Exercise · ZW-LIE-GROUP-003 · ★☆☆ Basic
What is the dimension of \((\mathbb{R},+)\) as a manifold?
Choose one option.
Show answer
- D. \(1\)
Explanation
The real line is a manifold of dimension 1, since each point has a neighborhood homeomorphic to an open interval of \(\mathbb{R}\). So its dimension is \(1\).
Review the concept: Lie group
A Lie group \(G\) is a group that is also a smooth manifold. In this structure, multiplication and inversion are smooth functions.
\[\begin{gathered} \\ m:G\times G\to G,\quad (g,h)\mapsto g\cdot h \\ i:G\to G,\quad g\mapsto g^{-1} \\ \end{gathered}\]
More practice problems: lie group
- ZW-LIE-GROUP-001 What must a Lie group satisfy besides being a group?
- ZW-LIE-GROUP-002 Besides multiplication, which other map must be smooth in a Lie group?
- ZW-LIE-GROUP-004 What is a smooth map between Lie groups that preserves the group operation called?
- ZW-LIE-GROUP-005 Which of these sets with its operation is a Lie group?
- ZW-LIE-GROUP-006 In S^1, what is the inverse of e^iθ?
- ZW-LIE-GROUP-007 In (ℝ,+), what is the inverse of a?
- ZW-LIE-GROUP-008 If G has dimension 3 and H is a closed Lie subgroup of dimension 1, what is the dimension of G/H?
- ZW-LIE-GROUP-009 Which map is the left translation L_g?