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A Study of Smooth Map on Manifolds Barishal University Journal Part 1, 5(1&2): 47-58 (2018) ISSN 2411-247X A STUDY OF SMOOTH MAP ON MANIFOLDS Shek Ahmed1,* Tanzila Yeasmin Nilu2 and Samima Akhter1 1Department of Mathematics, University of Barishal, Barishal-8200 2Department of Computer Science and Engineering, Green University of Bangladesh, Dhaka-1207 Abstract In this paper we present a theorem to determine characteristics of smooth map. For this purpose we studied some propositions and lemma related to smooth map on manifolds. The basic geometry of tangent space and definitions involving tangent bundle are discussed in our paper. We developed the notion of quotient manifold and sub manifold using the concept of smooth map. Finally, the theorem 6.1 on smooth map is generalized for finding the neighborhoods of surjective map. Keywords: Tangent space, Smooth map, Tangent bundle, Diffeomorphism, Manifolds. 1. Introduction A manifold is a topological space that locally resembles Euclidean space near each point. Although a manifold locally resembles Euclidean space, meaning that every point has a neighborhoods homeomorphism to an open subset of Euclidean space. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds naturally arise as a solution set of a system of equation and graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. This differentiable manifolds structure allows calculus to be done on manifolds. The study of manifolds combines many important areas of mathematics. It generalizes concept such as curves and surfaces as well as ideas from linear algebra and topology. There were several important results of manifolds between 18th and 19th century mathematics. The oldest of these was Non-Euclidean geometry, which considers spaces where Euclids parallel postulate fails. The Italian mathematician Saccheri (1733) *Corresponding author’s e-mail: [email protected] 47 Barishal University Journal Part 1, 5(1&2): 47-58 (2018) A Study of smooth map on manifolds first studied geometry and then Lobachevsky and Bolyai (1830) developed it. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space. These are called hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive curvature respectively. Gauss (1855) is the first to consider abstract spaces as mathematical objects in their own right. His theorem egregium gives a method for computing the curvature of a surface without considering the ambient space in which surface lies. Manifold theory has come to focus exclusively on these intrinsic properties while largely ignoring the extrinsic properties of the ambient space. Gauss (1805) and Monge (1807) first introduced differential geometry. The important contributions were made by many scientists in 19th century. Darboux and Bianchi (1896) collected and systematized the work. However, recently there are several researchers who worked on the development of several parts of manifolds. Ahmed et al., (2012) developed the characterization of vector fields on manifolds, Ali et al., (2012) worked on the exterior algebra with differential forms on manifolds. Ahmed et al., (2014) introduced the multi linear algebra and tensors with vector sub bundle on manifolds, Osman (2016) worked on basic integration on smooth manifolds and application map with Stokes theorem. In this paper we shall discuss the properties of tangent space, tangent bundle and developed the notion of quotient manifold and sub manifolds using the concept of smooth map. In this paper some necessary propositions related to smooth map are treated and the theorem 6.1 has been derived. 2. Tangent spaces Definition 2.1 (Flanders. 1963): For embedded sub manifolds 푀 ⊆ ℝ푛 , the tangent space 푇푎 푀 at 푎 ∈ 푀 can be defined as the set of all velocity vectors 푣 = 훾 (0), where 훾: 퐽 → 푀 is a smooth curve with 훾 0 = 푎, here 푗 ⊆ ℝ is an open interval around 0. 