First Steps in Differential Geometry: Riemannian, Contact, by Andrew McInerney

By Andrew McInerney

Differential geometry arguably deals the smoothest transition from the traditional college arithmetic series of the 1st 4 semesters in calculus, linear algebra, and differential equations to the better degrees of abstraction and evidence encountered on the higher department by means of arithmetic majors. this present day it's attainable to explain differential geometry as "the learn of buildings at the tangent space," and this article develops this viewpoint.

This booklet, in contrast to different introductory texts in differential geometry, develops the structure essential to introduce symplectic and phone geometry along its Riemannian cousin. the most target of this publication is to carry the undergraduate pupil who already has a fantastic beginning within the typical arithmetic curriculum into touch with the great thing about better arithmetic. specifically, the presentation the following emphasizes the implications of a definition and the cautious use of examples and structures as a way to discover these consequences.

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Example text

Then (Ψ2 ◦ Ψ1 )∗ = Ψ1∗ ◦ Ψ2∗ . Proof. Let T ∈ V3∗ and choose v ∈ V1 . 8 The Dual of a Vector Space, Forms, and Pullbacks 35 (Ψ2 ◦ Ψ1 )∗ (T )(v) = T ((Ψ2 ◦ Ψ1 )(v)) = T (Ψ2 (Ψ1 (v))), while on the other hand, (Ψ1∗ ◦ Ψ2∗ )(T )(v) = (Ψ1∗ (Ψ2∗ (T )))(v) = (Ψ1∗ (T ◦ Ψ2 ))(v) = ((T ◦ Ψ2 ) ◦ Ψ1 )(v) = T (Ψ2 (Ψ1 (v))). The construction of the dual space V ∗ is a special case of a more general construction. Suppose we are given several vector spaces V1 , . . , Vk . Recall (see Sect. 1) that the Cartesian product of V1 , .

7. Let V be a finite-dimensional vector space, and let T : V → V be a linear transformation. Then for any two bases B1 , B2 of V , we have det [T ]B1 ,B1 = det [T ]B2 ,B2 . Proof. 7 Constructing Subspaces II: Subspaces and Linear Transformations 27 [T ]B2 ,B2 = [Id]B2 ,B1 [T ]B1 ,B1 [Id]B1 ,B2 , −1 and that [Id]B2 ,B1 = [Id]B1 ,B2 , where Id : V → V is the identity transformation. For this reason, we refer to the determinant of the linear transformation T : V → V and write det(T ) to be the value of det(A), where A = [T ]B,B for any basis B of V .

N } is a basis for V ∗ . To show that B ∗ is linearly independent, suppose that c1 ε1 + · · · + cn εn = O (an equality of linear transformations). This means that for all v ∈ V , c1 ε1 (v) + · · · + cn εn (v) = O(v) = 0. In particular, for each i = 1, . . , n, setting v = ei gives 0 = c1 ε1 (ei ) + · · · + cn εn (ei ) = ci . Hence B ∗ is a linearly independent set. , T : V → R is a linear transformation. We need to find scalars c1 , . . , cn such that T = c1 ε1 + · · · + cn εn . Following the idea of the preceding argument for linear independence, define ci = T (ei ).

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