\end{eqnarray*}
Both are not degenerated. The following holds: h(G\vec{x},G\vec{y})=<\vec{x},G\vec{y}>=g(\vec{x},\vec{y}).
If we identify \cal V and \cal V^* with G, than g (or h)
gives an inner product on \cal V.
The inner product (,)_\Lambda on \Lambda^k(\cal V) is defined by:
(\Phi,\Psi)_\Lambda=\frac{1}{k!}(\Phi,\Psi)_{T^0_k(\cal V)}
The inner product of two vectors is than given by:
(\vec{x},\vec{y})=x^iy^i<\vec{c}_i,G\vec{c}_j>=g_{ij}x^ix^j
The matrix g_{ij} of G is given by
g_{ij}\hat{\vec{c}}^{~j}=G\vec{c}_i
The matrix g^{ij} of G^{-1} is given by:
g^{kl}\vec{c}_l=G^{-1}\hat{\vec{c}}^{~k}
For this metric tensor g_{ij} holds: g_{ij}g^{jk}=\delta_i^k.
This tensor can raise or lower indices:
x_j=g_{ij}x^i~~~,~~~x^i=g^{ij}x_j
and du^i=\hat{\vec{c}}^{~i}=g^{ij}\vec{c}_j.
Definition: let \cal U and \cal V be two finite dimensional vector
spaces with dimensions m and n. Let \cal U^*\times V^* be the cartesian
product of \cal U and \cal V. A function t:{\cal U^*\times V^*}\rightarrow I\hspace{-1mm}R;
(\hat{\vec{u}};\hat{\vec{v}}\,)\mapsto t(\hat{\vec{u}};\hat{\vec{v}}\,)=t^{\alpha\beta}u_\alpha u_\beta\in I\hspace{-1mm}R
is called a tensor if t is linear in \hat{\vec{u}} and \hat{\vec{v}}.
The tensors t form a vector space denoted by \cal U\otimes V.
The elements T\in\cal V\otimes V are called contravariant 2-tensors:
T=T^{ij}\vec{c}_i\otimes\vec{c}_j=T^{ij}\partial_i\otimes\partial_j. The
elements T\in\cal V^*\otimes V^* are called covariant 2-tensors:
T=T_{ij}\hat{\vec{c}}^{~i}\otimes\hat{\vec{c}}^{~j}=T_{ij}dx^i\otimes dx^j.
The elements T\in\cal V^*\otimes V are called mixed 2 tensors:
T=T_i^{.j}\hat{\vec{c}}^{~i}\otimes\vec{c}_j=T_i^{.j}dx^i\otimes\partial_j,
and analogous for T\in\cal V\otimes V^*.
The numbers given by
t^{\alpha\beta}=t(\hat{\vec{c}}^{~\alpha},\hat{\vec{c}}^{~\beta}\,)
with 1\leq\alpha\leq m and 1\leq\beta\leq n are the components of t.
Take \vec{x}\in\cal U and \vec{y}\in\cal V. Than the function
\vec{x}\otimes\vec{y}, definied by
(\vec{x}\otimes\vec{y})(\hat{\vec{u}},\hat{\vec{v}})=<\vec{x},\hat{\vec{u}}>_U<\vec{y},\hat{\vec{v}}>_V
is a tensor. The components are derived from: (\vec{u}\otimes\vec{v})_{ij}=u_iv^j.
The tensor product of 2 tensors is given by:
\begin{eqnarray*}
{2\choose0}~\mbox{form:}~&&(\vec{v}\otimes\vec{w})(\hat{\vec{p}},\hat{\vec{q}})=v^ip_iw^kq_k=T^{ik}p_iq_k\\
{0\choose2}~\mbox{form:}~&&(\hat{\vec{p}}\otimes\hat{\vec{q}})(\vec{v},\vec{w})=p_iv^iq_kw^k=T_{ik}v^iw^k\\
{1\choose1}~\mbox{form:}~&&(\vec{v}\otimes\hat{\vec{p}})(\hat{\vec{q}},\vec{w})=v^iq_ip_kw^k=T_k^iq_iw^k
\end{eqnarray*}
A tensor t\in{\cal V\otimes V} is called symmetric resp. antisymmetric if
\forall\hat{\vec{x}},\hat{\vec{y}}\in{\cal V^*} holds:
t(\hat{\vec{x}},\hat{\vec{y}}\,)=t(\hat{\vec{y}},\hat{\vec{x}}\,) resp.
t(\hat{\vec{x}},\hat{\vec{y}}\,)=-t(\hat{\vec{y}},\hat{\vec{x}}\,).
