Chapter 5

5-1. If \(M\) is a \(k\)-dimensional manifold-with-boundary, prove that \(\partial M\) is a \((k-1)\)-dimensional manifold and \(M-\partial M\) is a \(k\)-dimensional manifold.

5-3. (a) Let \(A \subset \mathbf{R}^{n}\) be an open set such that boundary \(A\) is an \((n-1)\) dimensional manifold. Show that \(N=A \cup\) boundary \(A\) is an \(n\)-dimensional manifold-with-boundary. (It is well to bear in mind the following example: if \(A=\left\\{x \in \mathbf{R}^{n}:|x|<1\right.\) or \(\left.1<|x|<2\right\\}\) then \(N=A \cup\) boundary \(A\) is a manifold-with- boundary, but \(\partial N \neq\) boundary \(A .)\) (b) Prove a similar assertion for an open subset of an \(n\)-dimensional manifold.

5-6. If \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}\), the graph of \(f\) is \(\\{(x, y): y=f(x)\\}\). Show that the graph of \(f\) is an \(n\)-dimensional manifold if and only if \(f\) is differentiable.

5-10. Suppose \(e\) is a collection of coordinate systems for \(M\) such that (1) For each \(x \in M\) there is \(f \in \mathcal{e}\) which is a coordinate system around \(x\); (2) if \(f, g \in e\), then det \(\left(f^{-1} \circ g\right)^{\prime}>0\). Show that there is a unique orientation of \(M\) such that \(f\) is orientation-preserving if \(f \in e\)

5-12. (a) If \(F\) is a differentiable vector field on \(M \subset \mathbf{R}^{n}\), show that there is an open set \(A \supset M\) and a differentiable vector field \(\tilde{F}\) on \(A\) with \(\tilde{F}(x)=F(x)\) for \(x \in M .\) Hint: Io this locally and use partitions of unity. (b) If \(M\) is closed, show that we can choose \(A=\mathbf{R}^{n}\).

5-14. If \(M \subset \mathbf{R}^{n}\) is an orientable \((n-1)\)-dimensional manifold, show that there is an open set \(A \subset \mathbf{R}^{n}\) and a differentiable \(g: A \rightarrow \mathbf{R}^{1}\) so that \(M=g^{-1}(0)\) and \(g^{\prime}(x)\) has rank 1 for \(x \in M .\) Hint: Problem 5-4 does this locally. Use the orientation to choose consistent local solutions and use partitions of unity.

5-15. Lel \(M\) be an \((n-1)\)-dimensional manifold in \(\mathbf{R}^{n} .\) Let \(M(\varepsilon)\) be the set of end points of normal vectors (in both directions) of length \(\varepsilon\) and suppose \(\varepsilon\) is small enough so that \(M(\varepsilon)\) is also an \((n-1)\)-dimensional manifold. Show that \(M(\varepsilon)\) is orientable (even if \(M\) is not). What is \(M(\varepsilon)\) if \(M\) is the Möbius strip?

5-16. Let \(g: A \rightarrow \mathbf{R}^{p}\) be as in Theorem 5-1. If \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}\) is differentiable and the maximum (or minimum) of \(f\) on \(g^{-1}(0)\) occurs at \(a\), show that there are \(\lambda_{1}, \ldots, \lambda_{p} \in \mathbf{R}\), such that (1) \(D_{j} f(a)=\sum_{i=1}^{n} \lambda_{i} D_{j} g^{i}(a) \quad j=1, \ldots, n\) Hint: This equation can be written \(d f(a)=\sum_{i=1}^{n} \lambda_{i} d g^{i}(a)\) and is obvious if \(g(x)=\left(x^{n-p+1}, \ldots, x^{n}\right)\) The maximum of \(f\) on \(g^{-1}(0)\) is sometimes called the maximum of \(f\) subject to the constraints \(g^{i}=0 .\) One can attempt to find \(a\) by solving the system of equations (1). In particular, if \(g: A \rightarrow \mathbf{R}\), we must solve \(n+1\) equations $$ \begin{aligned} D_{j} f(a) &=\lambda D_{j} g(a) \\ g(a) &=0 \end{aligned} $$ in \(n+1\) unknowns \(a^{1}, \ldots, a^{n}, \lambda\), which is often very simple if we leave the equation \(g(a)=0\) for last. This is Lagrange's method, and the useful but irrelevant \(\lambda\) is called a Lagrangian multiplier. The following problem gives a nice theoretical use for Lagrangian multipliers.

