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Differential Geometry - Lecture Notes

Differential Geometry - Lecture Notes - Diential Geometry...

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Diffential Geometry: Lecture Notes Dmitri Zaitsev D. Zaitsev: School of Mathematics, Trinity College Dublin, Dublin 2, Ireland E-mail address : [email protected]
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Contents Chapter 1. Introduction to Smooth Manifolds 5 1. Plain curves 5 2. Surfaces in R 3 7 3. Abstract Manifolds 9 4. Topology of abstract manifolds 12 5. Submanifolds 14 6. Differentiable maps, immersions, submersions and embeddings 16 Chapter 2. Basic results from Differential Topology 19 1. Manifolds with countable bases 19 2. Partition of unity 20 3. Regular and critical points and Sard’s theorem 22 4. Whitney embedding theorem 25 Chapter 3. Tangent spaces and tensor calculus 27 1. Tangent spaces 27 2. Vector fields and Lie brackets 31 3. Frobenius Theorem 33 4. Lie groups and Lie algebras 34 5. Tensors and differential forms 36 6. Orientation and integration of differential forms 37 7. The exterior derivative and Stokes Theorem 38 Chapter 4. Riemannian geometry 39 1. Riemannian metric on a manifold 39 2. The Levi-Civita connection 41 3. Geodesics and the exponential map 44 4. Curvature and the Gauss equation 46 3
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CHAPTER 1 Introduction to Smooth Manifolds Even things that are true can be proved. Oscar Wilde, The Picture of Dorian Gray 1. Plain curves Definition 1.1 . A regular arc or regular parametrized curve in the plain R 2 is any continuously differentiable map f : I R 2 , where I = ( a, b ) R is an open interval (bounded or unbounded: -∞ ≤ a < b ≤ ∞ ) such that the R 2 -valued derivative f 0 ( t ) is different from 0 = (0 , 0) for all t I . That is for every t I , f ( t ) = ( f 1 ( t ) , f 2 ( t )) R 2 and either f 0 1 ( t ) 6 = 0 or f 0 2 ( t ) 6 = 0. The variable t I is called the parameter of the arc. One may also consider closed intervals, in that case their endpoints require special treatment, we’ll see them as “boundary points”. Remark 1.2 . There is a meaningful theory of nondifferentiable merely continuous arcs (includ- ing exotic examples such as Peano curves covering a whole square in R 2 ) and of more restrictive injective continuous arcs (called Jordan curves) that is beyond the scope of this course. The assumption f 0 ( t ) 6 = 0 roughly implies that the image of f “looks smooth” and can be “locally approximated” by a line at each point. A map f with f 0 ( t ) 6 = 0 for all t is also called immersion . Example 1.3 . Without the assumption f 0 ( t ) 6 = 0 the image of f may look quite “unpleasant”. For instance, investigate the images of the following C maps: f ( t ) = ( t 2 , t 3 ) , f ( t ) := (0 , e 1 /t ) t < 0 (0 , 0) t = 0 ( e - 1 /t , 0) t > 0 . The first curve is called Neil parabola or semicubical parabola . Both maps are not regular at t = 0. Such a point is called a critical point or a singularity of the map f . Definition 1.4 . A regular curve is an equivalence class of regular arcs, where two arcs f : I R 2 and g : J R 2 are said to be equivalent if there exists a bijective continuously differential map ϕ : I J with ϕ 0 ( t ) > 0 for all t I (the inverse ϕ - 1 is then exists and is automatically continuously differentiable) such that f = g ϕ , i.e. f ( t ) = g ( ϕ ( t )) for all t . Sometimes a finite (or even countable) union of curves is also called a curve. A regular curve of class
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