A tensor t\in{\cal V^*\otimes V^*} is called symmetric resp. antisymmetric
if \forall\vec{x},\vec{y}\in{\cal V} holds:
t(\vec{x},\vec{y})=t(\vec{y},\vec{x}) resp.
t(\vec{x},\vec{y})=-t(\vec{y},\vec{x}). The linear transformations
\cal S and \cal A in \cal V\otimes W are defined by:
\begin{eqnarray*}
{\cal S}t(\hat{\vec{x}},\hat{\vec{y}}\,)&=&\frac{1}{2}(t(\hat{\vec{x}},\hat{\vec{y}})+t(\hat{\vec{y}},\hat{\vec{x}}\,))\\
{\cal A}t(\hat{\vec{x}},\hat{\vec{y}}\,)&=&\frac{1}{2}(t(\hat{\vec{x}},\hat{\vec{y}})-t(\hat{\vec{y}},\hat{\vec{x}}\,))
\end{eqnarray*}
Analogous in \cal V^*\otimes V^*. If t is symmetric resp. antisymmetric,
than {\cal S}t=t resp. {\cal A}t=t.
The tensors \vec{e}_i\vee\vec{e}_j=\vec{e}_i\vec{e}_j=2{\cal S}(\vec{e}_i\otimes\vec{e}_j),
with 1\leq i\leq j\leq n are a basis in \cal S(V\otimes V) with dimension
\frac{1}{2} n(n+1).
The tensors \vec{e}_i\wedge\vec{e}_j=2{\cal A}(\vec{e}_i\otimes\vec{e}_j),
with 1\leq i\leq j\leq n are a basis in \cal A(V\otimes V) with dimension
\frac{1}{2} n(n-1).
The complete antisymmetric tensor \varepsilon is given by:
\varepsilon_{ijk}\varepsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}.
The permutation-operators e_{pqr} are defined by:
e_{123}=e_{231}=e_{312}=1, e_{213}=e_{132}=e_{321}=-1, for all other
combinations e_{pqr}=0. There is a connection with the \varepsilon tensor:
\varepsilon_{pqr}=g^{-1/2}e_{pqr} and \varepsilon^{pqr}=g^{1/2}e^{pqr}.
Let \alpha\in\Lambda^k(\cal V) and \beta\in\Lambda^l(\cal V). Than
\alpha\wedge\beta\in\Lambda^{k+l}(\cal V) is defined by:
\alpha\wedge\beta=\frac{(k+l)!}{k!l!}{\cal A}(\alpha\otimes\beta)
If \alpha and \beta\in\Lambda^1(\cal V)={\cal V}^* holds:
\alpha\wedge\beta=\alpha\otimes\beta-\beta\otimes\alpha
The outer product can be written as: (\vec{a}\times\vec{b})_i=\varepsilon_{ijk}a^jb^k,
\vec{a}\times\vec{b}=G^{-1}\cdot*(G\vec{a}\wedge G\vec{b}).
Take \vec{a},\vec{b},\vec{c},\vec{d}\in I\hspace{-1mm}R^4. Than
(dt\wedge dz)(\vec{a},\vec{b})=a_0b_4-b_0a_4 is the oriented surface of the
projection on the tz-plane of the parallelogram spanned by \vec{a} and
\vec{b}.
Further
(dt\wedge dy\wedge dz)(\vec{a},\vec{b},\vec{c})=\det\left|\begin{array}{ccc}
a_0&b_0&c_0\\ a_2&b_2&c_2\\ a_4&b_4&c_4 \end{array}\right|
is the oriented 3-dimensional volume of the projection on the tyz-plane of
the parallelepiped spanned by \vec{a}, \vec{b} and \vec{c}.
(dt\wedge dx\wedge dy\wedge dz)(\vec{a},\vec{b},\vec{c},\vec{d})=\det(\vec{a},\vec{b},\vec{c},\vec{d})
is the 4-dimensional volume of the hyperparellelepiped spanned by
\vec{a}, \vec{b}, \vec{c} and \vec{d}.
