Category Archives: Differential Geometry

Vector Bundles

Let M be a differentiable manifold of dimension n. Consider an atlas \mathcal{U}=\{U_\alpha\}_{\alpha\in\mathcal{A}} along with coordinates x_\alpha^1,\cdots,x_\alpha^n in U_\alpha. For x=(x_\alpha^1(x),\cdots,x_\alpha^n(x))\in U_\alpha, a tangent vector is given by
v=\sum_{j=1}^nv_\alpha^j\frac{\partial}{\partial x_\alpha^j}.
If x\in U_\alpha\cap U_\beta, then v is also written as
v=\sum_{j=1}^nv_\beta^j\frac{\partial}{\partial x_\beta^j}.
Here, the change of coordinates is given by
v_\beta^j=vx_\beta^j=\sum_{k=1}^nv_\alpha^k\frac{\partial x_\beta^j}{\partial x_\alpha^k}.
For x\in U_\alpha\cap U_\beta and f=(f^1,\cdots,f^n)\in\mathbb{R}^n, define
\begin{align*} h_{\alpha\beta}(x)(f)&=\left(\sum_{k=1}^n\frac{\partial x_\beta^1}{\partial x_\alpha^k}f^k,\cdots,(\sum_{k=1}^n\frac{\partial x_\beta^n}{\partial x_\alpha^k}f^k\right)\\ &=\begin{pmatrix} \frac{\partial x_\beta^1}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^1}{\partial x_\alpha^n}\\ \vdots & \ddots & \vdots\\ \frac{\partial x_\beta^n}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^n}{\partial x_\alpha^n} \end{pmatrix}\begin{pmatrix} f^1\\ \vdots\\ f^n \end{pmatrix} \end{align*}
Hence h_{\alpha\beta}:U_\alpha\cap U_\beta\longrightarrow\mathrm{Aut(\mathbb{R}^n)}. The resulting bundle over M with fibre F=\mathbb{R}^n is called the tangent bundle of M and is denoted by TM. Note that TM is the set of all tangent vectors of M i.e. TM=\bigcup_{x\in M}T_xM. For each x\in U_\alpha, the fibre \pi^{-1}(x) of x\in M is T_xM\cong\{x\}\times\mathbb{R}^n. The local trivialization map h_\alpha:\pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times\mathbb{R}^n is given by
h_\alpha(v)=(x,(v_\alpha^1,\cdots,v_\alpha^n)),\ v\in T_xU_\alpha(=T_xM),\ x\in U_\alpha.

A fibre bundle (E,M,F,\pi) is called a vector bundle over M if each fibre F_x of x\in M is a vector space. So, a tangent bundle is a vector bundle. The tangent bundle TM is a differentiable manifold of dimension 2n with local coordinates in \pi^{-1}(U_\alpha) being (x_\alpha^1,\cdots,x_\alpha^n,v_\alpha^1,\cdots,v_\alpha^n). The Jacobian is given by
\begin{align*} J(x_\beta^1,\cdots,x_\beta^n;x_\alpha^1,\cdots,x_\alpha^n)&=\frac{\partial(x_\beta^1,\cdots,x_\beta^n)}{\partial(x_\alpha^1,\cdots,x_\alpha^n)}\\ &=\begin{pmatrix} \frac{\partial x_\beta^1}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^1}{\partial x_\alpha^n}\\ \vdots & \ddots & \vdots\\ \frac{\partial x_\beta^n}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^n}{\partial x_\alpha^n} \end{pmatrix}:U_\alpha\cap U_\beta\longrightarrow\mathrm{GL}(n,\mathbb{R}). \end{align*}
Let g_{\alpha\beta}=J(x_\beta^1,\cdots,x_\beta^n;x_\alpha^1,\cdots,x_\alpha^n) Then g_{\alpha\beta} satisfies
\begin{align*} g_{\alpha\alpha}(x)&=I_n;\\ g_{\beta\alpha}(x)&=g_{\alpha\beta}^{-1}(x),\ x\in U_\alpha\cap U_\beta;\\ g_{\alpha\beta}(x)g_{\beta\gamma}(x)g_{\gamma\alpha}(x)&=I_n,\ x\in U_\alpha\cap U_\beta\cap U_\gamma. \end{align*}
For x\in U_\alpha\cap U_\beta and f\in\mathbb{R}^n, h_{\alpha\beta}(x)(f)=g_{\alpha\beta}\cdot f. So, \mathrm{GL}(n,\mathbb{R}) acts on the fibre \mathbb{R}^n. The map g_{\alpha\beta} itself is often called a transition map.

