# Ngô Quốc Anh

## December 21, 2009

### The QE – Department of Mathematics, Rutgers University

Filed under: Đề Thi — Ngô Quốc Anh @ 12:58

The Mathematics Ph.D. program at Rutgers includes two qualifying examinations, a written exam and an oral exam . The written exam is taken first and covers advanced calculus, elementary topology (metric spaces, compactness, and related topics), and the material of 501 (real analysis), 503 (complex analysis), and 551 (algebra). It is offered twice a year, near the beginning of each semester.

The syllabus represents a common core of material required of all Rutgers Ph.D.’s. In particular, the exam is designed with the goal that a pass on this exam shows a level of mathematical knowledge and ability appropriate for teaching the central undergraduate classes in mathematics.

Each student is required to take the exam by the beginning of the student’s second year; the program director may allow a student who has entered with less preparation than the norm to take the exam a specified number of semesters later.

Students who fail this exam may take it again during the semester following the one in which the exam was failed. Students who fail on the second attempt or who do not take the exams on schedule (as determined by the program director) will not be allowed to continue in the Ph.D. program.

## December 20, 2009

### R-G: Einstein vacuum field equations

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 13:43

This is a short note concerning the so called Einstein vacuum field equations ${\rm Eins}(g)_{\alpha\beta}=0$ where ${\rm Eins}(g)$ is nothing but the Einstein curvature tensor defined in this topic. I have not discussed either  Einstein field equations or Einstein vacuum field equations yet, however, we can adop the following equation

$\displaystyle {\rm Ric}_{\alpha\beta} - {1 \over 2}g_{\alpha\beta} R + g_{\alpha\beta}\Lambda = {8 \pi G \over c^4} T_{\alpha\beta}$

to be the Einstein field equations (EFE) where ${\rm Ric}_{\alpha\beta}$ is the Ricci curvature tensor, $R$ the scalar curvature, $g_{\alpha\beta}$ the metric tensor, $\Lambda$ is the cosmological constant, $G$ is the gravitational constant, $c$ the speed of light, and $T_{\alpha\beta}$ the stress-energy tensor.

One can write the EFE in a more compact form by defining the Einstein tensor

$\displaystyle {\rm Eins}_{\alpha\beta} = {\rm Ric}_{\alpha\beta} - {1 \over 2}R g_{\alpha\beta}$,

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

$\displaystyle {\rm Eins}_{\alpha\beta} = {8 \pi G \over c^4} T_{\alpha\beta}$,

where the cosmological term has been absorbed into the stress-energy tensor as dark energy.

If the energy-momentum tensor $T_{\alpha\beta}$ is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting $T_{\alpha\beta}=0$ in the full field equations, the vacuum equations can be written as

$\displaystyle {\rm Ric}_{\alpha\beta} = {1 \over 2} R \, g_{\alpha\beta}$.

Taking the trace of this (contracting with $g_{\alpha\beta}$) and using the fact that $g^{\alpha\beta} g_{\alpha\beta} = 4$, we get

$\displaystyle R = {1 \over 2} R4 = 2 R$,

and thus

$\displaystyle R = 0$.

Substituting back, we get an equivalent form of the vacuum field equations

$\displaystyle {\rm Ric}_{\alpha\beta} = 0$.

The above equation is frequently used in the literature, sometimes, it is called the Einstein vacuum field equations, for example, we refer the reader to introduction part of the following paper due to James Isenberg. In the case of nonzero cosmological constant, the equations are

$\displaystyle {\rm Ric}_{\alpha\beta} = {1 \over 2}R g_{\alpha\beta} - \Lambda g_{\alpha\beta}$

which gives

$\displaystyle R = 4 \Lambda$

yielding the equivalent form

$\displaystyle {\rm Ric}_{\alpha\beta} = \Lambda g_{\alpha\beta}$.

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, ${\rm Ric}_{\alpha\beta} = 0$, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

## December 17, 2009

### R-G: Defining function

Filed under: Riemannian geometry — Ngô Quốc Anh @ 15:58

Let $M$ be a smooth manifold of dimension $n$.

Definition. If $S \subset M$ is an embedded submanifold, a smooth map $\Psi : M \to N$ such that $S$ is a regular level set of $\Psi$ is called a defining map for $S$. In other words,

$\displaystyle S=\Psi^{-1}(c)$

for some point $c \in N$. In particular, if $N=\mathbb R^{n-p}$ (so that $\Psi$ is a real-valued or vector-valued function), it is usually called a defining function.