\Lambda^k(\cal V) and \Lambda^{n-k}(\cal V) have the same dimension
because {n\choose k}={n\choose{n-k}} for 1\leq k\leq n.
Dim(\Lambda^n({\cal V}))=1. The choice of a basis means the choice of an oriented
measure of volume, a volume \mu, in \cal V. We can gauge \mu so that for
an orthonormal basis \vec{e}_i holds: \mu(\vec{e}_i)=1. This basis is than
by definition positive oriented if
\mu=\hat{\vec{e}}^{~1}\wedge \hat{\vec{e}}^{~2}\wedge...\wedge \hat{\vec{e}}^{~n}=1.
Because both spaces have the same dimension one can ask if there exists a
bijection between them. If \cal V has no extra structure this is not the
case. However, such an operation does exist if there is an inner product
defined on \cal V and the corresponding volume \mu. This is called
the Hodge star operator and denoted by *. The following holds:
\forall_{w\in\Lambda^k({\cal V})}\exists_{*w\in\Lambda^{k-n}({\cal V})}\forall_{\theta\in\Lambda^k({\cal V})}~~
\theta\wedge*w=(\theta,w)_\lambda\mu
For an orthonormal basis in I\hspace{-1mm}R^3 holds: the volume: \mu=dx\wedge dy\wedge dz,
*dx\wedge dy\wedge dz=1, *dx=dy\wedge dz, *dz=dx\wedge dy, *dy=-dx\wedge dz,
*(dx\wedge dy)=dz, *(dy\wedge dz)=dx, *(dx\wedge dz)=-dy.
For a Minkowski basis in I\hspace{-1mm}R^4 holds: \mu=dt\wedge dx\wedge dy\wedge dz,
G=dt\otimes dt-dx\otimes dx-dy\otimes dy-dz\otimes dz, and
*dt\wedge dx\wedge dy\wedge dz=1 and *1=dt\wedge dx\wedge dy\wedge dz.
Further *dt=dx\wedge dy\wedge dz and *dx=dt\wedge dy\wedge dz.
The directional derivative in point \vec{a} is given by:
{\cal L}_{\vec{a}}f=<\vec{a},df>=a^i\frac{\partial f}{\partial x^i}
The Lie-derivative is given by:
({\cal L}_{\vec{v}}\vec{w})^j=w^i\partial_iv^j-v^i\partial_iw^j
To each curvelinear coordinate system u^i we add a system of n^3
functions \Gamma^i_{jk} of \vec{u}, defined by
\frac{\partial^2\vec{x}}{\partial u^i\partial u^k}=\Gamma_{jk}^i\frac{\partial\vec{x}}{\partial u^i}
These are Christoffel symbols of the second kind. Christoffel symbols
are no tensors. The Christoffel symbols of the second kind are given by:
\left\{\begin{array}{@{}c@{}}i\\ jk \end{array}\right\}:=\Gamma^i_{jk}=
\left\langle\frac{\partial^2\vec{x}}{\partial u^k\partial u^j},dx^i\right\rangle
with \Gamma^i_{jk}=\Gamma^i_{kj}. Their transformation to a different
coordinate system is given by:
\Gamma_{j'k'}^{i'}=A_{i'}^iA_{j'}^jA_{k'}^k\Gamma^i_{jk}+A_i^{i'}(\partial_{j'}A_{k'}^i)
The first term in this expression is 0 if the primed coordinates are
cartesian.
There is a relation between Christoffel symbols and the metric:
\Gamma_{jk}^i=\frac{1}{2} g^{ir}(\partial_j g_{kr}+\partial_k g_{rj}-\partial_r g_{jk})
and \Gamma^\alpha_{\beta\alpha}=\partial_\beta(\ln(\sqrt{|g|})).
Lowering an index gives the Christoffel symbols of the first kind:
\Gamma^i_{jk}=g^{il}\Gamma_{jkl}.
