Math Problems Vault

Problem 3.3.10 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

(a)(b)

Let $X$ be a smooth vector field on a Riemannian manifold $M$ . The divergence of $X$ , denoted by $\operatorname {div}(X)$ , is defined by the function $\operatorname {trace}(\nabla X)$ .

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(a)

If $M$ is closed (i.e., compact without boundary), show that

\int _M \operatorname {div}(X) \, dv = 0.

(b)

If $M$ is compact with boundary $\partial M$ , show that

\int _M \operatorname {div}(X) \, dv = \int _{\partial M} \left\langle X, N \right\rangle \, ds,

where $N$ is the outer normal vector field of $\partial M$ .

Problem 3.3.14 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

(a)(b)(c)

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(a)

Let $f$ be a smooth function on a Riemannian manifold $M$ . Let $\operatorname {grad} f$ be the vector field defined by the equation

\left\langle \operatorname {grad} f, v \right\rangle _p = d_pf \, v, \qquad v \in T_pM.

Let $(x^1,\ldots ,x^n)$ be local coordinates around $p$ . Find the expression for $\operatorname {grad} f$ in terms of $x^1,\ldots ,x^n$ .

(b)

For a vector field $X$ define the divergence of $X$ , $\operatorname {div}(X)$ as the trace of the operator $Y \to D_Y X$ where $D$ is the Levi-Civita connection. Find the expression for the divergence of $X$ in a local coordinate system $(x^1,\ldots ,x^n)$ .

(c)

Use (a) and (b) to find the expression for the Laplacian $\Delta$ in local coordinates, where $\Delta$ acting on a smooth function $f$ is defined by $\Delta f = \operatorname {div}(\nabla f)$ .

Problem 3.3.15 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

Let $M$ be a compact connected Riemannian manifold without boundary. Let $f$ be a smooth function satisfying $\Delta f = 0$ . Show that $f = \text{const}$ .

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Problem 3.3.28 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

Let $M$ be a compact Riemannian manifold without boundary. For any $f \in C^\infty (M)$ , define $\nabla f \in \mathcal{X}(M)$ and $\Delta f \in C^\infty (M)$ as follows:

At any $p \in M$ , choose an orthonormal frame field $\{ e_1,\ldots ,e_n\}$ around $p$ and then define

(\nabla f)(p) = \sum _i (e_if)(e_i)

and

(\Delta f)(p) = -\sum _i [e_i(e_if) - (\nabla _{e_i}e_i)f].

Verify first that $\nabla f$ and $\Delta f$ are well defined (i.e., they do not depend on the choice of orthonormal frame) and then show that

\int f\Delta f = \int \left\| \nabla f\right\| ^2.

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Problem 3.3.31 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

Let $M^n$ be a Riemannian manifold. Show that whenever $f : M \to \mathbb {R}$ is a smooth function, there is a unique vector field $\nabla f$ (called the gradient of $f$ on $M$ ) such that

\left\langle \nabla f(p), \dot\gamma (t) \right\rangle = \frac{d}{dt} f(\gamma (t))

whenever $\gamma$ is a smooth curve in $M$ with $\gamma (t) = p$ .

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Problem 3.3.45 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

(a)(b)

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(a)

Consider $\mathbb {R}^{n}$ with the standard Euclidean metric and let $p \in \mathbb {R}^{n}$ be an arbitrary point. For any $x \in \mathbb {R}^{n}$ let $\rho_p(x)$ be the distance from $p$ to $x$ . Viewing $\rho_p(x)$ as a smooth function of $x$ away from $p$ , verify that $\left|\operatorname {grad}(\rho_p(x))\right|^2 = 1$ and that the integral curves of $\operatorname {grad}(\rho_p(x))$ are straight lines. (Here $\operatorname {grad}(\rho_p(x))$ refers to the usual gradient vector field of the function $\rho_p(x)$ .)

(b)

More generally, given a smooth function $f$ on a Riemannian manifold $(M,g_{ij})$ , define $\operatorname {grad}(f)$ to be the vector field given locally by

\sum _{i,j} \left(g^{ij}\frac{\partial f}{\partial x^j}\right)\frac{\partial }{\partial x^i}

show that if $\left|\operatorname {grad}(f)\right|^2=1$ , then the integral curves of the vector field $\operatorname {grad}(f)$ are geodesics.

Gradient, divergence, and Laplace-Beltrami operator

References