Ngô Quốc Anh

December 12, 2009

R-G: The Second Fundamental Form

Filed under: Riemannian geometry — Ngô Quốc Anh @ 18:02

In this section, we take a closer look at the curvature at a point of a curve $C$ on a surface $X$. Assuming that $C$ is  parameterized by arc length, we will see that the vector $X''(s)$ (which is equal to $\kappa \vec n$, where $\vec n$ is the principal normal to the curve $C$ at $p$, and $\kappa$ is the curvature) can be written as

$\displaystyle \kappa \vec n=\kappa_{\rm N} {\rm N} + \kappa_g \vec n_g$,

where ${\rm N}$ is the normal to the surface at $p$, and $\kappa_g \vec n_g$ is a tangential component normal to the curve.

The component $\kappa_{\rm N}$ is called the normal curvature. Computing it will lead to the second fundamental form, another very important quadratic form associated with a surface. The component $\kappa_g$ is called the geodesic curvature.

It turns out that it only depends on the first fundamental form, but computing it is quite complicated, and this will lead to the Christoffel symbols.

Definition 1. Given a surface $X$, given any curve $C: t \mapsto X(u(t), v(t))$ on $X$, for any point $p$ on $X$, the orthonormal frame $(\vec t, \vec n_g ,{\rm N})$ is defined such that

$\displaystyle\begin{gathered}\vec t = {X_u}u' + {X_v}v', \hfill \\{\rm N} = \frac{{{X_u} \times {X_v}}}{{\left\| {{X_u} \times {X_v}} \right\|}}, \hfill \\{\vec n_g} = {\rm N} \times t, \hfill \\ \end{gathered}$

where ${\rm N}$ is the normal vector to the surface $X$ at $p$. The vector $\vec n_g$ is called the geodesic normal vector.

Observe that $\vec n_g$ is the unit normal vector to the curve $C$ contained in the tangent space $T_p(X)$ at $p$. If we use the frame $(\vec t, \vec n_g ,N)$, we will see shortly that $\kappa \vec n$ can be written as

$\displaystyle \kappa \vec n=\kappa_{\rm N} {\rm N} + \kappa_g \vec n_g$,

The component $\kappa_{\rm N} {\rm N}$ is the orthogonal projection of $\kappa \vec n$ onto the normal direction ${\rm N}$, and for this reason $\kappa_{\rm N}$ is called the normal curvature of $C$ at $p$. The component $\kappa_g \vec n_g$ is the orthogonal projection of $\kappa \vec n$ onto the tangent space $T_p(X)$ at $p$.

We now show how to compute the normal curvature. This will uncover the second fundamental form. Since $X'=X_u u'+X_vv'$, using chain rule we get

$\displaystyle X''=X_{uu} (u')^2 + 2 X_{uv}u'v'+X_u u''+X_v v''$.

In order to decompose $X''$ into its normal component (along ${\rm N}$) and its tangential component, we use a neat trick suggested by Eugenio Calabi. Recall that

$\displaystyle (\vec u \times \vec v) \times \vec w=(\vec u \cdot \vec w)\vec v - (\vec w \cdot \vec v)\vec u$.

Using this identity we have

$\displaystyle\begin{gathered}(N \times ({X_{uu}}{(u')^2} + 2{X_{uv}}u'v' + {X_{vv}}{(v')^2})) \times {\rm N} \hfill \\ \qquad = \left( {{\rm N} \cdot {\rm N}} \right)({X_{uu}}{(u')^2} + 2{X_{uv}}u'v' + {X_{vv}}{(v')^2}) - \left( {{\rm N} \cdot ({X_{uu}}{{(u')}^2} + 2{X_{uv}}u'v' + {X_{vv}}{{(v')}^2})} \right){\rm N}. \hfill \\ \end{gathered}$

Since ${\rm N}$ is a unit vector we can write

$\displaystyle\begin{gathered}\kappa n = \left( {{\rm N} \cdot ({X_{uu}}{{(u')}^2} + 2{X_{uv}}u'v' + {X_{vv}}{{(v')}^2})} \right){\rm N} \hfill \\ \qquad \qquad+ ({\rm N} \times ({X_{uu}}{(u')^2} + 2{X_{uv}}u'v' + {X_{vv}}{(v')^2})) \times {\rm N} \hfill \\\qquad\qquad + {X_u}u'' + {X_v}v''. \hfill \\ \end{gathered}$

Thus, it is clear that the normal component is

$\displaystyle {\kappa _{\rm N}}{\rm N} = \left( {{\rm N} \cdot ({X_{uu}}{{(u')}^2} + 2{X_{uv}}u'v' + {X_{vv}}{{(v')}^2})} \right){\rm N}$,

and the normal curvature is given by

$\displaystyle {\kappa _{\rm N}} = {{\rm N} \cdot ({X_{uu}}{{(u')}^2} + 2{X_{uv}}u'v' + {X_{vv}}{{(v')}^2})}$.

