MTH3008 Weekly Problems 9

Original Documents: mth3008 weekly problem sheet 9.pdf / mth3008 weekly problem sheet 9 handwritten solutions.pdf / mth3008 weekly problem sheet 9 solutions.pdf

Vibes: More Christoffel-symbol marathon, now with a final payoff - the Riemann-Christoffel Tensor for two surfaces. Every problem reduces to “find the diagonal metric, read off $\partial g_{ii}/\partial x^k$, drop into 8.1’s four cases.” The curvature calculations in 9.4 and 9.6 follow the unit-sphere template from lecture 18.

Used Techniques:

• Christoffel Symbols formula restricted to orthogonal coordinates (8.1 cases).
• Raising with $g^{ii}=1/g_{ii}$ (diagonal).
• Riemann-Christoffel Tensor formula $R_{ijk}^r={\partial }_j{\Gamma }_{ki}^r-{\partial }_k{\Gamma }_{ij}^r+{\Gamma }_{ik}^p{\Gamma }_{pj}^r-{\Gamma }_{ij}^p{\Gamma }_{pk}^r$.
• In 2D, only $R_{212}^1$ is independent (all others either vanish or are related by antisymmetry).

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9.1. Christoffel Symbols for Exponential Coordinate Scaling

Question

For $\mathbf{r}=e^u{\mathbf{i}}_1+e^v{\mathbf{i}}_2+w{\mathbf{i}}_3$ with $(u,v,w)$ coords, find all ${\Gamma }_{ijk}$ and ${\Gamma }_{jk}^i$.

Basis. ${\mathbf{e}}_1=e^u{\mathbf{i}}_1$, ${\mathbf{e}}_2=e^v{\mathbf{i}}_2$, ${\mathbf{e}}_3={\mathbf{i}}_3$. Orthogonal.

Metric. $g_{11}=e^{2u}$, $g_{22}=e^{2v}$, $g_{33}=1$. Inverse: $g^{11}=e^{-2u}$, $g^{22}=e^{-2v}$, $g^{33}=1$.

Nonzero metric derivatives. $\partial g_{11}/\partial u=2e^{2u}$, $\partial g_{22}/\partial v=2e^{2v}$; every other derivative is zero.

First-kind (8.1 case $i=j=k$):

$$\boxed{{\Gamma }_{111}=e^{2u},{\Gamma }_{222}=e^{2v}.}$$

(No ${\Gamma }_{iik}$ or ${\Gamma }_{ijj}$ survive because every cross-derivative $\partial g_{ii}/\partial x^{k\ne i}$ vanishes.)

Second-kind:

$$\boxed{{\Gamma }_{11}^1=e^{-2u}\cdot e^{2u}=1,{\Gamma }_{22}^2=e^{-2v}\cdot e^{2v}=1.}$$

All others zero.

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9.2. Christoffel Symbols for a Mixed Exponential-Trigonometric-Logarithmic Map

Question

For $\mathbf{r}(u,v,w)=(e^u,\sin v,\ln (1+w))$, find all Christoffel symbols.

Basis. ${\mathbf{e}}_1=e^u{\mathbf{i}}_1$, ${\mathbf{e}}_2=\cos v{\mathbf{i}}_2$, ${\mathbf{e}}_3=\frac{1}{1+w}{\mathbf{i}}_3$. Orthogonal.

Metric. $g_{11}=e^{2u}$, $g_{22}={\cos }^{2}v$, $g_{33}=(1+w{)}^{-2}$. Inverse: $g^{11}=e^{-2u}$, $g^{22}={\sec }^{2}v$, $g^{33}=(1+w{)}^2$.

Nonzero metric derivatives. $\partial g_{11}/\partial u=2e^{2u}$; $\partial g_{22}/\partial v=-2\sin v\cos v$; $\partial g_{33}/\partial w=-2(1+w{)}^{-3}$.

First-kind - only diagonal cases survive:

$$\boxed{{\Gamma }_{111}=e^{2u},{\Gamma }_{222}=-\sin v\cos v,{\Gamma }_{333}=-\frac{1}{(1+w{)}^3}.}$$

Second-kind:

$$\boxed{{\Gamma }_{11}^1=1,{\Gamma }_{22}^2=-\tan v,{\Gamma }_{33}^3=-\frac{1}{1+w}.}$$

All others zero.

