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.

**Read or Download A First Course in Geometric Topology and Differential Geometry PDF**

**Best differential geometry books**

**Global Analysis: Differential Forms in Analysis, Geometry, and Physics**

This booklet is dedicated to differential varieties and their functions in a variety of parts of arithmetic and physics. Well-written and with lots of examples, this introductory textbook originated from classes on geometry and research and provides a conventional mathematical procedure in a lucid and extremely readable sort.

**Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor**

A important challenge in differential geometry is to narrate algebraic homes of the Riemann curvature tensor to the underlying geometry of the manifold. the complete curvature tensor is generally particularly tough to accommodate. This booklet offers effects in regards to the geometric effects that persist with if quite a few traditional operators outlined when it comes to the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and better order generalizations) are assumed to have consistent eigenvalues or consistent Jordan basic shape within the applicable domain names of definition.

**Stochastic Calculus in Manifolds**

Addressed to either natural and utilized probabilitists, together with graduate scholars, this article is a pedagogically-oriented creation to the Schwartz-Meyer second-order geometry and its use in stochastic calculus. P. A. Meyer has contributed an appendix: "A brief presentation of stochastic calculus" proposing the root of stochastic calculus and therefore making the e-book larger obtainable to non-probabilitists additionally.

**Additional info for A First Course in Geometric Topology and Differential Geometry **

**Sample text**

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.