5-17. (a) Let \(T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}\) be self-adjoint with matrix \(A-\left(a_{i j}\right)\), so that \(a_{i j}=a_{j i} . \quad\) If \(f(x)=\langle T x, x\rangle=\Sigma a_{i j} x^{i} x^{j}\), show that \(D_{k} f(x)=\) \(2 \sum_{j-1}^{n} a_{k j} x^{j} .\) By considering the maximum of \(\langle T x, x\rangle\) on \(S^{n-1}\) show that there is \(x \in S^{n-1}\) and \(\lambda \in \mathbf{R}\) with \(T x=\lambda x\). (b) If \(V=\left\\{y \in \mathbf{R}^{n}:\langle x, y\rangle=0\right\\}\), show that \(T(V) \subset V\) and \(T: V \rightarrow V\) is self-adjoint. (c) Show that \(T\) has a basis of eigenvectors.

5-20. If \(\omega\) is a \((k-1)\)-form on a compact \(k\)-dimensional manifold \(M\), prove that \(\int M d \omega=0 .\) Give a counterexample if \(M\) is not compact.

5-22. If \(M_{1} \subset \mathbf{R}^{n}\) is an \(n\)-dimensional manifold-with- boundary and \(M_{2} \subset M_{1}-\partial M_{1}\) is an \(n\)-dimensional manifold-with-boundary, and \(M_{1}, M_{2}\) are compact, prove that $$ \int_{\partial M_{1}} \omega=\int_{\lambda M_{2}} \omega $$ where \(\omega\) is an \((n-1)\)-form on \(M_{1}\), and \(\partial M_{1}\) and \(\partial M_{2}\) have the orientations induced by the usual orientations of \(M_{1}\) and \(M_{2}\). Hint: Find a manifold-with-boundary \(M\) such that \(\partial M=\partial M_{1} \cup \partial M_{2}\) and such that the induced orientation on \(\partial M\) agrees with that for \(\partial M_{1}\) on \(\partial M_{1}\) and is the negative of that for \(\partial M_{2}\) on \(\partial M_{2}\)

5-23. If \(M\) is an oriented one-dimensional manifold in \(\mathbf{R}^{n}\) and \(c:[0,1] \rightarrow M\) is orientation-preserving, show that $$ \int_{[0,1]} c^{*}(d s)=\int_{[0,1]} \sqrt{\left[\left(c^{1}\right)^{\prime}\right]^{2}+\cdots+\left[\left(c^{n}\right)^{\prime}\right]^{2}} $$

5-24. If \(M\) is an \(n\)-dimensional manifold in \(\mathbf{R}^{n}\), with the usual orientation, show that \(d V=d x^{1} \wedge \ldots \wedge d x^{n}\), so that the volume of \(M\), as defined in this section, is the volume as defined in Chapter 3 . (Note that this depends on the numerical factor in the definition of \(\left.\omega \wedge \eta_{1}\right)\)

5-27. If \(T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}\) is a norm preserving linear transformation and \(M\) is a \(k\)-dimensional manifold in \(\mathbf{R}^{n}\), show that \(M\) has the same volume as \(T(M)\).

5-28. (a) If \(M\) is a \(k\)-dimensional manifold, show that an absolute \(k\)-tensor \(|d V|\) can be defined, even if \(M\) is not orientable, so that the volume of \(M\) can be defined as \(\int_{M}|d V|\) (b) If \(c:[0,2 \pi] \times(-1,1) \rightarrow \mathbf{R}^{3}\) is defined by \(c(u, v)=\) $$ (2 \cos u+v \sin (u / 2) \cos u, 2 \sin u+v \sin (u / 2) \sin u, v \cos u / 2) $$ show that \(c([0,2 \pi] \times(-1,1))\) is a Möbius strip and find its area.

5-29. If there is a nowhere-zero \(k\)-form on a \(k\)-dimensional manifold \(M\), show that \(M\) is orientable.