Letting

$\displaystyle L={\rm N} \cdot X_{uu}, \quad M={\rm N} \cdot X_{uv}, \quad N={\rm N}\cdot X_{vv}$

we have

$\displaystyle {\kappa _N} = L(u')^2+2Mu'v'+N(v')^2$.

Recalling that

$\displaystyle N = \frac{{{X_u} \times {X_v}}}{{\left\| {{X_u} \times {X_v}} \right\|}}$,

using the Lagrange identity

$\displaystyle (\vec u \cdot \vec v)^2+\|\vec u \times \vec v\|^2 = \|\vec u\|^2 \| vec v\|^2$,

we see that

$\displaystyle \| X_u \times X_v \| = \sqrt{EG-F^2}$,

and $L={\rm N} \cdot X_{uu}$ can be writtne as

$\displaystyle L = \frac{{\left( {{X_u} \times {X_v}} \right) \cdot {X_{uu}}}}{{\sqrt {EG - {F^2}} }} = \frac{{\left( {{X_u},{X_v},{X_{uu}}} \right)}}{{\sqrt {EG - {F^2}} }}$,

where $(\cdot, \cdot, \cdot)$ is the determinant of three vectors. Some authors (including Gauss himself and Darboux) use the notation

$\displaystyle \begin{gathered} D = \left( {{X_u},{X_v},{X_{uu}}} \right), \hfill \\ D' = \left( {{X_u},{X_v},{X_{uv}}} \right), \hfill \\ D'' = \left( {{X_u},{X_v},{X_{vv}}} \right), \hfill \\ \end{gathered}$

and we also have

$\displaystyle L = \frac{D}{{\sqrt {EG - {F^2}} }}, \quad M = \frac{{D'}}{{\sqrt {EG - {F^2}} }}, \quad N = \frac{{D''}}{{\sqrt {EG - {F^2}} }}$.

These expressions were used by Gauss to prove his famous Theorema Egregium.

Definition 2. Given a surface $X$, for any point $p = X(u, v)$ on $X$, letting

$\displaystyle L = {\rm N} \cdot X_{uu}, \quad M = {\rm N} \cdot X_{uv}, \quad N = {\rm N}\cdot X_{vv}$,

where $N$ is the unit normal at $p$, the quadratic form

$\displaystyle (x, y) \mapsto Lx^2+2Mxy+Ny^2$

is called the second fundamental form of $X$ at $p$. It is often denoted as $\mathrm I\!\mathrm I_p$ and in matrix form, we have

$\displaystyle {\mathrm I\!\mathrm I_p}\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} x & y\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} L & M\\ M & N\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x\\ y\\ \end{array} } \right)$.

For a curve $C$ on the surface $X$ (parameterized by arc length), the quantity $\kappa_N$ given by the formula

$\displaystyle \kappa_N = L(u')^2 + 2Mu'v' + N(v')^2$

is called the normal curvature of $C$ at $p$.

The second fundamental form was introduced by Gauss in 1827. Unlike the first fundamental form, the second fundamental form is not necessarily positive or definite.

Gaussian curvature. The Gaussian curvature of a surface is given by

$\displaystyle K = \frac{\det \mathrm I\!\mathrm I}{\det \mathrm I} = \frac{ LN-M^2}{EG-F^2 }$,

Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that $K$ is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.

R-G: The First Fundamental Form (Riemannian Metric)

Filed under: Riemannian geometry — Ngô Quốc Anh @ 16:42

Given a curve $C$ on a surface $X$ parametrized by two parameters $u, v$, we first compute the element of arc length of the curve $C$. For this, we need to compute the square norm of the tangent vector $\dot C (t)$. The square norm of the tangent vector $\dot C (t)$ to the curve $C$ at $p$ is

$\displaystyle \| \dot C \|^2 = (X_u \dot u + X_v \dot v) \cdot (X_u \dot u + X_v \dot v)$,

where $\cdot$ is the inner product in $\mathbb R^3$, and thus,

$\displaystyle \| \dot C \|^2 = (X_u\cdot X_u)\dot u^2 + 2(X_u\cdot X_v) \dot u\dot v +(X_v \cdot X_v) \dot v^2$.

Following common usage, we let

$\displaystyle E = X_u \cdot X_u, \quad F = X_u \cdot X_v, \quad G = X_v \cdot X_v$,

and

$\displaystyle \| \dot C \|^2 = E \dot u^ 2 + 2F\dot u \dot v+ G\dot v^2$.

Euler already obtained this formula in 1760. Thus, the map

$\displaystyle (x, y) \mapsto Ex^2 + 2Fxy + Gy^2$

is a quadratic form on $\mathbb R^2$, and since it is equal to $\| \dot C \|^2$, it is positive definite. This quadratric form plays a major role in the theory of surfaces, and deserves an official definition.