If the transition map h_\alpha\beta is the group action of a Lie group G on the fibre F, the fibre bundle (E,M,F,\pi) is called a G-bundle and the Lie group G is called a structure group. The tangent bundle TM is also a G-bundle with structure group \mathrm{GL}(n,\mathbb{R}).

Differentiable Manifolds and Tangent Spaces

In \mathbb{R}^n, there is a globally defined orthonormal frame
E_{1p}=(1,0,\cdots,0)_p,\ E_{2p}=(0,1,0,\cdots,0)_p,\cdots,E_{np}=(0,\cdots,0,1)_p.
For any tangent vector X_p\in T_p(\mathbb{R}^n), X_p=\sum_{i=1}^n\alpha^iE_{ip}. Note that the coefficients \alpha^i are the ones that distinguish tangent vectors in T_p(\mathbb{R}^n). For a differentiable function f, the directional derivative X_p^\ast f of f with respect to X_p is given by
X_p^\ast f=\sum_{i=1}^n\alpha^i\left(\frac{\partial f}{\partial x_i}\right).
We identify each X_p with the differential operator
X_p^\ast=\sum_{i=1}^n\alpha^i\frac{\partial}{\partial x_i}:C^\infty(p)\longrightarrow\mathbb{R}.
Then the frame fields E_{1p},E_{2p},\cdots,E_{np} are identified with
\left(\frac{\partial}{\partial x_1}\right)_p,\left(\frac{\partial}{\partial x_2}\right)_p,\cdots,\left(\frac{\partial}{\partial x_n}\right)_p
respectively. Unlike \mathbb{R}^n, we cannot always have a globally defined frame on a differentiable manifold. So it is necessary for us to use local coordinate neighborhoods that are homeomorphic to \mathbb{R}^n and the associated frames \frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\cdots,\frac{\partial}{\partial x_n}.

Example. The points (x,y,z) are represented in terms of the spherical coordinates (\phi,\theta) as
x=\sin\phi\cos\theta,y=\sin\phi\sin\theta,z=\cos\phi,\ 0\leq\phi\leq\pi,\ 0\leq\theta\leq 2\pi.
By chain rule, one finds the standard basis \frac{\partial}{\partial\phi},\frac{\partial}{\partial\theta} for T_\ast S^2:
\begin{align*} \frac{\partial}{\partial\phi}&=\cos\phi\cos\theta\frac{\partial}{\partial x}+\cos\phi\sin\theta\frac{\partial}{\partial y}-\sin\phi\frac{\partial}{\partial z},\\ \frac{\partial}{\partial\theta}&=-\sin\phi\sin\theta\frac{\partial}{\partial x}+\sin\phi\cos\theta\frac{\partial}{\partial y}. \end{align*}
The frame field is not globally defined on S^2 since \frac{\partial}{\partial\theta} at \phi=0,\pi. More generally, the following theorem holds.

Frame field on 2-sphere

Theorem. [Hairy Ball Theorem] If n is even, a non-vanishing C^\infty vector field on S^n does not exist i.e. a C^\infty vector field on S^n must take zero value at some point of S^n.

The Hairy Ball Theorem tells us why we have ball spots on our heads. It can be also stated as “you cannot comb a hairy ball flat.” There may also be a meteorological implication of this theorem. It may implicate that there must be at least one spot on earth where there is no wind at all. No-wind spot may be the eye of a hurricane. So, as long as there is wind (and there always is) on earth, there must be a hurricane somewhere at all times.

