Submanifolds via regular values

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Problem 3.2.24 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 3.2 "Differential Geometry of Surfaces"

(a)(b)

Let $M$ be a 2-dimensional manifold smoothly embedded in $\mathbb {R}^{3}$ with unit normal $n$ .

(a)

Prove that for each $p \in M$ there exist an open neighborhood $U_p$ of $p$ in $\mathbb {R}^{3}$ and a smooth function $F : U_p \to \mathbb {R}$ such that $F^{-1}(0) = U_p \cap M$ .

(b)

Find $F$ if $M$ is the graph of a smooth function $f : \mathbb {R}^{2} \to \mathbb {R}$ .

Problem 3.3.13 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

A = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a,b,c,d \in \mathbb {R}, \, ad-bc=1\right\} .

Show that

(a)

$A$ is a differentiable manifold.

(b)

$A$ is a Lie group with the standard matrix multiplication as a product.

Problem 3.3.17 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"

Regard $\mathbb {R}^{9}$ as the space of all $3\times 3$ matrices with real entries. Does the subset

\Sigma = \{ A \in \mathbb {R}^{9} : \det (A) = 0\}

form a smooth submanifold of $\mathbb {R}^{9}$ ?

Problem 3.3.19 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"

Prove that

\left\{ (x_1,x_2,x_3,x_4) : \operatorname {rank}\begin{pmatrix} x_1 & x_2 \\ x_3 & x_4 \end{pmatrix} = 1\right\}

is a three-dimensional submanifold of $\mathbb {R}^{4}$ .

Problem 3.3.21 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 $M = \{ (x,y) : x,y \in \mathbb {R}^{3}, \left\| x\right\| =1, \left\| y\right\| =1, \left\langle x, y \right\rangle =0\}$ .

(a)

Show that $M$ is a smooth compact embedded submanifold of $\mathbb {R}^{6}$ and explain how $M$ can be identified with the unit tangent bundle of $S^2$ .

(b)

Show that $M$ is orientable.

Problem 3.3.25 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 $X$ denote a submanifold of Euclidean space $\mathbb {E}^n$ , and set

\begin{align} U_\varepsilon X & := \{ x+v : x \in X, v \in N_xX, \left|v\right|<\varepsilon \} , \\ B(X,\varepsilon ) & := \{ y \in \mathbb {E}^n : \left|y-x\right|<\varepsilon \text{ for some } x \in X\} . \end{align}

Show that $U_\varepsilon X \subset B(X,\varepsilon )$ for all $\varepsilon$ . Show that the two are not generally equal. (Consider examples of 1-dimensional submanifold in $\mathbb {E}^2$ .) Can you give conditions which imply equality?

Problem 3.3.29 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 $p(x_1,x_2,x_3,x_4) = (x_1^2+x_2^2)(x_3^2+x_4^2)$ and $q(x_1,x_2,x_3,x_4) = x_1^2+x_2^2+x_3^2+x_4^2$ . Define

S_{a,b} = \{ x \in \mathbb {R}^{4} \mid p(x)=a \text{ and } q(x)=b\} .

For what $a,b \geq 0$ , is $S_{a,b}$ a manifold? Explain.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.