Musical isomorphism

In linear algebra, a finite-dimensional vector space is isomorphic to its dual space (the space of linear functionals mapping the vector space to its base field), but not canonically isomorphic to it. This is to say that given a fixed basis for the vector space, there is a natural way to go back and forth between vectors and linear functionals: vectors are represented in the basis by column vectors, and linear functionals are represented in the basis by row vectors, and one can go back and forth by transposing. However, without a fixed basis, there is no way to go back and forth between vectors and linear functionals. This is what is meant by that there is no canonical isomorphism.

On the other hand, a finite-dimensional vector space $V$ endowed with a non-degenerate bilinear form $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is canonically isomorphic to its dual. The canonical isomorphism $V\; \backslash to\; V^*$ is given by

: $v\; \backslash mapsto\; \backslash langle\; v,\; \backslash cdot\; \backslash rangle$.

The non-degeneracy of $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ means exactly that the above map is an isomorphism. An example is where $V\; =\; \backslash mathbb\; R^n$ and $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is the dot product.

In a basis $e\_i$, the canonical isomorphism above is represented as follows. Let $g\_\{ij\}\; =\; \backslash langle\; e\_i,e\_j\; \backslash rangle$ be the components of the non-degenerate bilinear form and let $g^\{ij\}$ be the components of the inverse matrix to $g\_\{ij\}$. Let $e^i$ be the dual basis of $e\_i$. A vector $v$ is written in the basis as $v\; =\; v^i\; e\_i$ using Einstein summation notation, i.e., $v$ has components $v^i$ in the basis. The canonical isomorphism applied to $v$ gives an element of the dual, which is called a covector. The covector has components $v\_i$ in the dual basis given by contracting with $g$:

:$v\_i\; =\; g\_\{ij\}v^j.$

This is what is meant by lowering the index. Conversely, contracting a covector $\backslash alpha\; =\; \backslash alpha\_i\; e^i$ with the inverse of $g$ gives a vector with components

:$\backslash alpha^i\; =\; g^\{ij\}\backslash alpha\_j.$

in the basis $e\_i$. This process is called raising the index.

Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in $g\_\{ij\}$ and $g^\{ij\}$ being inverses:

:$g^\{ij\}g\_\{jk\}=g\_\{kj\}g^\{ji\}=\{\backslash delta^i\}\_k=\{\backslash delta\_k\}^i$

where $\backslash delta^i\_j$ is the Kronecker delta or identity matrix.

The musical isomorphisms are the global version of the canonical isomorphism $v\; \backslash mapsto\; \backslash langle\; v,\; \backslash cdot\; \backslash rangle$ and its inverse for the tangent bundle and cotangent bundle of a (pseudo-)Riemannian manifold $(M,g)$. They are canonical isomorphisms of vector bundles which are at any point {{math|p}} the canonical isomorphism applied to the tangent space of {{math|M}} at {{math|p}} endowed with the inner product $g\_p$.

Because every smooth manifold can be (non-canonically) endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a smooth manifold is (non-canonically) isomorphic to its dual.

Let {{math|(M, g)}} be a (pseudo-)Riemannian manifold. At each point {{mvar|p}}, the map {{math|g{{sub|p}}}} is a non-degenerate bilinear form on the tangent space {{math|T{{sub|p}}M}}. If {{mvar|v}} is a vector in {{math|T{{sub|p}}M}}, its flat is the covector

: $v^\backslash flat\; =\; g\_p(v,\backslash cdot)$

in {{math|T{{su|lh=1em|p=∗|b=p }}M}}. Since this is a smooth map that preserves the point {{mvar|p}}, it defines a morphism of smooth vector bundles $\backslash flat\; :\; \backslash mathrm\{T\}M\; \backslash to\; \backslash mathrm\{T\}^*M$. By non-degeneracy of the metric, $\backslash flat$ has an inverse $\backslash sharp$ at each point, characterized by

: $g\_p(\backslash alpha^\backslash sharp,\; v)\; =\; \backslash alpha(v)$

for {{mvar|α}} in {{math|T{{su|lh=1em|p=∗|b=p }}M}} and {{mvar|v}} in {{math|T{{sub|p}}M}}. The vector $\backslash alpha^\backslash sharp$ is called the sharp of {{mvar|α}}. The sharp map is a smooth bundle map $\backslash sharp\; :\; \backslash mathrm\{T\}^*M\; \backslash to\; \backslash mathrm\{T\}M$.

Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each {{mvar|p}} in {{mvar|M}}, there are mutually inverse vector space isomorphisms between {{math|T{{sub|p }}M}} and {{math|T{{su|lh=1em|p=∗|b=p }}M}}.

The flat and sharp maps can be applied to vector fields and covector fields by applying them to each point. Hence, if {{mvar|X}} is a vector field and {{mvar|ω}} is a covector field,

: $X^\backslash flat\; =\; g(X,\backslash cdot)$

and

: $g(\backslash omega^\backslash sharp,\; X)\; =\; \backslash omega(X)$.

Suppose {{math|{e{{sub|i}}}{{null}}}} is a moving tangent frame (see also smooth frame) for the tangent bundle {{math|TM}} with, as dual frame (see also dual basis), the moving coframe (a moving tangent frame for the cotangent bundle $\backslash mathrm\{T\}^*M$; see also coframe) {{math|{e{{sup|i}}}{{null}}}}. Then the pseudo-Riemannian metric, which is a 2-covariant tensor field, can be written locally in this coframe as {{math|g {{=}} g{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}} using Einstein summation notation.

Given a vector field {{math|X {{=}} X{{i sup|i }}e{{sub|i}}}} and denoting {{math|1=g{{sub|ij}} X{{i sup|i}} = X{{sub|j}}}}, its flat is

: $X^\backslash flat\; =\; g\_\{ij\}\; X^i\; \backslash mathbf\{e\}^j\; =\; X\_j\; \backslash mathbf\{e\}^j$.

This is referred to as lowering an index, because the components of {{mvar|X}} are written with an upper index {{math|X{{i sup|i}}}}, whereas the components of $X^\backslash flat$ are written with a lower index {{math|X{{sub|j}}}}.

In the same way, given a covector field {{math|1=ω = ω{{sub|i}} e{{sup|i}}}} and denoting {{math|1=g{{sup|ij}} ω{{sub|i}} = ω{{sup|j}}}}, its sharp is

: $\backslash omega^\backslash sharp\; =\; g^\{ij\}\; \backslash omega\_i\; \backslash mathbf\{e\}\_j\; =\; \backslash omega^j\; \backslash mathbf\{e\}\_j$,

where {{math|g{{i sup|ij}}}} are the components of the inverse metric tensor (given by the entries of the inverse matrix to {{math|g{{sub|ij}}}}). Taking the sharp of a covector field is referred to as raising an index.

The musical isomorphisms may also be extended, for each {{math|r, s, k}}, to an isomorphism between the bundle

:$\backslash bigotimes\_\{i=1\}^s\; \{\backslash rm\; T\}\; M\; \backslash otimes\; \backslash bigotimes\_\{j=1\}^\{r\}\; \{\backslash rm\; T\}^*\; M$

of $(r,s)$ tensors and the bundle of $(r-k,s+k)$ tensors. Here {{mvar|k}} can be positive or negative, so long as {{math|r - k ≥ 0}} and {{math|s + k ≥ 0}}.

Lowering an index of an $(r,s)$ tensor gives a $(r-1,s+1)$ tensor, while raising an index gives a $(r+1,s-1)$. Which index is to be raised or lowered must be indicated.