Example 1. The sphere $\mathbb S^n$ is an embeded submanifold of $\mathbb R^{n+1}$. The sphere is easily seen to be a regular level set of the function $f:\mathbb R^{n+1} \to \mathbb R$ given by $f(x)=|x|^2$ since $df=2 \sum_i x^i dx^i$ vanishes only at the origin.

Definition. More generally, if $U$ is an open subset of $M$ and $\Psi:U \to N$ is a smooth map such that $S \cap U$ is a regular level set of $\Psi$, then $\Psi$ is called a local defining map (or local defining function) for $S$.

Example 2. The smooth map $X:\mathbb R^2 \to \mathbb R^3$ given by

$\displaystyle X(\varphi, \theta)=\big( (2+\cos \varphi) \cos \theta, (2+\cos \varphi) \sin \theta, \sin \varphi\big)$

is an immersion of $\mathbb R^2$ into $\mathbb R^3$ whose image, denoted by $D$, is the doughnut-shaped shape surface obtained by revolving the circle $(y-2)^2+z^2=1$ around the $z$-axis, a point $(x,y,z)$ is in $D$ if and only if it satisfies $(r-2)^2+z^2=1$ where $r=\sqrt{x^2+y^2}$ is the distance from the $z$-axis. Thus $D$ is the zero set of the function $\Psi(x,y,z)=(r-2)^2+z^2-1$ which is smooth on $\mathbb R^3$ minus the $z$-axis. A straightforward computation shows that $lated d\Psi$ does not vanish on $D$, so $\Psi$ is a global defining function for $D$.

### Schwarz’s Lemma, Schwarz-Pick theorem, and some applications involving inequalities

Filed under: Các Bài Tập Nhỏ, Giải tích 7 (MA4247), Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 10:52

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk.

Schwarz’s Lemma: Let $D=\{z : |z|<1\}$ be the open unit disk in the complex plane $\mathbb C$. Let $f : D \to \overline D$ be a holomorphic function with $f(0)=0$. The Schwarz lemma states that under these circumstances $|f(z)| \leq |z|$ for all $z \in D$, and $|f'(0)| \leq 1$. Moreover, if the equality $|f(z)|=|z|$ holds for any $z \ne 0$, or $|f'(0)|=1$ then $f$ is a rotation, that is, $f(z)=az$ with $|a=1$.

This lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove; however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No similar result exists for real functions, of course. To prove the lemma, one applies the maximum modulus principle to the function $\frac{f(z)}{z}$.

Proof: Let $g(z)=\frac{f(z)}{z}$. The function $g(z)$ is holomorphic in $D$ (excluding $0$) since $f(0)=0$ and $f$ is holomorphic. Let $D_r$ be a closed disc within $D$ with radius $r$. By the maximum modulus principle,

$\displaystyle |g(z)| = \frac{|f(z)|}{|z|} \leq \frac{|f(z_r)|}{|z_r|} \le \frac{1}{r}$

for all $z$ in $D_r$ and all $z_r$ on the boundary of $D_r$. As $r$ approaches $1$ we get $|g(z)| \leq 1$. Moreover, if there exists a $z_0$ in $D$ such that $g(z_0)=1$. Then, applying the maximum modulus principle to $g$, we obtain that $g$ is constant, hence $f(z)=kz$, where $k$ is constant and $|k|=1$. This is also the case if $|f'(0)|=1$.

A variant of the Schwarz lemma can be stated that is invariant under analytic automorphisms on the unit disk, i.e. bijective holomorphic mappings of the unit disc to itself. This variant is known as the Schwarz-Pick theorem (after Georg Pick):

Schwarz-Pick theorem: Let $f : D \to D$ be holomorphic. Then, for all $z_1, z_2 \in D$,

$\displaystyle\left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right| \le \frac{\left|z_1-z_2\right|}{\left|1-\overline{z_1}z_2\right|}$

and, for all $z \in D$

$\displaystyle\frac{\left|f'(z)\right|}{1-\left|f(z)\right|^2} \le \frac{1}{1-\left|z\right|^2}$.

