B1.8

Differentiable Manifolds | Part II, 2003

State the Implicit Function Theorem and outline how it produces submanifolds of Euclidean spaces.

Show that the unitary group U(n)GL(n,C)U(n) \subset G L(n, \mathbb{C}) is a smooth manifold and find its dimension.

Identify the tangent space to U(n)U(n) at the identity matrix as a subspace of the space of n×nn \times n complex matrices.

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