A First Course in Geometric Topology and Differential by Ethan D. Bloch

By Ethan D. Bloch

The distinctiveness of this article in combining geometric topology and differential geometry lies in its unifying thread: the inspiration of a floor. With quite a few illustrations, routines and examples, the coed involves comprehend the connection among glossy axiomatic process and geometric instinct. The textual content is stored at a concrete point, 'motivational' in nature, averting abstractions. a few intuitively attractive definitions and theorems bearing on surfaces within the topological, polyhedral, and delicate instances are offered from the geometric view, and element set topology is particular to subsets of Euclidean areas. The therapy of differential geometry is classical, facing surfaces in R3 . the fabric here's obtainable to math majors on the junior/senior point.

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This m is called the dimension of S at p. Otherwise p is called a singular point of S. S is called an m−dimensional (regular ) submanifold of M if every point of S is regular of the same dimension m. Remarks. a) We shall summarize the definition of “p is regular point of S” by saying that S is given by the equations xm+1 = · · · , xn = 0 locally around p. The number m is independent of the choice of the (xi ): if S is also given by ˜ ˜ x ˜m+1 = ··· = x ˜n = 0, then maps (x1 , · · · , xm ) ↔ (˜ x1 , · · · , x ˜m ) which relate the ˜ coordinates of the points of S in U U are inverses of each other and smooth.

Let Pn be the set of all one–dimensional subspaces of Rn+1 (lines through the origin). Let F : S n → Rn be the map which associates to each p ∈ S n the line p = Rp. (a) Show that Pn admits a unique manifold structure so that the map F is a local diffeomorphism. (b)Show that the manifold structure on Pn defined in problem 10 is the same as the one defined in part (a). 3. VECTORS AND DIFFERENTIALS 37 12. Generalize problem 10 to the set Gk,n of k–dimensional subspaces of Rn . [Suggestion. Consider the map φ which intersects a k–dimensional subspace P ∈Gk,n with the coordinate (n − k)–plane with equation x1 = x2 = · · · = xk = 1 and similar maps using other coordinate (n − k)–planes of this type.

Let M be a manifold, f 1 , · · · , f k smooth functions on M . Let S be the set of points p ∈ M satisfying f 1 (p) = 0, · · · , f k (p) = 0. Suppose the differentials df 1 , · · · , df k are linearly independent at every point of S. (a)S is a submanifold of M of dimension m = n − k and its tangent space at p ∈ S is the subspace of vectors v ∈ Tp M satisfying the linear equations (df 1 )p (v) = 0, · · · , (df k )p (v) = 0. (b)I f f k+1 , · · · , f n are any m = n − k additional smooth functions on M so that all n differentials df 1 , · · · , df n are linearly independent at p, then their restrictions to S, denoted t1 := f k+1 |S , · · · , tm := f k+m |S , form a coordinate system around p on S.

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