## December 15, 2009

### R-G: Codazzi equations in classical differential geometry

Filed under: Riemannian geometry — Ngô Quốc Anh @ 21:49

In classical differential geometry of surfaces, the Codazzi-Mainardi equations are expressed via the second fundamental form $(L, M, N)$

$\displaystyle\begin{gathered}{L_v} - {M_u} = L\Gamma _{12}^1 + M(\Gamma _{12}^2 - \Gamma _{11}^1) - N\Gamma _{11}^2, \hfill \\{M_v} - {N_u} = L\Gamma _{22}^1 + M(\Gamma _{22}^2 - \Gamma _{12}^1) - N\Gamma _{12}^2. \hfill \\\end{gathered}$

Consider a parametric surface in Euclidean space,

$\displaystyle\mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v))$

where the three component functions depend smoothly on ordered pairs $(u,v)$ in some open domain $U$ in the $uv$-plane. Assume that this surface is regular, meaning that the vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ are linearly independent. Complete this to a basis $\{\mathbf{r}_u,\mathbf{r}_v,\mathbf{n}\}$, by selecting a unit vector $\mathbf{n}$ normal to the surface. The unit vector $\mathbf{n}$ is nothing but

$\displaystyle\mathbf{n}=\frac{\mathbf{r}_u \times \mathbf{r}_v}{\|\mathbf{r}_u \times \mathbf{r}_v\|}$.

It is possible to express the second partial derivatives of $\mathbf{r}$ using the Christoffel symbols and the second fundamental form.

$\displaystyle\begin{gathered}{{\mathbf{r}}_{uu}} = \Gamma _{11}^1{{\mathbf{r}}_u} + \Gamma _{11}^2{{\mathbf{r}}_v} + L{\mathbf{n}}, \hfill \\{{\mathbf{r}}_{uv}} = \Gamma _{12}^1{{\mathbf{r}}_u} + \Gamma _{12}^2{{\mathbf{r}}_v} + M{\mathbf{n}}, \hfill \\{{\mathbf{r}}_{vv}} = \Gamma _{22}^1{{\mathbf{r}}_u} + \Gamma _{22}^2{{\mathbf{r}}_v} + N{\mathbf{n}}. \hfill \\ \end{gathered}$

Clairaut’s theorem states that partial derivatives commute

$\displaystyle\left(\bold{r}_{uu}\right)_v=\left(\bold{r}_{uv}\right)_u$

If we differentiate $\mathbf{r}_{uu}$ with respect to $v$ and $\mathbf{r}_{uv}$ with respect to $u$, we get

$\displaystyle \begin{gathered}{\left( {\Gamma _{11}^1} \right)_v}{{\mathbf{r}}_u} + \Gamma _{11}^1{{\mathbf{r}}_{uv}} + {\left( {\Gamma _{11}^2} \right)_v}{{\mathbf{r}}_v} + \Gamma _{11}^2{{\mathbf{r}}_{vv}} + {L_v}{\mathbf{n}} + L{{\mathbf{n}}_v} \hfill \\ \qquad = {\left( {\Gamma _{12}^1} \right)_u}{{\mathbf{r}}_u} + \Gamma _{12}^1{{\mathbf{r}}_{uu}} + {\left( {\Gamma _{12}^2} \right)_u}{{\mathbf{r}}_v} + \Gamma _{12}^2{{\mathbf{r}}_{uv}} + {M_u}{\mathbf{n}} + M{{\mathbf{n}}_u} \hfill \\ \end{gathered}$

Now substitute the above expressions for the second derivatives and equate the coefficients of $\mathbf{n}$

$\displaystyle M \Gamma_{11}^1 + N \Gamma_{11}^2 + L_v = L \Gamma_{12}^1 + M \Gamma_{12}^2 + M_u$

Rearranging this equation gives the first Codazzi equation. The second equation may be derived similarly.

## December 13, 2009

### R-G: The Einstein Tensor, 2

Filed under: Riemannian geometry — Ngô Quốc Anh @ 20:53

In this topic, we shall give a natural way to construct the Einstein tensor. Let us apply the Ricci contraction to the Bianchi identities

$\displaystyle g^{\alpha \mu} \big( R_{\alpha \beta \mu;\lambda} + R_{\alpha \lambda\beta ;\mu} + R_{\alpha \mu\lambda;\beta} \big)= 0$.

Since $g_{\alpha \beta; \mu}=0$ and $g^{\alpha \beta; \mu}=0$, we can take $g_{\alpha \mu}$ in and out of covariant derivatives at will. We get

$\displaystyle g^{\alpha \mu} \big( R^\mu_{\beta \mu \nu ;\lambda} + R^\mu_{\beta\lambda\mu;\nu} + R^\mu_{\beta\nu\lambda;\mu} \big)= 0$.

Using the antisymmetry on the indices $\mu$ and $\lambda$ we get

$\displaystyle R^\mu_{\beta \mu \nu ;\lambda} - R^\mu_{\beta\mu\lambda;\nu} + R^\mu_{\beta\nu\lambda;\mu} = 0$

so

$\displaystyle R_{\beta\nu ;\lambda} - R_{\beta\lambda;\nu} + R^\mu_{\beta\nu\lambda;\mu} = 0$.