For instance, consider the {{nowrap|(0, 2)}} tensor {{math|X {{=}} X{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}}. Raising the second index, we get the {{nowrap|(1, 1)}} tensor

:$X^\backslash sharp\; =\; g^\{jk\}\; X\_\{ij\}\; \backslash ,\; \{\backslash rm\; e\}^i\; \backslash otimes\; \{\backslash rm\; e\}\_k\; .$

In other words, the components $X\_i^k$ of $X^\backslash sharp$ are given by

:$X\_i^k\; =\; g^\{jk\}\; X\_\{ij\}.$

Similar formulas are available for tensors of other orders. For example, for a $(0,n)$ tensor {{mvar|X}}, all indices are raised by:

:$X^\{j\_1j\_2\backslash cdots\; j\_n\}\; =\; g^\{j\_1i\_1\}g^\{j\_2i\_2\}\backslash cdots\; g^\{j\_ni\_n\}X\_\{i\_1i\_2\backslash cdots\; i\_n\}.$

For a $(n,0)$ tensor {{mvar|X}}, all indices are lowered by:

:$X\_\{j\_1j\_2\backslash cdots\; j\_n\}\; =\; g\_\{j\_1i\_1\}g\_\{j\_2i\_2\}\backslash cdots\; g\_\{j\_ni\_n\}X^\{i\_1i\_2\backslash cdots\; i\_n\}.$

For a mixed tensor of order $(n,m)$, all lower indices are raised and all upper indices are lowered by

:$\{X\_\{p\_1p\_2\backslash cdots\; p\_n\}\}^\{q\_1q\_2\backslash cdots\; q\_m\}\; =\; g\_\{p\_1i\_1\}g\_\{p\_2i\_2\}\backslash cdots\; g\_\{p\_ni\_n\}g^\{q\_1j\_1\}g^\{q\_2j\_2\}\backslash cdots\; g^\{q\_mj\_m\}\{X^\{i\_1i\_2\backslash cdots\; i\_n\}\}\_\{j\_1j\_2\backslash cdots\; j\_m\}.$

Well-formulated expressions are constrained by the rules of Einstein summation notation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we can immediately see that an expression such as $g\_\{ij\}v^iu^j$

is well formulated while $g\_\{ij\}v\_iu\_j$ is not.

In the context of exterior algebra, an extension of the musical operators may be defined on {{math|⋀V}} and its dual {{math|⋀V ^*}}, and are again mutual inverses:{{sfn|Vaz|da Rocha|2016|loc=pp. 48, 50}}

:$\backslash flat\; :\; \backslash bigwedge\_\{i=1\}^k\; V\; \backslash to\; \backslash bigwedge\_\{i=1\}^k\; V^*,$

:$\backslash sharp\; :\; \backslash bigwedge\_\{i=1\}^k\; V^*\; \backslash to\; \backslash bigwedge\_\{i=1\}^k\; V,$

defined by

:$(X\; \backslash wedge\; \backslash ldots\; \backslash wedge\; Z)^\backslash flat\; =\; X^\backslash flat\; \backslash wedge\; \backslash ldots\; \backslash wedge\; Z^\backslash flat,$

:$(\backslash alpha\; \backslash wedge\; \backslash ldots\; \backslash wedge\; \backslash gamma)^\backslash sharp\; =\; \backslash alpha^\backslash sharp\; \backslash wedge\; \backslash ldots\; \backslash wedge\; \backslash gamma^\backslash sharp.$

In this extension, in which {{math|{{music|flat}}}} maps k-vectors to k-covectors and {{math|{{music|sharp}}}} maps k-covectors to k-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:

$$Y^\backslash sharp\; =\; (\; Y\_\{i\_1\; \backslash dots\; i\_j\}\; \backslash mathbf\{e\}^\{i\_1\}\; \backslash otimes\; \backslash dots\; \backslash otimes\; \backslash mathbf\{e\}^\{i\_j\})^\backslash sharp\; =\; g^\{i\_1\; r\_1\}\; \backslash dots\; g^\{i\_j\; r\_s\}\; \backslash ,\; Y\_\{i\_1\; \backslash dots\; i\_k\}\; \backslash ,\; \backslash mathbf\{e\}\_\{r\_1\}\; \backslash otimes\; \backslash dots\; \backslash otimes\; \backslash mathbf\{e\}\_\{r\_s\}.$$