The covariant derivative \nabla_j of a vector, covector and of
rank-2 tensors is given by:
\begin{eqnarray*}
\nabla_ja^i &=&\partial_ja^i+\Gamma^i_{jk}a^k\\
\nabla_ja_i &=&\partial_ja_i-\Gamma^k_{ij}a_k\\
\nabla_\gamma a^\alpha_\beta &=&\partial_\gamma a^\alpha_\beta -\Gamma^\varepsilon_{\gamma\beta} a^\alpha_\varepsilon+\Gamma^\alpha_{\gamma\varepsilon}a_\beta^\varepsilon\\
\nabla_\gamma a_{\alpha\beta}&=&\partial_\gamma a_{\alpha\beta}-\Gamma^\varepsilon_{\gamma\alpha}a_{\varepsilon\beta}-\Gamma^\varepsilon_{\gamma\beta}a_{\alpha\varepsilon}\\
\nabla_\gamma a^{\alpha\beta}&=&\partial_\gamma a^{\alpha\beta}+\Gamma^\alpha_{\gamma\varepsilon}a^{\varepsilon\beta}+\Gamma^\beta_{\gamma\varepsilon}a^{\alpha\varepsilon}
\end{eqnarray*}
Ricci's theorem:
\nabla_\gamma g_{\alpha\beta}=\nabla_\gamma g^{\alpha\beta}=0
is given by:
{\rm grad}(f)=G^{-1}df=g^{ki}\frac{\partial f}{\partial x^i}\frac{\partial}{\partial x^k}
is given by:
{\rm div}(a^i)=\nabla_ia^i=\frac{1}{\sqrt{g}}\partial_k(\sqrt{g}\,a^k)
is given by:
{\rm rot}(a)=G^{-1}\cdot*\cdot d\cdot G\vec{a}=-\varepsilon^{pqr}\nabla_qa_p=\nabla_qa_p-\nabla_pa_q
is given by:
\Delta(f)={\rm div~grad}(f)=*d*df=\nabla_ig^{ij}\partial_jf=g^{ij}\nabla_i\nabla_jf=
\frac{1}{\sqrt{g}}\frac{\partial}{\partial x^i}\left(\sqrt{g}\,g^{ij}\frac{\partial f}{\partial x^j}\right)
We limit ourselves to I\hspace{-1mm}R^3 with a fixed orthonormal basis. A point is
represented by the vector \vec{x}=(x^1,x^2,x^3). A space curve is a
collection of points represented by \vec{x}=\vec{x}(t). The arc length of a
space curve is given by:
s(t)=\int\limits_{t_0}^t\sqrt{\left(\frac{dx}{d\tau}\right)^2+\left(\frac{dy}{d\tau}\right)^2+\left(\frac{dz}{d\tau}\right)^2}d\tau
The derivative of s with respect to t is the length of the vector d\vec{x}/dt:
\left(\frac{ds}{dt}\right)^2=\left(\frac{d\vec{x}}{dt},\frac{d\vec{x}}{dt}\right)
The osculation plane in a point P of a space curve is the limiting
position of the plane through the tangent of the plane in point P and a point
Q when Q approaches P along the space curve. The osculation plane is
parallel with \dot{\vec{x}}(s). If \ddot{\vec{x}}\neq0 the osculation
plane is given by:
\vec{y}=\vec{x}+\lambda\dot{\vec{x}}+\mu\ddot{\vec{x}}~~~\mbox{so}~~~
\det(\vec{y}-\vec{x},\dot{\vec{x}},\ddot{\vec{x}}\,)=0
In a bending point holds, if \dddot{\vec{x}}\neq0:
\vec{y}=\vec{x}+\lambda\dot{\vec{x}}+\mu\dddot{\vec{x}}
The tangent has unit vector \vec{\ell}=\dot{\vec{x}}, the
main normal unit vector \vec{n}=\ddot{\vec{x}} and the
binormal \vec{b}=\dot{\vec{x}}\times\ddot{\vec{x}}. So the main normal
lies in the osculation plane, the binormal is perpendicular to it.
Let P be a point and Q be a nearby point of a space curve \vec{x}(s).