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9.3. Orthogonal Coordinates with Given Scale Factors

Question

Orthogonal system $(a,b,c)$ with $h_1=e^{a+b}$, $h_2=\sqrt{1+a^2}$, $h_3=ab$. Find all Christoffel symbols.

Metric. $g_{ii}=h_i^2$:

$$g_{11}=e^{2(a+b)},g_{22}=1+a^2,g_{33}=a^2{b}^2.$$

$g^{11}=e^{-2(a+b)}$, $g^{22}=1/(1+a^2)$, $g^{33}=1/(a^2{b}^2)$.

Nonzero derivatives.

• $\partial g_{11}/\partial a=\partial g_{11}/\partial b=2e^{2(a+b)}$
• $\partial g_{22}/\partial a=2a$
• $\partial g_{33}/\partial a=2a{b}^2$, $\partial g_{33}/\partial b=2a^{2}b$

First-kind (8.1 cases), nonzero only:

$${\Gamma }_{111}=e^{2(a+b)},{\Gamma }_{112}={\Gamma }_{121}=e^{2(a+b)},{\Gamma }_{122}=-a,{\Gamma }_{133}=-a{b}^2,$$ $${\Gamma }_{211}=-e^{2(a+b)},{\Gamma }_{212}={\Gamma }_{221}=a,{\Gamma }_{233}=-a^{2}b,$$ $${\Gamma }_{313}={\Gamma }_{331}=a{b}^2,{\Gamma }_{323}={\Gamma }_{332}=a^{2}b.$$

Second-kind via ${\Gamma }_{jk}^i={\Gamma }_{ijk}/g_{ii}$ (no sum, diagonal metric):

$$\boxed{{\Gamma }_{11}^1=1,{\Gamma }_{12}^1={\Gamma }_{21}^1=1,{\Gamma }_{22}^1=-a{e}^{-2(a+b)},{\Gamma }_{33}^1=-a{b}^2{e}^{-2(a+b)},}$$ $$\boxed{{\Gamma }_{11}^2=-\frac{e^{2(a+b)}}{1+a^2},{\Gamma }_{12}^2={\Gamma }_{21}^2=\frac{a}{1+a^2},{\Gamma }_{33}^2=-\frac{a^{2}b}{1+a^2},}$$ $$\boxed{{\Gamma }_{13}^3={\Gamma }_{31}^3=\frac{1}{a},{\Gamma }_{23}^3={\Gamma }_{32}^3=\frac{1}{b}.}$$

All others zero.

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9.4. Christoffel Symbols and Curvature of the Unit Sphere

Question

Sphere parametrised by $\mathbf{r}=(\sin a\cos b,\sin a\sin b,\cos a)$, $0<a<\pi$, $0<b<2\pi$. (1) Compute ${\Gamma }_{jk}^i$. (2) Show the surface has non-zero curvature.

1) Christoffel symbols. As a 2D surface (only $a,b$ matter).

${\mathbf{e}}_1=(\cos a\cos b,\cos a\sin b,-\sin a)$, ${\mathbf{e}}_2=(-\sin a\sin b,\sin a\cos b,0)$.

Metric: $g_{11}=1$, $g_{22}={\sin }^{2}a$, $g_{12}=0$. Inverse: $g^{11}=1$, $g^{22}={\csc }^{2}a$.

Nonzero derivative: $\partial g_{22}/\partial a=2\sin a\cos a$.

First-kind: ${\Gamma }_{122}=-\sin a\cos a$, ${\Gamma }_{212}={\Gamma }_{221}=\sin a\cos a$.

Second-kind:

$$\boxed{{\Gamma }_{22}^1=-\sin a\cos a,{\Gamma }_{12}^2={\Gamma }_{21}^2=\cot a.}$$

All others zero.

2) Riemann-Christoffel tensor. Compute the key component $R_{212}^1={\partial }_1{\Gamma }_{22}^1-{\partial }_2{\Gamma }_{21}^1+{\Gamma }_{22}^p{\Gamma }_{p1}^1-{\Gamma }_{21}^p{\Gamma }_{p2}^1$:

• $\partial {\Gamma }_{22}^1/\partial a=-{\cos }^{2}a+{\sin }^{2}a=-\cos 2a$.
• $\partial {\Gamma }_{21}^1/\partial b=0$ (since ${\Gamma }_{21}^1=0$).
• ${\Gamma }_{22}^p{\Gamma }_{p1}^1$: both $p=1$ and $p=2$ terms vanish (${\Gamma }_{11}^1=0$ and ${\Gamma }_{22}^2=0$).
• ${\Gamma }_{21}^p{\Gamma }_{p2}^1$: $p=2$ contributes ${\Gamma }_{21}^2{\Gamma }_{22}^1=\cot a\cdot (-\sin a\cos a)=-{\cos }^{2}a$.