This works not just for k-vectors in the context of linear algebra but also for k-forms in the context of a (pseudo-)Riemannian manifold:

:$\backslash flat\; :\; \backslash bigwedge\_\{i=1\}^k\; \{\backslash rm\; T\}\; M\; \backslash to\; \backslash bigwedge\_\{i=1\}^k\; \{\backslash rm\; T\}^*\; M,$

:$\backslash sharp\; :\; \backslash bigwedge\_\{i=1\}^k\; \{\backslash rm\; T\}^*\; M\; \backslash to\; \backslash bigwedge\_\{i=1\}^k\; \{\backslash rm\; T\}\; M,$

More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.

Given a {{nowrap|(0, 2)}} tensor {{math|X {{=}} X{{sub|ij }}e{{sup|i}} ⊗ e{{i sup|j}}}}, we define the trace of {{mvar|X}} through the metric tensor {{mvar|g}} by

$$\backslash operatorname\{tr\}\_g\; (\; X\; )\; :=\; \backslash operatorname\{tr\}\; (\; X^\backslash sharp\; )\; =\; \backslash operatorname\{tr\}\; (\; g^\{jk\}\; X\_\{ij\}\; \backslash ,\; \{\backslash bf\; e\}^i\; \backslash otimes\; \{\backslash bf\; e\}\_k\; )\; =\; g^\{ij\}\; X\_\{ij\}\; .$$

Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.

The trace of an $(r,\; s)$ tensor can be taken in a similar way, so long as one specifies which two distinct indices are to be traced. This process is also called contracting the two indices. For example, if {{mvar|X}} is an $(r,\; s)$ tensor with {{math|r > 1}}, then the indices $i\_1$ and $i\_2$ can be contracted to give an $(r-2,\; s)$ tensor with components

:$X^\{i\_3i\_4\; \backslash cdots\; i\_r\}\_\{j\_1j\_2\backslash cdots\; j\_s\}\; =\; g\_\{i\_1i\_2\}\; X^\{i\_1i\_2\backslash cdots\; i\_r\}\_\{j\_1j\_2\backslash cdots\; j\_s\}.$

The covariant 4-position is given by

:$X\_\backslash mu\; =\; (-ct,\; x,\; y,\; z)$

with components:

:$X\_0\; =\; -ct,\; \backslash quad\; X\_1\; =\; x,\; \backslash quad\; X\_2\; =\; y,\; \backslash quad\; X\_3\; =\; z$

(where {{mvar|x}},{{mvar|y}},{{mvar|z}} are the usual Cartesian coordinates) and the Minkowski metric tensor with metric signature (− + + +) is defined as

:$\backslash eta\_\{\backslash mu\; \backslash nu\}\; =\; \backslash eta^\{\backslash mu\; \backslash nu\}\; =\; \backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}

in components:

:$\backslash eta\_\{00\}\; =\; -1,\; \backslash quad\; \backslash eta\_\{i0\}\; =\; \backslash eta\_\{0i\}\; =\; 0,\backslash quad\; \backslash eta\_\{ij\}\; =\; \backslash delta\_\{ij\}\backslash ,(i,j\; \backslash neq\; 0).$

To raise the index, multiply by the tensor and contract:

:$X^\backslash lambda\; =\; \backslash eta^\{\backslash lambda\backslash mu\}X\_\backslash mu\; =\; \backslash eta^\{\backslash lambda\; 0\}X\_0\; +\; \backslash eta^\{\backslash lambda\; i\}X\_i$

then for {{math|λ {{=}} 0}}:

:$X^0\; =\; \backslash eta^\{00\}X\_0\; +\; \backslash eta^\{0i\}X\_i\; =\; -X\_0$

and for {{math|λ {{=}} j {{=}} 1, 2, 3}}:

:$X^j\; =\; \backslash eta^\{j0\}X\_0\; +\; \backslash eta^\{ji\}X\_i\; =\; \backslash delta^\{ji\}X\_i\; =\; X\_j\; \backslash ,.$

So the index-raised contravariant 4-position is:

:$X^\backslash mu\; =\; (ct,\; x,\; y,\; z)\backslash ,.$

This operation is equivalent to the matrix multiplication

:$\backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

-ct \\

x \\

y \\

z

\end{pmatrix} = \begin{pmatrix}

ct \\

x \\

y \\

z

\end{pmatrix}.

