) which are only defined along a map** $f: K \rightarrow M$, between two manifolds K and M. The rigorous formulation of such a definition requires first to introduce the notion of pull-back of a vector bundle [41].

Definition D.8

Let $\pi : \mathbb {E}\rightarrow M$ be a vector bundle and $f: K \rightarrow M$ be a smooth map**. Then the set

$$\begin{aligned} f^{*}\mathbb {E}:= \bigsqcup _{k \in K} E_{f(k)} \subset \mathbb {E}\end{aligned}$$

is a vector bundle above K, referred to as the pull-back by f of the vector bundle $\mathbb {E}$. A section of this bundle is therefore a map** $s: K \rightarrow \mathbb {E}$, such as $\pi (s(k)) = f(k)$.

Example D.9

A vector field defined along a curve $c: I \rightarrow M$ is a curve $X: I \rightarrow TM$, such that $X(t) \in T_{c(t)}M$, for any $t \in I$.

Example D.10

The Lagrangian velocity $\varvec{V}(t): \mathcal {B}\rightarrow T\mathcal {E}$, at time t, is a section of the pullback bundle $p(t)^{*}T\mathcal {E}$.

Proposition D.11

Let $\pi : \mathbb {E}\rightarrow M$ be a vector bundle, equipped with a covariant derivative $\nabla$ and $f: K \rightarrow M$, a smooth map**. Then, there exists a unique covariant derivative, denoted $f^{*}\nabla$, on the vector bundle $f^{*}\mathbb {E}$, called the pull-back of $\nabla$, and such that

$$\begin{aligned} (f^{*}\nabla )_{X} (f \circ s) = f^{*}\left( \nabla _{Tf.X} s\right) , \end{aligned}$$

for any section s of $\mathbb {E}$ and any vector field X on K.

Example D.12

Consider the case where $K = \mathcal {B}$, $M=\mathcal {E}$, $\mathbb {E}= T\mathcal {E}$ is the tangent vector bundle to $\mathcal {E}$ and

$$\begin{aligned} f = p: \mathcal {B}\rightarrow \mathcal {E}, \end{aligned}$$

is an embedding of $\mathcal {B}$ in $\mathcal {E}$. Consider a local coordinate system $(X^{I})$ on $\mathcal {B}$ and a local coordinate system $(x^{i})$ on $\mathcal {E}$, where Christoffel’s symbols are written $\Gamma _{ij}^{k}$. Let $\varvec{V}$ be a section of $p^{*}T\mathcal {E}$, i.e. a map**

$$\begin{aligned} \varvec{V}: \mathcal {B}\rightarrow T\mathcal {E}, \quad \text {such that} \quad \pi (X(X)) = p(X), \end{aligned}$$

then, we have

$$\begin{aligned} {[}(p^{*}\nabla )_{\partial _{X^{I}}} V]^{k} = \partial _{X^{I}} V^{k} + \Gamma _{ij}^{k} \frac{\partial p^{i}}{\partial _{X^{I}}} V^{j}. \end{aligned}$$

Example D.13

Let $(M,{\varvec{\gamma }})$ be a Riemannian manifold and $\nabla$ the associated covariant derivative. Let $c: I \rightarrow M$ be curve. Then, the pull-back of $\nabla$ by c, usually noted $D_{t}$ is defined on the vector space of vector fields defined along c. It is characterized by the following properties:

• (1) for any function $f: I \rightarrow \mathbb {R}$,

  $$\begin{aligned} D_{t}(fX)(t) = f^{\prime }(t)\, X(t) + f(t)\, D_{t}(X)(t); \end{aligned}$$

• (2) If $X(t) = \tilde{X}(c(t))$ where $\tilde{X}$ is a vector field on M, then

  $$\begin{aligned} (D_{t}X)(t) = (\nabla _{c^{\prime }(t)}\tilde{X})((t)). \end{aligned}$$

In a local coordinate system $(x^{i})$ of M, we have:

$$\begin{aligned} (D_{t}X)^{k} = \partial _{t}X^{k} + \Gamma _{ij}^{k} (\partial _{t}x^{i})X^{j}. \end{aligned}$$

Remark D.14

On an infinite dimensional manifold, equipped with a covariant derivative, the curvature is defined along a parameterized surface c(s, t), and writes

$$\begin{aligned} R(\partial _s, \partial _{t})X = D_{s}D_{t}X - D_{t}D_{s}X. \end{aligned}$$