These equations are called the contracted Bianchi identities. Let us now contract a second time on the indices $\beta$ and $\nu$

$\displaystyle g^{\beta\nu} \big( R_{\beta\nu ;\lambda} - R_{\beta\lambda;\nu} + R^\mu_{\beta\nu\lambda;\mu}\big) = 0$.

This gives

$\displaystyle R^\nu_{\nu ;\lambda} - R^\nu_{\lambda;\nu} + R^{\mu\nu}_{\nu\lambda;\mu} = 0$

so

$\displaystyle R_{;\lambda} - 2R^\mu_{\lambda;\mu} = 0$

or

$\displaystyle 2R^\mu_{\lambda;\mu}-R_{;\lambda} = 0$.

Since $R_{;\lambda} =g^\mu_\lambda R_{;\mu}$, we get

$\displaystyle \big(R^\mu_{\lambda}-\frac{1}{2}g^\mu_\lambda R\big)_{;\mu }= 0$.

Raising the index $\lambda$ with $g^{\lambda \nu}$ we get

$\displaystyle \big(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\big)_{;\mu }= 0$.

Defining

$\displaystyle E^{\mu\nu}= R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R$

we get

$\displaystyle {E^{\mu\nu}}_{;\mu }=0$.

Theorem. The tensor $E^{\mu\nu}$ is divergence free in the sense that

$\displaystyle {E^{\mu\nu}}_{;\mu }=0$.

The tensor $E^{\mu\nu}$ is constructed only from the Riemann tensor and the metric, and it is automatically divergence free as an identity. It is called the Einstein tensor, since its importance for gravity was first understood by Einstein. Some authors denote the Einstein tensor by $E_{\mu\nu}$. We will see later that Einstein’s field equations for General Relativity in the vacuum case are

$\displaystyle E^{\mu\nu}=\frac{8 \pi G}{c^4}T^{\mu\nu}$

where $T^{\mu\nu}$ is the stress- energy tensor. The Bianchi identities then imply

$\displaystyle {T^{\mu\nu}}_{;\mu}=0$

which is the conservation of energy and momentum.

## 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

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

## December 10, 2009

### R-G: Curvature

Filed under: Riemannian geometry — Ngô Quốc Anh @ 13:17

In this topic we discuss Gauss’ work to explain the concept of curvature.

The curvature of a curve in a plane is determined by how fast its unit normal vector $n$ (or the tangent vector for that matter) changes as we move along the curve. A measure of curvature is the ratio of the small change $|dn|$ in the unit normal vector to the distance $ds$ moved by the point on the curve.

A straight line has zero curvature because the unit normals are all parallel and do not change. A circle of radius $R$ has curvature $\frac{1}{R}$ because for the distance $ds$ that the point $P$ moves, the unit normal vector changes by an angle $\frac{ds}{R}$ so $|dn| = \frac{ds}{R}$.

Gauss defined the curvature of a surface analogously.

• Curvature of plane curves

For a plane curve $C$, the mathematical definition of curvature uses a parametric representation of $C$ with respect to the arc length parametrization. It can be computed given any regular parametrization by a more complicated formula given below.

Curvature: Let $\gamma (s)$ be a regular parametric curve, where $s$ is the arc length, or natural parameter. This determines the unit tangent vector $T$, the unit normal vector $N(s)$, the curvature $\kappa (s)$, the oriented or signed curvature $k(s)$, and the radius of curvature $R(s)$ at each point:

$\displaystyle T(s)=\gamma'(s),\quad T'(s)=\kappa (s)N(s),\quad \kappa (s) = \|\gamma''(s)\| = \left|k(s)\right|, \quad R(s)=\frac{1}{\kappa(s)}$.

Local expressions. For a plane curve given parametrically as $c(t) = (x(t),y(t))$, the curvature is

$\displaystyle\kappa = \frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}$,

and the signed curvature $k$ is

$\displaystyle k = \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}$.

For the less general case of a plane curve given explicitly as $y = f(x)$ the curvature is

$\displaystyle\kappa = \frac{|y''|}{(1+y'^2)^{3/2}}$.

Slightly abusing notation, the signed curvature may also be written in this way as

$\displaystyle k=\frac{y''}{(1+y'^2)^{3/2}}$

with the understanding that the curve is traversed in the direction of increasing $x$.

• Curvature of space curves

For a parametrically defined space curve as $c(t) = (x(t),y(t),z(t))$, its curvature is:

$\displaystyle F[x,y,z]=\frac{\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}$.