It has been known that all odd-dimensional spheres have at least one non-vanishing C^\infty vector field and that only spheres S^1, S^3, S^7 have a C^\infty field of basis. For instance, there are three mutually perpendicular unit vector fields on S^3\subset\mathbb{R}^4 i.e. a frame field: Let S^3=\{(x^1,x^2,x^3,x^4)\in\mathbb{R}^4: \sum_{i=1}^4(x^i)^2=1\}. Then
\begin{align*} X&=-x^2\frac{\partial}{\partial x^1}+x^2\frac{\partial}{\partial x^2}+x^4\frac{\partial}{\partial x^3}-x^3\frac{\partial}{\partial x^4},\\ Y&=-x^3\frac{\partial}{\partial x^1}-x^4\frac{\partial}{\partial x^2}+x^1\frac{\partial}{\partial x^3}+x^2\frac{\partial}{\partial x^4},\\ Z&=-x^4\frac{\partial}{\partial x^1}+x^3\frac{\partial}{\partial x^2}-x^2\frac{\partial}{\partial x^3}+x^1\frac{\partial}{\partial x^4} \end{align*}
form an orthonormal basis of C^\infty vector fields on S^3.

Fibre Bundles

A fibre bundle is an object (E,M,F,\pi) consisting of

  1. The total space E;
  2. The base space M with an open covering \mathcal{U}=\{U_\alpha\}_{\alpha\in\mathcal{A}};
  3. The fibre F and the projection map E\stackrel{\pi}{ \longrightarrow}M.

The simplest case is E=M\times F. In this case, the bundle is called a trivial bundle. In general the total space may be too complicated for us to understand, so it would be nice if we can always find smaller parts that are simple enough for us to understand such as trivial bundles. For this reason, we want the fibre bundle to have the additional property: For each U_\alpha\in\mathcal{U}, there exists a homeomorphism h_\alpha : \pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times F. Such a homeomorphism h_\alpha is called a local trivialization. For each x\in M, F_x:=\pi^{-1}(x) is homeomorphic to \{x\}\times F. F_x is called the fibire of x.

Let x\in U_\alpha\cap U_\beta. Then F_x^\alpha\subset\pi^{-1}(U_\alpha) and F_x^\beta\subset\pi^{-1}(U_\beta) may not be the same. However, the two fibres are homeomorphic. For each x\in M, denote by h_{\alpha\beta}(x) the homeomorphism from F_x^\alpha to F_x^\beta. Then for each x\in M, h_{\alpha\beta}(x)\in\mathrm{Aut}(F) where \mathrm{Aut}(F) is the group of homeomorphisms from F to itself i.e. the automorphism group of F. The map h_{\alpha\beta}: U_\alpha\cap U_\beta\longrightarrow\mathrm{Aut}(F) is called a transition map. Note that for U_\alpha,U_\beta\in\mathcal{U} with U_\alpha\cap U_\beta\ne\emptyset, h_\alpha\circ h_\beta^{-1}:(U_\alpha\cap U_\beta)\times F\longrightarrow (U_\alpha\cap U_\beta)\times F satisfies
h_\alpha\circ h_\beta^{-1}(x,f)=(x,h_{\alpha\beta}(x)f)
for any x\in U_\alpha\cap U_\beta, f\in F

Structural Equations

Definition. The dual 1-forms \theta_1,\theta_2,\theta_3 of a frame E_1,E_2,E_3 on \mathbb{E}^3 are defined by
\theta_i(v)=v\cdot E_i(p),\ v\in T_p\mathbb{E}^3.
Clearly \theta_i is linear.

Example. The dual 1-forms of the natural frame U_1,U_2,U_3 are dx_1, dx_2, dx_3 since
dx_i(v)=v_i=v\cdot U_i(p)
for each v\in T_p\mathbb{E}^3.

For any vector field V on \mathbb{E}^3,
V=\sum_i\theta_i(V)E_i.
To see this, let us calculate for each V(p)\in T_p\mathbb{E}^3
\begin{align*} \sum_i\theta_i(V(p))E_i(p)&=\sum_i(V(p)\cdot E_i(p))E_i(p)\\ &=\sum_iV_i(p)E_i(p)\\ &=V(p). \end{align*}

Lemma. Let \theta_1,\theta_2,\theta_3 be the dual 1-forms of a frame E_1, E_2, E_3. Then any 1-form \phi on \mathbb{E}^3 has a unique expression
\phi=\sum_i\phi(E_i)\theta_i.

