Lie groups in a Riemannian/differential-geometric context

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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 3.3.35 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"

If $\phi : G_1 \to G_2$ is a Lie group homomorphism, show that for all $v \in LG_1$ , we have

\phi (\exp (v)) = \exp (\phi _*v).

Problem 3.3.37 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 $G$ be a Riemannian manifold with a global frame-field $\{ e_i\}_{i=1}^n$ .

(a)

Show that any connection on $G$ is completely determined by its effect on the frame field, i.e., by the vector fields $\nabla_{e_i}e_j$ , $i,j=1,\ldots ,n$ .

(b)

Show that when $G$ is a Lie group with a bi-invariant metric $\left\langle \cdot , \cdot \right\rangle$ , and the frame-field is left-invariant, we characterize the Levi-Civita connection on $(G, \left\langle \cdot , \cdot \right\rangle )$ by setting,

\nabla _{e_i}e_j = \frac12 [e_i,e_j]

for all $i,j=1,\ldots ,n$ .

Problem 3.3.39 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"

Show that the sphere $S^{2n}$ is not the underlying topological space of any Lie group.

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.