Given two vectors, $X^\backslash mu$ and $Y^\backslash mu$, we can write down their (pseudo-)inner product in two ways:

:$\backslash eta\_\{\backslash mu\backslash nu\}X^\backslash mu\; Y^\backslash nu.$

By lowering indices, we can write this expression as

:$X\_\backslash mu\; Y^\backslash mu.$

In matrix notation, the first expression can be written as

:

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

Y^0 \\

Y^1 \\

Y^2 \\

Y^3\end{pmatrix}

while the second is, after lowering the indices of $X^\backslash mu$,

:

Y^0 \\

Y^1 \\

Y^2 \\

Y^3\end{pmatrix}.

For a (0,2) tensor,{{cite book |title=Tensor Calculus |first=D. C. |last=Kay |series=Schaum’s Outlines |publisher=McGraw Hill |location=New York |year=1988 |isbn=0-07-033484-6 }} twice contracting with the inverse metric tensor and contracting in different indices raises each index:

:$A^\{\backslash mu\backslash nu\}=g^\{\backslash mu\backslash rho\}g^\{\backslash nu\backslash sigma\}A\_\{\backslash rho\; \backslash sigma\}.$

Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:

:$A\_\{\backslash mu\backslash nu\}=g\_\{\backslash mu\backslash rho\}g\_\{\backslash nu\backslash sigma\}A^\{\backslash rho\backslash sigma\}$

Let's apply this to the theory of electromagnetism.

The contravariant electromagnetic tensor in the {{math|(+ − − −)}} signature is given byNB: Some texts, such as: {{cite book | author=Griffiths, David J. | authorlink = David J. Griffiths | title=Introduction to Elementary Particles | publisher=Wiley, John & Sons, Inc | year=1987 | isbn=0-471-60386-4}}, will show this tensor with an overall factor of −1. This is because they used the negative of the metric tensor used here: {{math|(− + + +)}}, see metric signature. In older texts such as Jackson (2nd edition), there are no factors of {{mvar|c}} since they are using Gaussian units. Here SI units are used.

:$F^\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{pmatrix\}$

0 & -\frac{E_x}{c} & -\frac{E_y}{c} & -\frac{E_z}{c} \\

\frac{E_x}{c} & 0 & -B_z & B_y \\

\frac{E_y}{c} & B_z & 0 & -B_x \\

\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}.

In components,

:$F^\{0i\}\; =\; -F^\{i0\}\; =\; -\; \backslash frac\{E^i\}\{c\}\; ,\backslash quad\; F^\{ij\}\; =\; -\; \backslash varepsilon^\{ijk\}\; B\_k$

To obtain the covariant tensor {{mvar|F_αβ}}, contract with the inverse metric tensor:

:$\backslash begin\{align\}$

F_{\alpha\beta} & = \eta_{\alpha\gamma} \eta_{\beta\delta} F^{\gamma\delta} \\

& = \eta_{\alpha 0} \eta_{\beta 0} F^{0 0} + \eta_{\alpha i} \eta_{\beta 0} F^{i 0}

+ \eta_{\alpha 0} \eta_{\beta i} F^{0 i} + \eta_{\alpha i} \eta_{\beta j} F^{i j}