Proof. Let V be any vector field on \mathbb{E}^3. Then
\begin{align*} \sum_i\phi(E_i)\theta_i(V)&=\sum_i\phi(E_i)\theta_i(V)\\ &=\phi(\sum_i\theta_i(V)E_i)\ \mbox{by linearity of $phi$}\\ &=\phi(V). \end{align*}
Let A=(a_{ij}) be the attitude matrix of a frame field E_1, E_2, E_3, i.e.
\begin{equation}\label{eq:frame}E_i=\sum_ja_{ij}U_j,\ i=1,2,3.\end{equation}
Clearly \theta_i=\sum_j\theta_i(U_j)dx_j. On the other hand,
\theta_i(U_j)=E_i\cdot U_j=\left(\sum_ka_{ik}U_k\right)\cdot U_j=a_{ij}. Hence the dual formulation of \eqref{eq:frame} is
\begin{equation}\label{eq:dualframe}\theta_i=\sum_ja_{ij}dx_j.\end{equation}

Theorem. [Cartan Structural Equations] Let E_1, E_2, E_3 be a frame field on \mathbb{E}^3 with dual 1-forms \theta_1, \theta_2, \theta_3 and connection forms \omega_{ij}, i,j=1,2,3. Then

  1. The First Structural Equations: d\theta_i=\sum_j\omega_{ij}\wedge\theta_j.
  2. The Second Structural Equations: d\omega_{ij}=\sum_k\omega_{ik}\wedge\omega_{kj}.

Proof. The exterior derivative of \eqref{eq:dualframe} is
d\theta_i=\sum_jda_{ij}\wedge dx_j. Since \omega=dA\cdot{}^tA and {}^tA=A^{-1} (recall that A is an orthogonal matrix), dA=\omega\cdot A, i.e.
da_{ij}=\sum_k\omega_{ik}a_{kj}.
So,
\begin{align*} d\theta_i&=\sum_j\left\{\left(\sum_k\omega_{ik}a_{kj}\right)\wedge dx_j\right\}\\ &=\sum_k\left\{\omega_{ik}\wedge\sum_j a_{kj}dx_j\right\}\\ &=\sum_k\omega_{ik}\wedge\theta_k. \end{align*}

From \omega=dA\cdot{}^tA,
\begin{equation}\label{eq:connectform}\omega_{ij}=\sum_kda_{ik}a_{jk}.\end{equation}
The exterior derivative of \eqref{eq:connectform} is
\begin{align*} d\omega_{ij}&=\sum_k da_{jk}\wedge d_{ik}\\ &=-\sum_k da_{ik}\wedge da_{jk}, \end{align*}
i.e.
\begin{align*} d\omega&=-dA\wedge{}^t(dA)\\ &=-(\omega\cdot A)\cdot({}^tA\cdot{}^t\omega)\\ &=-\omega\cdot (A\cdot{}^tA)\cdot{}^t\omega\\ &=-\omega\cdot{}^t\omega\ \ \ (A\cdot{}^tA=I)\\ &=\omega\cdot\omega.\ \ \ (\mbox{$\omega$ is skew-symmetric.}) \end{align*}
This is equivalent to the second structural equations.

Example. [Structural Equations for the Spherical Frame Field] Let us first calculate the dual forms and connection forms.

