Geometria Riemanniana Teste 1 1911 PDF

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Geometria Riemanniana Teste...

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Riemannian Geometry 1st Test - November 15, 2019 LMAC and MMA Duration: 180 minutes Show your calculations 1. Recall that the u(n) is the space of anti-hermitean matrices and that SU (n) = {M ∈ U (n) : det M = 1}. a) Prove that SU (n) is a submanifold of U (n) by showing that it is the inverse image of a regular value of a map f : U (n) → S 1 . What is su(n) and what is its dimension? b) Compute the left invariant vector field corresponding to   i 0 ∈ su(2). 0 −i

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Check directly that the vector field you construct belongs to X (SU (2)). 2. Consider the upper half plane, H, with the area form ω=

dx ∧ dy y2

and the disc D = {(x, y) ∈ H : x2 + (y − 2) 2 < 1}. a) Compute the q area of D. q Suggestion: integrate first in y and use R cos2 θ 1+a tan θ − θ. arctan dθ = 1+a a a+sin2 θ Ra b) Use Stokes’ Theorem to write D ω as an integral of a one form on ∂D, and write this integral using the parameterization (x, y) = (cos θ, 2 + sin θ). p c) Let ρ := x2 + (y − 2)2 . Check that the frame E1 =

E2 =

y ((2 − y)∂x + x∂y ), ρ y (−x∂x + (2 − y)∂y ), ρ

is orthonormal for the metric with line element ds2 =

dx2 + dy 2 , y2

and that E1 is tangent to the boundary of D. Write the dual frame (ω 1 , ω 2).

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2

RG Test - 15.11.19 d) Check that

(2) 2 dx ∧ dy, ρy2 x = − 2 dx ∧ dy. ρy

dω 1 = dω 2

e) Compute the one form ω12 such that

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dω 1 = − ω 2 ∧ ω12 , dω 2 =

ω 1 ∧ ω12.

f ) The geodesic curvature of the boundary of D is given by kg = ω12(E1 ). Check that kg = 2. g) The Gaussian curvature, K, of H is identically equal to −1. Calculate Z Z Kω + kg ds. D

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∂D

3. Prove that every closed form in Ω1 (S 2 ) is exact.

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