Differential Geometry and Topology: With a View to Dynamical by Keith Burns

By Keith Burns

Available, concise, and self-contained, this e-book bargains an exceptional creation to 3 similar topics: differential geometry, differential topology, and dynamical structures. issues of unique curiosity addressed within the ebook contain Brouwer's mounted aspect theorem, Morse concept, and the geodesic flow.Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential types, and integration on manifolds give you the beginning for lots of purposes in dynamical platforms and mechanics. The authors additionally speak about the Gauss-Bonnet theorem and its implications in non-Euclidean geometry versions. The differential topology point of the ebook facilities on classical, transversality conception, Sard's theorem, intersection thought, and fixed-point theorems. the development of the de Rham cohomology builds additional arguments for the robust connection among the differential constitution and the topological constitution. It additionally furnishes the various instruments beneficial for a whole figuring out of the Morse thought. those discussions are through an creation to the speculation of hyperbolic platforms, with emphasis at the crucial function of the geodesic move. the combination of geometric conception, topological concept, and urban purposes to dynamical structures set this e-book aside. With fresh, transparent prose and powerful examples, the authors' intuitive technique creates a remedy that's understandable to relative newcomers, but rigorous sufficient for people with extra heritage and adventure within the box.

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We now discuss the orientability of manifolds with boundary. 6 If a manifold with boundary M is orientable, then so is ∂M . PROOF Assume that p is a boundary point. If φ : U → V is a local parametrization of M near p with U ⊆ {(x1 , . . , xm ) | xm ≥ 0}, then Tp (∂M ) coincides with the image of {(x1 , . . , xm ) | xm = 0} through (dφ)p . The tangent space Tp (∂M ) is a co-dimension 1 subspace of Tp M . It separates Tp M into two kinds of vectors: the images through (dφ)p of vectors (x1 , . .

Xm , 0, . . , 0) ∈ Rn is an immersion (referred as the canonical immersion), and also an embedding. If m ≥ n, the mapping (x1 , . . , xn , xn+1 , . . , xm ) ∈ Rm → (x1 , . . , xn ) ∈ Rn is a submersion (referred as the canonical submersion). The following theorem states that any immersion is locally the same as the canonical immersion. 5 (Immersion theorem) If f : M → N is an immersion, then for every point p ∈ M there exist a local parametrization φ : U → V near p and a local parametrization ¯ → V¯ near f (p) such that ψ:U (ψ −1 ◦ f ◦ φ)(x1 , .

Thus, the points in Vα ∩ f −1 (−∞, a] correspond through φ to the points in U ∩ Hm . 4 Let f : M → N be a smooth map from the m-manifold with boundary M to the n-manifold without boundary N . If q ∈ N is a regular value for both f and f |∂M , then f −1 (q) is an (m − n)-manifold with boundary ∂(f −1 (q)) equal to f −1 (q) ∩ ∂M . PROOF By using local coordinates, we can reduce the problem to the special case when f : Hm → Rn and q ∈ f (∂Hm ). 2 implies that f −1 (q) is a smooth (n − m)-manifold near p.

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