From the spherical coordinates
\begin{align*} x_1&=\rho\cos\varphi\cos\theta,\\ x_2&=\rho\cos\varphi\sin\theta,\\ x_3&=\rho\sin\varphi, \end{align*}
we obtain differentials
\begin{align*} dx_1&=\cos\varphi\cos\theta d\rho-\rho\sin\varphi\cos\theta d\varphi-\rho\cos\varphi\sin\theta d\theta,\\ dx_2&=\cos\varphi\sin\theta d\rho-\rho\sin\varphi\sin\theta d\varphi+\rho\cos\varphi\cos\theta d\theta,\\ dx_3&=\sin\varphi d\rho+\rho\cos\varphi d\varphi. \end{align*}
From the spherical frame field F_1, F_2, F_3 discussed here, we find its attitude matrix
A=\begin{pmatrix} \cos\varphi\cos\theta & \cos\varphi\sin\theta & \sin\varphi\\ -\sin\theta & \cos\theta & 0\\ -\sin\varphi\cos\theta & -\sin\varphi\sin\theta & \cos\varphi \end{pmatrix}.
Thus by (2) we find the dual 1-forms
\begin{align*} \begin{pmatrix} \theta_1\\ \theta_2\\ \theta_3 \end{pmatrix}&=\begin{pmatrix} \cos\varphi\cos\theta & \cos\varphi\sin\theta & \sin\varphi\\ -\sin\theta & \cos\theta & 0\\ -\sin\varphi\cos\theta & -\sin\varphi\sin\theta & \cos\varphi \end{pmatrix}\begin{pmatrix} dx_1\\ dx_2\\ dx_3 \end{pmatrix}\\ &=\begin{pmatrix} d\rho\\ \rho\cos\theta d\theta\\ \rho d\varphi \end{pmatrix}. \end{align*}
\begin{align*} &dA=\\ &\begin{bmatrix} -\sin\varphi\cos\theta d\varphi-\cos\varphi\sin\theta d\theta & -\sin\varphi\sin\theta d\varphi+\cos\varphi\cos\theta d\theta & \cos\varphi d\varphi\\ -\cos\theta d\theta & -\sin\theta d\theta & 0\\ -\cos\varphi\cos\theta d\varphi+\sin\varphi\sin\theta d\theta & -\cos\varphi\sin\theta d\varphi-\sin\varphi\sin\theta d\theta & -\sin\varphi d\varphi \end{bmatrix}\end{align*}
and so,
\begin{align*} \omega&=\begin{pmatrix} 0 & \omega_{12} & \omega_{13}\\ -\omega_{12} & 0 & \omega_{23}\\ -\omega_{13} & -\omega_{23} & 0 \end{pmatrix}\\ &=dA\cdot{}^tA\\ &=\begin{pmatrix} 0 & \cos\varphi d\theta & d\varphi\\ -\cos\varphi d\theta & 0 & \sin\varphi d\theta\\ -d\varphi & -\sin\varphi d\theta & 0 \end{pmatrix}. \end{align*}
From these dual 1-forms and connections forms one can immediately verify the first and the second structural equations.

Tensors I

Tensors may be considered as a generalization of vectors and covectors. They are extremely important quantities for studying differential geometry and physics.

Let M^n be an n-dimensional differentiable manifold. For each x\in M^n, let E_x=T_xM^n, i.e. the tangent space to M^n at x. We denote the canonical basis of E by \partial=\left(\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^n}\right) and its dual basis by \sigma=dx=(dx^1,\cdots,dx^n), where x^1,\cdots,x^n are local coordinates. The canonical basis \frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^1} also simply denoted by \partial_1,\cdots,\partial_n.

Covariant Tensors

Definition. A covariant tensor of rank r is a multilinear real-valued function
Q:E\times E\times\cdots\times E\longrightarrow\mathbb{R}
of r-tuples of vectors. A covariant tensor of rank r is also called a tensor of type (0,r) or shortly (0,r)-tensor. Note that the values of Q must be independent of the basis in which the components of the vectors are expressed. A covariant vector (also called covector or a 1-form) is a covariant tensor of rank 1. An important of example of covariant tensor of rank 2 is the metric tensor G:
G(v,w)=\langle v,w\rangle=\sum_{i,j}g_{ij}v^iw^j.