Let \Delta\varphi be the angle between the tangents in P and Q and let
\Delta\psi be the angle between the osculation planes (binormals) in P and
Q. Then the curvature \rho and the torsion \tau in P are
defined by:
\rho^2=\left(\frac{d\varphi}{ds}\right)^2=\lim_{\Delta s\rightarrow0}\left(\frac{\Delta\varphi}{\Delta s}\right)^2~~~,~~~
\tau^2=\left(\frac{d\psi}{ds}\right)^2
and \rho>0. For plane curves \rho is the ordinary curvature and
\tau=0. The following holds:
\rho^2=(\vec{\ell},\vec{\ell})=(\ddot{\vec{x}},\ddot{\vec{x}}\,)~~~\mbox{and}~~~
\tau^2=(\dot{\vec{b}},\dot{\vec{b}})
Frenet's equations express the derivatives as linear combinations of these
vectors:
\dot{\vec{\ell}}=\rho\vec{n}~~,~~\dot{\vec{n}}=-\rho\vec{\ell}+\tau\vec{b}~~,~~
\dot{\vec{b}}=-\tau\vec{n}
From this follows that \det(\dot{\vec{x}},\ddot{\vec{x}},\dddot{\vec{x}}\,)=\rho^2\tau.
Some curves and their properties are:
| Screw line | \tau/\rho=constant |
| Circle screw line | \tau=constant, \rho=constant |
| Plane curves | \tau=0 |
| Circles | \rho=constant, \tau=0 |
| Lines | \rho=\tau=0 |
A surface in I\hspace{-1mm}R^3 is the collection of end points of the vectors
\vec{x}=\vec{x}(u,v), so x^h=x^h(u^\alpha). On the surface are 2 families
of curves, one with u=constant and one with v=constant.
The tangent plane in a point P at the surface has basis:
\vec{c}_1=\partial_1\vec{x}~~~\mbox{and}~~~\vec{c}_2=\partial_2\vec{x}
Let P be a point of the surface \vec{x}=\vec{x}(u^\alpha). The following
two curves through P, denoted by u^\alpha=u^\alpha(t),
u^\alpha=v^\alpha(\tau), have as tangent vectors in P
\frac{d\vec{x}}{dt}=\frac{du^\alpha}{dt}\partial_\alpha\vec{x}~~~,~~~
\frac{d\vec{x}}{d\tau}=\frac{dv^\beta}{d\tau}\partial_\beta\vec{x}
The first fundamental tensor of the surface in P is the inner product
of these tangent vectors:
\left(\frac{d\vec{x}}{dt},\frac{d\vec{x}}{d\tau}\right)=
(\vec{c}_\alpha,\vec{c}_\beta)\frac{du^\alpha}{dt}\frac{dv^\beta}{d\tau}
The covariant components w.r.t.\ the basis
\vec{c}_\alpha=\partial_\alpha\vec{x} are:
g_{\alpha\beta}=(\vec{c}_\alpha,\vec{c}_\beta)
For the angle \phi between the parameter curves in P: u=t,v=constant and
u=constant, v=\tau holds:
\cos(\phi)=\frac{g_{12}}{\sqrt{g_{11}g_{22}}}
For the arc length s of P along the curve u^\alpha(t) holds:
ds^2=g_{\alpha\beta}du^\alpha du^\beta
This expression is called the line element.
The 4 derivatives of the tangent vectors
\partial_\alpha\partial_\beta\vec{x}=\partial_\alpha\vec{c}_\beta are each
linear independent of the vectors \vec{c}_1, \vec{c}_2 and \vec{N}, with
\vec{N} perpendicular to \vec{c}_1 and \vec{c}_2. This is written as:
\partial_\alpha\vec{c}_\beta=\Gamma^\gamma_{\alpha\beta}\vec{c}_\gamma+h_{\alpha\beta}\vec{N}
This leads to:
\Gamma^\gamma_{\alpha\beta}=(\vec{c}^{~\gamma},\partial_\alpha\vec{c}_\beta)~~~,~~~
h_{\alpha\beta}=(\vec{N},\partial_\alpha\vec{c}_\beta)=\frac{1}{\sqrt{\det|g|}}\det(\vec{c}_1,\vec{c}_2,\partial_\alpha\vec{c}_\beta)
A curve on the surface \vec{x}(u^\alpha) is given by:
u^\alpha=u^\alpha(s), than \vec{x}=\vec{x}(u^\alpha(s)) with s the
arc length of the curve. The length of \ddot{\vec{x}} is the curvature
\rho of the curve in P. The projection of \ddot{\vec{x}} on the surface
is a vector with components
p^\gamma=\ddot{u}^\gamma+\Gamma^\gamma_{\alpha\beta}\dot{u}^\alpha\dot{u}^\beta
of which the length is called the geodetic curvature of the curve in p.