5-30. (a) If \(f:[0,1] \rightarrow \mathbf{R}\) is differentiable and \(c:[0,1] \rightarrow \mathbf{R}^{2}\) is defined by \(c(x)=(x, f(x))\), show that \(c([0,1])\) has length \(\int_{0}^{1} \sqrt{1+\left(f^{\prime}\right)^{2}} .\) (b) Show that this length is the least upper bound of lengths of inscribed broken lines. Hint: If \(0=t_{0} \leq t_{1} \leq \cdots \leq t_{n}=1\), then $$ \begin{aligned} \left|c\left(t_{i}\right)-c\left(t_{i-1}\right)\right| &=\sqrt{\left(t_{i}-t_{i-1}\right)^{2}+\left(f\left(t_{i}\right)-f\left(t_{i-1}\right)\right)^{2}} \\\ &=\sqrt{\left(t_{i}-t_{i-1}\right)^{2}+f^{\prime}\left(s_{i}\right)^{2}\left(t_{i}-t_{i-1}\right)^{2}} \end{aligned} $$ for some \(s_{i} \in\left[t_{i-1}, t_{i}\right]\).

5-31. Consider the 2 -form \(\omega\) defined on \(\mathbf{R}^{3}-0\) by $$ \omega=\frac{x d y \wedge d z+y d z \wedge d x+z d x \wedge d y}{\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}} $$ (a) Show that \(\omega\) is closed. (b) Show that $$ \omega(p)\left(v_{p}, w_{p}\right)=\frac{\langle v \times w, p\rangle}{|p|^{3}} $$ For \(r>0\) let \(S^{2}(r)=\left\\{x \in \mathbf{R}^{3}:|x|=r\right\\} . \quad\) Show that \(\omega\) restricted to the tangent space of \(S^{2}(r)\) is \(1 / r^{2}\) times the volume element, and that \(\int S^{2}(r) \omega=4 \pi . \quad\) Conclude that \(\omega\) is not exact. Nevertheless we denote \(\omega\) by \(d \Theta\) since, as we shall see, \(d \Theta\) is the analogue of the 1 -form \(d \theta\) on \(\mathbf{R}^{2}-0\) (c) If \(v_{p}\) is a tangent vector such that \(v=\lambda p\) for some \(\lambda \in \mathbf{R}\) show that \(d \Theta(p)\left(v_{p}, w_{p}\right)=0\) for all \(w_{p}\). If a two-dimensional manifold \(M\) in \(\mathbf{R}^{3}\) is part of a generalized cone, that is, \(M\) is the union of segments of rays through the origin, show that \(\int_{M} d \theta=0\) (d) Let \(M \subset \mathbf{R}^{3}-0\) be a compact two-dimensional manifoldwith-boundary such that every ray through 0 intersects \(M\) at most once (Figure 5-10). The union of those rays through 0 which intersect. \(M\), is a solid cone \(C(M) .\) The solid angle subtended by \(M\) is defined as the area of \(C(M) \cap S^{2}\), or equivalently as \(1 / r^{2}\) times the area of \(C(M) \cap S^{2}(r)\) for \(r>0\). Prove that the solid angle subtended by \(M\) is \(\left|\int_{M} d \Theta\right| .\) Hint: Choose \(r\) small enough so that there is a three-dimensional manifold-with-boundary \(N\) (as in Figure \(5-10)\) such that \(\partial N\) is the union of \(M\) and \(C(M) \cap S^{2}(r)\), and a part of a generalized cone. (Actually, \(N\) will be a manifold-

5-35. Applying the generalized divergence theorem to the set \(M=\) \(\left\\{x \in \boldsymbol{R}^{n}:|x| \leq a\right\\}\) and \(F(x)=x_{x}\), find the volume of \(S^{n-1}=\) \(\left\\{x \in R^{n}:|x|=1\right\\}\) in terms of the \(n\)-dimensional volume of \(B_{n}=\) \(\left\\{x \in \mathbf{R}^{n}:|x| \leq 1\right\\} . \quad\) (This volume is \(\pi^{n / 2} /(n / 2) !\) if \(n\) is even and \(2^{(n+1) / 2} \pi^{(n-1) / 2} / 1 \cdot 3 \cdot 5 \cdot \ldots\) if \(n\) is odd.)