In componenents, by multilinearity
\begin{align*} Q(v_1\cdots,v_r)&=Q\left(\sum_{i_1}v_1^{i_1}\partial_{i_1},\cdots,\sum_{i_r}v_r^{i_r}\partial_{i_r}\right)\\ &=\sum_{i_1,\cdots,i_r}v_1^{i_1}\cdots v_r^{i_r}Q(\partial_{i_1},\cdots,\partial_{i_r}). \end{align*}
Denote Q(\partial_{i_1},\cdots,\partial_{i_r}) by Q_{i_1,\cdots,i_r}. Then
Q(v_1\cdots,v_r)=\sum_{i_1,\cdots,i_r}Q_{i_1,\cdots,i_r}v_1^{i_1}\cdots v_r^{i_r}.\ \ \ \ \ \mbox{(1)}
Using the Einstein’s convention, (1) can be shortly written as
Q(v_1\cdots,v_r)=Q_{i_1,\cdots,i_r}v_1^{i_1}\cdots v_r^{i_r}.
The set of all covariant tensors of rank r forms a vector space over \mathbb{R}. The number of components in such a tensor is n^r. The vector space of all covariant r-th rank tensors is denoted by
E^\ast\otimes E^\ast\otimes\cdots\otimes E^\ast=\otimes^r E^\ast.

If \alpha,\beta\in E^\ast, i.e. covectors, we can form the 2nd rank covariant tensor, the tensor product \alpha\otimes\beta of \alpha and \beta: Define \alpha\otimes\beta: E\times E\longrightarrow\mathbb{R} by
\alpha\otimes\beta(v,w)=\alpha(v)\beta(w).
If we write \alpha=a_idx^i and \beta=b_jdx^j, then
(\alpha\otimes\beta)_{ij}=\alpha\otimes\beta(\partial_i,\partial_j)=\alpha(\partial_i)\beta(\partial_j)=a_ib_j.

Contravariant Tensors

A contravariant vector, i.e. an element of E can be considered as a linear functional v: E^\ast\longrightarrow\mathbb{R} defined by
v(\alpha)=\alpha(v)=a_iv^i,\ \alpha=a_idx^i\in E^\ast.

Definition. A contravariant tensor of rank s is a multilinear real-valued function T on s-tuples of covectors
T:E^\ast\times E^\ast\times\cdots\times E^\ast\longrightarrow\mathbb{R}. A contravariant tensor of rank s is also called a tensor of type (s,0) or shortly (s,0)-tensor.
For 1-forms \alpha_1,\cdots,\alpha_s
T(\alpha_1,\cdots,\alpha_s)=a_{1_{i_1}}\cdots a_{s_{i_s}}T^{i_1\cdots i_s}
where
T^{i_1\cdots i_s}:=T(dx^{i_1},\cdots,dx^{i_s}).
The space of all contravariant tensors of rank s is denoted by
E\otimes E\otimes\cdots\otimes E:=\otimes^s E.
Contravariant vectors are contravariant tensors of rank 1. An example of a contravariant tensor of rank 2 is the inverse of the metric tensor G^{-1}=(g^{ij}):
G^{-1}(\alpha,\beta)=g^{ij}a_ib_j.

Given a pair v,w of contravariant vectors, we can form the tensor product v\otimes w in the same manner as we did for covariant vectors. It is the 2nd rank contravariant tensor with components (v\otimes w)^{ij}=v^jw^j. The metric tensor G and its inverse G^{-1} may be written as
G=g_{ij}dx^i\otimes dx^j\ \mbox{and}\ G^{-1}=g^{ij}\partial_i\otimes\partial_j.

Mixed Tensors

Definition. A mixed tensor, r times covariant and s times contravariant, is a real multilinear function W
W: E^\ast\times E^\ast\times\cdots\times E^\ast\times E\times E\times\cdots\times E\longrightarrow\mathbb{R}
on s-tuples of covectors and r-tuples of vectors. It is also called a tensor of type (s,r) or simply (s,r)-tensor. By multilinearity
W(\alpha_1,\cdots,\alpha_s, v_1,\cdots, v_r)=a_{1_{i_1}}\cdots a_{s_{i_s}}W^{i_1\cdots i_s}{}_{j_1\cdots j_r}v_1^{j_1}\cdots v_r^{j_r}
where
W^{i_1\cdots i_s}{}_{j_1\cdots j_r}:=W(dx^{i_1},\cdots,dx^{i_s},\partial_{j_1},\cdots,\partial_{j_r}).

