- #1

aalma

- 46

- 1

Hello :),

I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##.

Definitions I know:

Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components ##(˙γ1,..., ˙γn)##.

Two (smooth) curves ##γ,γ': (− ,) → M## with ##γ(0) = γ'(0) = x## are called equivalent if in a certain chart containing x (or in any chart) they define the same tangent vector at ##x##.

We define ##T_x(M)## to be the set of equivalence classes of curves as above. We note that ##T_x(M)## is a vector space: Choose a chart ##a : U → R^n## containing x. Then the assignment ##γ → d/dt (a ◦ γ)(0)## defines a bijection from ##T_x(M)## to ##R^n##. Different charts define different bijections, but they differ by the Jacobi matrix: if ##b : U → R^n## is another chart, the bijections from ##Tx(M)## to ##R^n## differ by the Jacobi matrix at ##a(x)## of the transformation ##b◦a^{−1} : a(U) → b(U)##.

Example: Let ##G = SO(n,R)## and let ##γ## be a curve in ##G## with ##γ(0) = 1##. Then we can write ##γ(s) = 1 + sA + o(s^2)## where ##A ∈ M_{n× n}## and we want to describe possible values for ##A##. Obviously ##γ(s)^t = 1 + sA^t + o(s^2)## so ##1 = γ(s)γ(s)^t = 1 + s(A + A^t) + o(s^2)## which implies that ##A## is skewsymmetric. In the opposite direction, if ##A## is skew-symmetric, ##γ(s) = e^{sA}## is orthogonal and ##e^{sA} = 1 + sA + o(s^2)##. We have proven that ##T_1(G)## in this case is the vector space of skew-symmetric matrices.

Can you please suggest a way to compute such tangent spaces.

I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##.

Definitions I know:

Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components ##(˙γ1,..., ˙γn)##.

Two (smooth) curves ##γ,γ': (− ,) → M## with ##γ(0) = γ'(0) = x## are called equivalent if in a certain chart containing x (or in any chart) they define the same tangent vector at ##x##.

We define ##T_x(M)## to be the set of equivalence classes of curves as above. We note that ##T_x(M)## is a vector space: Choose a chart ##a : U → R^n## containing x. Then the assignment ##γ → d/dt (a ◦ γ)(0)## defines a bijection from ##T_x(M)## to ##R^n##. Different charts define different bijections, but they differ by the Jacobi matrix: if ##b : U → R^n## is another chart, the bijections from ##Tx(M)## to ##R^n## differ by the Jacobi matrix at ##a(x)## of the transformation ##b◦a^{−1} : a(U) → b(U)##.

Example: Let ##G = SO(n,R)## and let ##γ## be a curve in ##G## with ##γ(0) = 1##. Then we can write ##γ(s) = 1 + sA + o(s^2)## where ##A ∈ M_{n× n}## and we want to describe possible values for ##A##. Obviously ##γ(s)^t = 1 + sA^t + o(s^2)## so ##1 = γ(s)γ(s)^t = 1 + s(A + A^t) + o(s^2)## which implies that ##A## is skewsymmetric. In the opposite direction, if ##A## is skew-symmetric, ##γ(s) = e^{sA}## is orthogonal and ##e^{sA} = 1 + sA + o(s^2)##. We have proven that ##T_1(G)## in this case is the vector space of skew-symmetric matrices.

Can you please suggest a way to compute such tangent spaces.

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