This remains the same if the surface is curved and the line element remains the
same. The projection of \ddot{\vec{x}} on \vec{N} has length
p=h_{\alpha\beta}\dot{u}^\alpha\dot{u}^\beta
and is called the normal curvature of the curve in P. The theorem
of Meusnier states that different curves on the surface with the same tangent
vector in P have the same normal curvature.
A geodetic line of a surface is a curve on the surface for which in each
point the main normal of the curve is the same as the normal on the surface.
So for a geodetic line is in each point p^\gamma=0, so
\frac{d^2u^\gamma}{ds^2}+\Gamma^\gamma_{\alpha\beta}\frac{du^\alpha}{ds}\frac{du^\beta}{ds}=0
The covariant derivative \nabla/dt in P of a vector field of a surface along
a curve is the projection on the tangent plane in P of the normal derivative
in P.
For two vector fields \vec{v}(t) and \vec{w}(t) along the same curve of
the surface follows Leibniz' rule:
\frac{d(\vec{v},\vec{w})}{dt}=\left(\vec{v},\frac{\nabla\vec{w}}{dt}\right)+\left(\vec{w},\frac{\nabla\vec{v}}{dt}\right)
Along a curve holds:
\frac{\nabla}{dt}(v^\alpha\vec{c}_\alpha)=\left(\frac{dv^\gamma}{dt}+\Gamma^\gamma_{\alpha\beta}\frac{du^\alpha}{dt}v^\beta\right)\vec{c}_\gamma
The Riemann tensor R is defined by:
R^\mu_{\nu\alpha\beta}T^\nu=\nabla_\alpha\nabla_\beta T^\mu-\nabla_\beta\nabla_\alpha T^\mu
This is a 1\choose 3 tensor with n^2(n^2-1)/12 independent components not
identically equal to 0. This tensor is a measure for the curvature of the
considered space. If it is 0, the space is a flat manifold. It has the
following symmetry properties:
R_{\alpha\beta\mu\nu}=R_{\mu\nu\alpha\beta}=-R_{\beta\alpha\mu\nu}=-R_{\alpha\beta\nu\mu}
The following relation holds:
[\nabla_\alpha,\nabla_\beta]T_\nu^\mu=R_{\sigma\alpha\beta}^\mu T_\nu^\sigma+R_{\nu\alpha\beta}^\sigma T_\sigma^\mu
The Riemann tensor depends on the Christoffel symbols through
R^\alpha_{\beta\mu\nu}=\partial_\mu\Gamma^\alpha_{\beta\nu}-\partial_\nu\Gamma^\alpha_{\beta\mu}+\Gamma^\alpha_{\sigma\mu}\Gamma^\sigma_{\beta\nu}-\Gamma^\alpha_{\sigma\nu}\Gamma^\sigma_{\beta\mu}
In a space and coordinate system where the Christoffel symbols are 0 this
becomes:
R^\alpha_{\beta\mu\nu}=\frac{1}{2} g^{\alpha\sigma}(\partial_\beta\partial_\mu g_{\sigma\nu}-\partial_\beta\partial_\nu g_{\sigma\mu}+\partial_\sigma\partial_\nu g_{\beta\mu}-\partial_\sigma\partial_\mu g_{\beta\nu})
The Bianchi identities are: \nabla_\lambda R_{\alpha\beta\mu\nu}+\nabla_\nu R_{\alpha\beta\lambda\mu} +\nabla_\mu R_{\alpha\beta\nu\lambda}=0.
The Ricci tensor is obtained by contracting the Riemann tensor:
R_{\alpha\beta}\equiv R_{\alpha\mu\beta}^\mu, and is symmetric in its
indices: R_{\alpha\beta}=R_{\beta\alpha}. The Einstein tensor G is
defined by: G^{\alpha\beta}\equiv R^{\alpha\beta}-\frac{1}{2} g^{\alpha\beta}.
It has the property that \nabla_\beta G^{\alpha\beta}=0. The Ricci-scalar is
R=g^{\alpha\beta}R_{\alpha\beta}.