A 2nd rank mixed tensor may arise from a linear operator A: E\longrightarrow E. Define W_A: E^\ast\times E\longrightarrow\mathbb{R} by W_A(\alpha,v)=\alpha(Av). Let A=(A^i{}_j) be the matrix associated with A, i.e. A(\partial_j)=\partial_i A^i{}_j. Let us calculate the component of W_A:
W_A^i{}_j=W_A(dx^i,\partial_j)=dx^i(A(\partial_j))=dx^i(\partial_kA^k{}_j)=\delta^i_kA^k{}_j=A^i{}_j.
So the matrix of the mixed tensor W_A is just the matrix associated with A. Conversely, given a mixed tensotr W, once convariant and once contravariant, we can define a linear transformation A such that W(\alpha,v)=\alpha(A,v). We do not distinguish between a linear transformation A and its associated mixed tensor W_A. In components, W(\alpha,v) is written as
W(\alpha,v)=a_iA^i{}_jv^j=aAv.

The tensor product w\otimes\beta of a vector and a covector is the mixed tensor defined by
(w\otimes\beta)(\alpha,v)=\alpha(w)\beta(v). The associated transformation is can be written as
A=A^i{}_j\partial_i\otimes dx^j=\partial_i\otimes A^i{}_jdx^j.

For math undergraduates, different ways of writing indices (raising, lowering, and mixed) in tensor notations can be very confusing. Main reason is that in standard math courses such as linear algebra or elementary differential geometry (classical differential geometry of curves and surfaces in \mathbb{E}^3) the matrix of a linear transformation is usually written as A_{ij}. Physics undergraduates don’t usually get a chance to learn tensors in undergraduate physics courses. In order to study more advanced differential geometry or physics such as theory of special and general relativity, and field theory one must be able to distinguish three different ways of writing matrices A_{ij}, A^{ij}, and A^i{}_j. To summarize, A_{ij} and A^{ij} are bilinear forms on E and E^\ast, respectively that are defined by
A_{ij}v^iv^j\ \mbox{and}\ A^{ij}a_ib_j\ (\mbox{respectively}). A^i{}_j is the matrix of a linear transformation A: E\longrightarrow E.

Let (E,\langle\ ,\ \rangle) be an inner product space. Given a linear transformation A: E\longrightarrow E (i.e. a mixed tensor), one can associate a bilinear covariant bilinear form A’ by
A'(v,w):=\langle v,Aw\rangle=v^ig_{ij}A^j{}_k w^k. So we see that the matrix of A’ is
A’_{ik}=g_{ij}A^j{}_k. The process can be said as “we lower the index j, making it a k, by mans of the metric tensor g_{ij}.” In tensor analysis one uses the same letter, i.e. instead of A’, one writes
A_{ik}:=g_{ij}A^j{}_k. This is clearly a covariant tensor. In general, the components of the associated covariant tensor A_{ik} differ from those of the mixed tensor A^i{}_j. But if the basis is orthonormal, i.e. g_{ij}=\delta^i_j then they coincide. That is the reason why we simply write A_{ij} without making any distiction in linear algebra or in elementary differential geometry.

Similarly, one may associate to the linear transformation A a contravariant bilinear form
\bar A(\alpha,\beta)=a_iA^i{}_jg^{jk}b_k whose matrix components can be written as
A^{ik}=A^i{}_jg^{jk}.

Note that the metric tensor g_{ij} represents a linear map from E to E^\ast, sending the vector with components v^j into the covector with components g_{ij}v^j. In quantum mechanics, the covector g_{ij}v^j is denoted by \langle v| and called a bra vector, while the vector v^j is denoted by |v\rangle and called a ket vector. Usually the inner product on E
\langle\ ,\ \rangle:E\times E\longrightarrow\mathbb{R};\ \langle v,w\rangle=g_{ij}v^iw^j is considered as a covariant tensor of rank 2. But in quantum mechanics \langle v,w\rangle is not considered as a covariant tensor g_{ij} of rank 2 acting on a pair of vectors (v,w), rather it is regarded as the braket \langle v|w\rangle, a bra vector \langle v| acting on a ket vector |w\rangle.