Definition. Given a surface $X$, for any point $p =X(u, v)$ on $X$, letting

$\displaystyle E = X_u \cdot X_u, \quad F = X_u \cdot X_v, \quad G = X_v \cdot X_v$,

$\displaystyle (x, y) \mapsto Ex^2 + 2Fxy + Gy^2$

is called the first fundamental form of $X$ at $p$. It is often denoted as $\mathrm I_p$, and in matrix form, we have

$\displaystyle {I_p}\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} x & y\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} E & F\\ F & G\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x\\ y\\ \end{array} } \right)$.

The symmetric bilinear form $\varphi_I$ associated with $I$ is an inner product on the tangent space at $p$, such that

$\displaystyle {\varphi_I}\left( ({x_1,y_1}),(x_2,y_2) \right) = \left( {\begin{array}{*{20}{c}} x_1 & y_1\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} E & F\\ F & G\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x_2\\ y_2\\ \end{array} } \right)$.

This inner product is also denoted as ${\left\langle {\left( {{x_1},{y_1}} \right),{\text{ }}\left( {{x_2},{y_2}} \right)} \right\rangle _p}$. The inner product $\varphi_I$ can be used to determine the angle of two curves passing through $p$, i.e., the angle $\theta$ of the tangent vectors to these two curves at $p$. We have

$\displaystyle \cos \theta = \frac{{\left\langle {\left( {{\dot u_1},{\dot v_1}} \right),{\text{ }}\left( {{\dot u_2},{\dot v_2}} \right)} \right\rangle }}{{\sqrt {I\left( {{\dot u_1},{\dot v_1}} \right)} \sqrt {I\left( {{\dot u_2},{\dot v_2}} \right)} }}$.

For example, the angle between the $u$-curve and the $v$-curve passing through $p$ (where $u$ or $v$ is constant) is given by

$\displaystyle\cos \theta= \frac{F}{{\sqrt {EG} }}$.

Thus, the $u$-curves and the $v$-curves are orthogonal iff $F(u, v) =0$ on $\Omega$.

Example 1. The helicoid is the surface defined over $\mathbb R \times \mathbb R$ such that

$\displaystyle x=u \cos v, y=u \sin v, z=v$.

This is the surface generated by a line parallel to the $xOy$ plane, touching the $z$ axis, and also touching an helix of axis $Oz$. It is easily verified that $(E, F,G) = (1, 0, u^2+ 1)$. The figure below shows a portion of helicoid corresponding to $0 \leq v \leq 2\pi$ and $-2\leq u \leq 2$.

Example 2. The catenoid is the surface of revolution defined over $\mathbb R \times \mathbb R$ such that

$\displaystyle x=\cosh u \cos v, y=\cosh u \sin v, z=u$.

It is the surface obtained by rotating a catenary around the $z$-axis. It is easily verified that $(E, F,G) = (\cosh^2 u, 0, \cosh^2 u)$. The figure below shows a portion of catenoid corresponding to $0 \leq v \leq 2\pi$ and $-2\leq u \leq 2$.

We will see how the first fundamental form relates to the curvature of curves on a surface.

R-G: Traceless Ricci tensor, Einstein tensor, and Schouten tensor

Filed under: Riemannian geometry — Ngô Quốc Anh @ 13:47

Traceless Ricci tensor

Traceless Ricci tensor is defined to be

$\displaystyle {B_{ij}} = {R_{ij}} - \frac{1}{n}S{g_{ij}}$.

Clearly, the trace of traceless Ricci tensor $B$ is nothing but

$\displaystyle g^{ij}B_{ij}$

which can be estimated as follows

$\displaystyle\begin{gathered}{g^{ij}}{B_{ij}} = {g^{ij}}\left( {{R_{ij}} - \frac{1}{n}S{g_{ij}}} \right) \hfill \\\qquad= {g^{ij}}{R_{ij}} - \frac{1}{n}{g^{ij}}S{g_{ij}} \hfill \\\qquad= {g^{ij}}{R_{ij}} - \frac{1}{n}\delta_{ij}S \hfill \\\qquad = 0. \hfill\end{gathered}$

This is why we call $B$ the traceless Ricci tensor.

Einstein tensor

The Einstein tensor $E$ is a rank-$2$ tensor defined over Riemannian manifolds. In index-free notation it is defined as

$\displaystyle E={\rm Ric}-\frac{1}{2}gR$.

In component form, the previous equation reads as

$\displaystyle E_{ij} = R_{ij} - {1\over2} g_{ij}R$.

The Einstein tensor is symmetric

$\displaystyle E_{ij} = E_{ji}$

and, like the stress-energy tensor, divergenceless

$\displaystyle E_{ij,j} = 0$.

The divergenceless property will be proved elsewhere later.

Schouten tensor

The Schouten tensor $S$ is defined as

$\displaystyle S={\rm Ric}-\frac{1}{2(n-1)}gR$.

In component form, the previous equation reads as

$\displaystyle E_{ij} = R_{ij} - {1 \over{2(n-1)}} g_{ij}R$.