Given a function $r(t)$ with values in $R^3$, the curvature at a given value of t is

$\displaystyle \kappa = \frac{|\dot{r} \times \ddot{r}|}{|\dot{r}|^3}$

where $\dot{r}$ and $\ddot{r}$ correspond to the first and second derivatives of $r(t)$, respectively, and $\times$ is the cross (vector) product. (Note that this formula is the vector notation of $F[x,y,z]$ above.)

• Curves on surfaces

When a one dimensional curve lies on a two dimensional surface embedded in three dimensions $R^3$, further measures of curvature are available, which take the surface’s unit-normal vector, u into account. These are the normal curvature, geodesic curvature and geodesic torsion.

Any non-singular curve on a smooth surface will have its tangent vector $T$ lying in the tangent plane of the surface orthogonal to the normal vector. The normal curvature, $k_n$, is the curvature of the curve projected onto the plane containing the curve’s tangent $T$ and the surface normal $u$; the geodesic curvature, $k_g$, is the curvature of the curve projected onto the surface’s tangent plane; and the geodesic torsion (or relative torsion), $\tau_r$, measures the rate of change of the surface normal around the curve’s tangent.

Let the curve be a unit speed curve and let $t=u \times T$ so that $T, u, t$ form an orthonormal basis: the Darboux frame. The above quantities are related by

$\displaystyle\begin{pmatrix} T'\\ t'\\ u' \end{pmatrix} = \begin{pmatrix} 0&\kappa_g&k_n\\ -\kappa_g&0&\tau_r\\ -\kappa_n&-\tau_r&0 \end{pmatrix} \begin{pmatrix} T\\ t\\ u \end{pmatrix}$.

Principal curvature: All curves with the same tangent vector will have the same normal curvature, which is the same as the curvature of the curve obtained by intersecting the surface with the plane containing $T$ and $u$.

Taking all possible tangent vectors then the maximum and minimum values of the normal curvature at a point are called the principal curvatures, $\kappa_1$ and $\kappa_2$, and the directions of the corresponding tangent vectors are called principal directions.

• Curvature of surfaces (two dimensions)

Gaussian curvature: In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. Here we adopt the convention that a curvature is taken to be positive if the curve turns in the same direction as the surface’s chosen normal, otherwise negative.

The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures. It has the dimension of 1/length2 and is positive for spheres, negative for one-sheet hyperboloids and zero for planes. It determines whether a surface is locally convex (when it is positive) or locally saddle (when it is negative).

Symbolically, the Gaussian curvature $K$ is defined as

$\displaystyle K= \kappa_1 \kappa_2$

where $\kappa_1$ and $\kappa_2$ are the principal curvatures. It is also given by

$\displaystyle K= \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)e_1, e_2\rangle}{\det g}$,

where $\nabla_i = \nabla_{e_i}$ is the covariant derivative and $g$ is the metric tensor. At a point $p$ on a regular surface in $\mathbb R^3$, the Gaussian curvature is also given by

$\displaystyle K(p) = \det(S(p))$,

where $S$ is the shape operator.

The above definition of Gaussian curvature is extrinsic in that it uses the surface’s embedding in $\mathbb R^3$ normal vectors, external planes etc. Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; the cylinder has extrinsic curvature, but no intrinsic curvature.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss‘ celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point $P$ is the following: imagine an ant which is tied to $P$ with a short thread of length $r$. She runs around $P$ while the thread is completely stretched and measures the length $C(r)$ of one complete trip around $P$. If the surface were flat, she would find $C(r)=2\pi r$. On curved surfaces, the formula for $C(r)$ will be different, and the Gaussian curvature $K$ at the point $P$ can be computed by the Bertrand–Diquet–Puiseux theorem as

$\displaystyle K = \mathop {\lim }\limits_{r \to 0} (2\pi r - {\text{C}}(r))\frac{3}{{\pi {r^3}}}$.

The integral of the Gaussian curvature over the whole surface is closely related to the surface’s Euler characteristic; see the Gauss-Bonnet theorem. The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss-Bonnet theorem is Descartes’ theorem on total angular defect. Because curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.

Mean curvature: The mean curvature is equal to the sum of the principal curvatures, $\kappa_1+\kappa_2$, over 2, that is

$\displaystyle H=\frac{\kappa_1+\kappa_2}{2}$.

It has the dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface such as a soap film, has mean curvature zero and a soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

• Curvature of surfaces (three dimensions)

By extension of the former argument, a space of three or more dimensions can be intrinsically curved; the full mathematical description is described at curvature of Riemannian manifolds (Riemann curvature tensor, Sectional curvature, Ricci curvature, Scalar curvature, Einstein curvature tensor, Weyl curvature tensor).

Older Posts »