\end{align}

and since {{math|F⁰⁰ {{=}} 0 }} and {{math|F⁰ⁱ {{=}} − Fⁱ⁰}}, this reduces to

:$F\_\{\backslash alpha\backslash beta\}\; =\; \backslash left(\backslash eta\_\{\backslash alpha\; i\}\; \backslash eta\_\{\backslash beta\; 0\}\; -\; \backslash eta\_\{\backslash alpha\; 0\}\; \backslash eta\_\{\backslash beta\; i\}\; \backslash right)\; F^\{i\; 0\}\; +\; \backslash eta\_\{\backslash alpha\; i\}\; \backslash eta\_\{\backslash beta\; j\}\; F^\{i\; j\}$

Now for {{math|α {{=}} 0}}, {{math|β {{=}} k {{=}} 1, 2, 3}}:

:$\backslash begin\{align\}$

F_{0k} & = \left(\eta_{0i} \eta_{k0} - \eta_{00} \eta_{ki} \right) F^{i0} + \eta_{0i} \eta_{kj} F^{ij} \\

& = \bigl(0 - (-\delta_{ki}) \bigr) F^{i0} + 0 \\

& = F^{k0} = - F^{0k} \\

\end{align}

and by antisymmetry, for {{math|α {{=}} k {{=}} 1, 2, 3}}, {{math|β {{=}} 0}}:

:$F\_\{k0\}\; =\; -\; F^\{k0\}$

then finally for {{math|α {{=}} k {{=}} 1, 2, 3}}, {{math|β {{=}} l {{=}} 1, 2, 3}};

:$\backslash begin\{align\}$

F_{kl} & = \left(\eta_{ k i} \eta_{ l 0} - \eta_{ k 0} \eta_{ l i} \right) F^{i 0} + \eta_{ k i} \eta_{ l j} F^{i j} \\

& = 0 + \delta_{ k i} \delta_{ l j} F^{i j} \\

& = F^{k l} \\

\end{align}

The (covariant) lower indexed tensor is then:

:$F\_\{\backslash alpha\backslash beta\}\; =\; \backslash begin\{pmatrix\}$

0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\

-\frac{E_x}{c} & 0 & -B_z & B_y \\

-\frac{E_y}{c} & B_z & 0 & -B_x \\

-\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}

This operation is equivalent to the matrix multiplication

:$\backslash begin\{pmatrix\}$

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}

\begin{pmatrix}

0 & -\frac{E_x}{c} & -\frac{E_y}{c} & -\frac{E_z}{c} \\

\frac{E_x}{c} & 0 & -B_z & B_y \\

\frac{E_y}{c} & B_z & 0 & -B_x \\

\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}

\begin{pmatrix}

-1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}

=\begin{pmatrix}

0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\

-\frac{E_x}{c} & 0 & -B_z & B_y \\

-\frac{E_y}{c} & B_z & 0 & -B_x \\

-\frac{E_z}{c} & -B_y & B_x & 0

\end{pmatrix}.

• Duality (mathematics)
• {{section link|Dual space|Bilinear products and dual spaces}}
• Einstein notation
• Flat (music) and Sharp (music) about the signs {{music|flat}} and {{music|sharp}}
• Hodge star operator
• Metric tensor
• Vector bundle

{{Reflist}}

• {{cite book|last=Lee|first=J. M.|title=Introduction to Smooth manifolds|year=2003|series=Springer Graduate Texts in Mathematics|isbn=0-387-95448-1|volume=218}}
• {{cite book|last=Lee|first=J. M.|title=Riemannian Manifolds – An Introduction to Curvature|year=1997|publisher=Springer Verlag|series=Springer Graduate Texts in Mathematics|volume=176|isbn=978-0-387-98322-6}}
• {{cite book|last1=Vaz|first1=Jayme|last2=da Rocha|first2=Roldão|year=2016|title=An Introduction to Clifford Algebras and Spinors |publisher=Oxford University Press|isbn=978-0-19-878-292-6}}

{{Riemannian geometry}}

{{tensors}}

{{Manifolds}}

Category:Differential geometry

Category:Riemannian geometry

Category:Riemannian manifolds

Category:Symplectic geometry