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

By Andrew McInerney

Differential geometry arguably deals the smoothest transition from the normal collage 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 top department through arithmetic majors. this day it really is attainable to explain differential geometry as "the learn of buildings at the tangent space," and this article develops this aspect of view.

This booklet, in contrast to different introductory texts in differential geometry, develops the structure essential to introduce symplectic and make contact with geometry along its Riemannian cousin. the most target of this publication is to convey the undergraduate scholar who already has a superb beginning within the usual arithmetic curriculum into touch with the wonderful thing about greater arithmetic. specifically, the presentation right here emphasizes the implications of a definition and the cautious use of examples and structures so that it will discover these effects.

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First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergraduate Texts in Mathematics)

Differential geometry arguably deals the smoothest transition from the traditional collage 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 top department by means of arithmetic majors. at the present time it really is attainable to explain differential geometry as "the learn of buildings at the tangent space," and this article develops this perspective.

Extra info for First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergraduate Texts in Mathematics)

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Vk is the set of ordered k-tuples of vectors V1 × · · · × Vk = {(v1 , . . , vk ) | vi ∈ Vi for all i = 1, . . , k} . The set V = V1 × · · · × Vk can be given the structure of a vector space by defining vector addition and scalar multiplication componentwise. 8. Let V1 , . . , Vk and W be vector spaces. A function T : V1 × · · · × Vk → W is multilinear if it is linear in each component: T (x1 + y, x2 , . . , xk ) = T (x1 , x2 , . . , xk ) + T (y, x2 , . . , xk ), .. T (x1 , x2 , . .

The fact that sT1 is also linear for every s ∈ R is proved similarly. Note that the ⊓ zero “vector” O ∈ V ∗ is defined by O(v) = 0 for all v ∈ V . The space V ∗ is called the dual vector space to V . Elements of V ∗ are variously called dual vectors, linear one-forms, or covectors. The proof of the following theorem, important in its own right, includes a construction that we will rely on often: the basis dual to a given basis. 2. Suppose that V is a finite-dimensional vector space. Then dim(V ) = dim(V ∗ ).

Wn ). 11 that G0 is a bilinear form on Rn . The reader may verify property (I2). , when v = 0. 2 can be generalized to any finite-dimensional vector space V . Starting with any basis B = {e1 , . . , en } for V , define GB (v, w) = v1 w1 + · · · + vn wn , where v = v1 e1 + · · · + vn en and w = w1 e1 + · · · + wn en . This function GB is well defined because of the unique representation of v and w in the basis B. 3. Every finite-dimensional vector space carries an inner product structure. 9 Geometric Structures I: Inner Products 39 Of course, there is no unique inner product structure on a given vector space.

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