Therefore:

$$R_{212}^1=-\cos 2a-0+0-(-{\cos }^{2}a)=-\cos 2a+{\cos }^{2}a=\boxed{{\sin }^{2}a}.$$

Since ${\sin }^{2}a\ne 0$ on the open interval $0<a<\pi$, the sphere has non-zero curvature - it is not Euclidean. ✓

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9.5. Surface of Revolution with Unit-Speed Meridians

Question

$S$ parametrised by $\mathbf{r}(x,y)=(f(x)\cos y,f(x)\sin y,g(x))$, with $(f^{\prime }{)}^2+(g^{\prime }{)}^2=1$. Find all Christoffel symbols.

Basis.

$${\mathbf{e}}_1=(f^{\prime }\cos y,f^{\prime }\sin y,g^{\prime }),{\mathbf{e}}_2=(-f\sin y,f\cos y,0).$$

Metric. Using $(f^{\prime }{)}^2+(g^{\prime }{)}^2=1$:

$$g_{11}=(f^{\prime }{)}^2+(g^{\prime }{)}^2=1,g_{22}=f^2,g_{12}=0.$$

Orthogonal; $g^{11}=1$, $g^{22}=1/f^2$.

Nonzero derivative. $\partial g_{22}/\partial x=2f{f}^{\prime }$.

First-kind: ${\Gamma }_{122}=-f{f}^{\prime }$, ${\Gamma }_{212}={\Gamma }_{221}=f{f}^{\prime }$.

Second-kind:

$$\boxed{{\Gamma }_{22}^1=-f{f}^{\prime },{\Gamma }_{12}^2={\Gamma }_{21}^2=\frac{f^{\prime }}{f}.}$$

All others zero.

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9.6. Curvature of the Surface of Revolution

Question

Using the Riemann-Christoffel tensor, determine whether the surface $S$ of problem 9.5 has zero curvature (assuming $f\ne 0$).

Apply the formula with $r=1,i=2,j=1,k=2$:

$$R_{212}^1=\frac{\partial {\Gamma }_{22}^1}{\partial x}-\frac{\partial {\Gamma }_{21}^1}{\partial y}+{\Gamma }_{22}^p{\Gamma }_{p1}^1-{\Gamma }_{21}^p{\Gamma }_{p2}^1.$$

• $\partial {\Gamma }_{22}^1/\partial x=-(f^{\prime }{f}^{\prime }+f{f}^{\prime \prime })=-(f^{\prime }{)}^2-f{f}^{\prime \prime }$.
• $\partial {\Gamma }_{21}^1/\partial y=0$.
• ${\Gamma }_{22}^p{\Gamma }_{p1}^1$: all contributions vanish (${\Gamma }_{11}^1={\Gamma }_{21}^1={\Gamma }_{22}^2=0$).
• ${\Gamma }_{21}^p{\Gamma }_{p2}^1$: only $p=2$ contributes, ${\Gamma }_{21}^2{\Gamma }_{22}^1=(f^{\prime }/f)(-f{f}^{\prime })=-(f^{\prime }{)}^2$.

Therefore:

$$R_{212}^1=(-(f^{\prime }{)}^2-f{f}^{\prime \prime })-0+0-(-(f^{\prime }{)}^2)=\boxed{-f{f}^{\prime \prime }}.$$

Conclusion. Since $f\ne 0$, the surface is Euclidean (flat) iff $f^{\prime \prime }=0$, i.e. iff $f$ is linear in $x$. Generically $f^{\prime \prime }\ne 0$, so the surface has non-zero curvature.

When is a surface of revolution flat?

$f^{\prime \prime }=0$ means $f(x)=Ax+B$. Combined with $(f^{\prime }{)}^2+(g^{\prime }{)}^2=1$, this constraints $g^{\prime }=\pm \sqrt{1-A^2}$, a constant. Two special cases:

• $A=0$: $f$ constant ⇒ cylinder (radius $B$).
• $A\ne 0$: cone (straight slanted meridian).

Both are developable (flat) surfaces - consistent with zero curvature. Everything else (sphere, torus, paraboloid, …) has curvature.