푛 It turns out that 푇푎 푀 becomes a vector subspace of ℝ . Example 2.1 (Flanders. 1963) Consider the sphere 푆푛 ⊆ ℝ푛+1, given as the set of 푥 such that ∥ 푥 ∥2= 1. A curve 훾 푡 lies in 푆푛 if and only if ∥ 훾 푡 ∥= 1. Taking the derivative of the equation 훾 푡 . 훾 푡 = 1 at 푡 = 0. We obtain after dividing by 2 and using 훾 0 = 푛+1 푎, 푎훾 0 = 0. That is 푇푎 푀 consist of vectors 푣 ∈ ℝ that are orthogonal to 푎 ∈ 3 푆푛 ℝ {0}. It is easily seen that every such vector 푣 is of the form 훾 (0), hence that 푇푎 = (ℝ푝)⊥, hence the hyperplane orthogonal to the line through 푎. Definition 2.2 : Let 푀 be a manifold and 푎 ∈ 푀. The tangent space 푇푎 푀 is the set of all linear maps 푣: 퐶∞ (푀) → ℝ of the form 푑 푣 푓 = | 푓(훾 푡 ) 푑푡 푡=0 48 Barishal University Journal Part 1, 5(1&2): 47-58 (2018) Ahmed et al. ∞ For smooth curve 훾 ∈ 퐶 (퐽, 푀) with 훾 0 = 푎. The elements 푣 ∈ 푇푎 푀 are called the tangent vectors to 푀 at 푎. The following local coordinate description makes it clear that 푇푎 푀 is linear subspace of the vector space 퐿(퐶∞ 푀 , ℝ) of linear maps 퐶∞ (푀) → ℝ. The dimension of linear maps equal to the dimension of 푀. Theorem 2.1 (Narasimhan. 1968) : Let (푈, 휑) be a coordinate chart around 푎. A linear ∞ map 푣: 퐶 (푀) → ℝ is in 푇푎 푀 if and only if it has the form 휕(푓표휑−1) 푣 푓 = 푚 푝푖 | (2.1) 푖=1 휕푢푖 푢=휑(푎) for some 푝 = (푝1, … , 푝푚 ) ∈ ℝ푛 . Proof. Given a linear map 푣 of this form. Let 훾 : ℝ → 휑(푈) be a curve with 훾 푡 = 휑 푎 + 푡푝 for |푡| sufficiently small. Let 훾 = 휑−1표 훾 . Then 푑 푑 −1 | 푓 훾 푡 = | 푓표휑 ( 휑 푎 + 푡푝) 푑푡 푡=0 푑푡 푡=0 휕(푓표휑−1) = 푚 푝푖 | (2.2) 푖=1 휕푢푖 푢=휑(푎) by the chain rule. Conversely, given any curve 훾 with 훾 0 = 푎, let 훾 = 휑 표 훾 be the corresponding curve in 휑(푈). Then 훾 = 휑(푎) and 푑 푑 −1 | 푓 훾 푡 = | 푓표휑 (훾 푡 ) 푑푡 푡=0 푑푡 푡=0 휕(푓표휑−1) = 푚 푝푖 | , (2.3) 푖=1 휕푢푖 푢=훾 푎 푑훾 where 푎 = | . We can use this result as an alternative definition of the tangent 푑푡 푡=0 space. Definition 2.3 (Brickell et al., 1970) : Let 푀 be a smooth manifold, and let 풴(푀) be the ring of smooth functions on 푀. A tangent space of 푀 at a point 푎 ∈ 푀 is a linear map 휉: 풴(푀) → ℝ such that 휉 푓푔 = 휉 푓 푔 푎 + 푓 푎 휉 푔 , 푓, 푔 ∈ 풴(푀). The tangent vectors form a real vector space, 푇푎 (푀), under the linear operations 휆휉 + 휇휂 푓 = 휆휉 푓 + 휇휂 푓 , 휆, 휇 ∈ ℝ, 휉, 휂 ∈ 푇푎 푀 , 푓 ∈ 풴(푀) (2.4) 푇푎 (푀) is called the tangent space of 푀 at a. 49 Barishal University Journal Part 1, 5(1&2): 47-58 (2018) A Study of smooth map on manifolds 3. Smooth functions on manifolds A real-valued function on an open subset 푀 ⊆ ℝ푛 is called smooth if it is infinitely differentiable. The notion of smooth functions on open subsets of Euclidean spaces carries over to manifolds. A function is smooth if its expression in local coordinates is smooth. Definition 3.1 (Brickell et al., 1970) : A function 푓: 푀 → ℝ on a manifold 푀 is called smooth if for all charts (푈, 휑) of the function 푓표휑−1: 휑(푈) → ℝ (3.1) is smooth. The set of smooth functions on 푀 is denoted by 퐶∞ 푀 . Example 3.1 The height function 푓: 푆2 → ℝ, (푥, 푦, 푧) → 푧 is smooth. In fact, we see that ∞ 3 for any smooth function 푕 ∈ 퐶 ℝ , the restriction 푓 = 푕|푆2 is again smooth. Lemma 3.1 (Olum. 1953) : Smooth functions 푓 ∈ 퐶∞ 푀 are continuous. Proof. For every open subset 푗 ⊆ ℝ, the pre-image 푓−1(퐽) ⊆ 푀 is open. We have to show that for every (푈, 휑), the set 휑(푈 ∩ 푓−1(퐽)) ⊆ ℝ푛 is open. But this subset coincides with the pre-image of 퐽 under the map 푓표휑−1: 휑(푈) → ℝ, which is a smooth function on an open subset of ℝ푛 and these are continuous. 4. The derivative of a smooth map Let 휑: 푀 → 푁 be a smooth map. Recall that 휑 induces a homomorphism 휑∗: 풴(푀) ← 풴(푁) given by ∗ 휑 푓 푥 = 푓 휑 푥 , 푓 ∈ 풴 푁 , 푥 ∈ 푀. (4.1) ∗ Lemma 4.1 (Narasimhan. 1968) : Let 휉 ∈ 푇푎 푀 . Then 휉 ∘ 휑 ∈ 푇휑(푎) 푁 , and the ∗ correspondence 휉 ⟼ 휉 ∘ 휑 defines a linear map from 푇푎 푀 to 푇휑(푎) 푁 . Proof. 휉 ∘ 휑∗ is a linear map from 풴(푁) to ℝ. Moreover, ∗ ∗ ∗ ∗ ∗ 휉 ∘ 휑 푓푔 = 휉 휑 푓. 휑 푔 = 휉 휑 푓 . 푔 휑 푎 + 푓 휑 푎 . 휉(휑 푔) (4.2) ∗ ∗ (푓, 푔 ∈ 풴 푁 ) and so 휉 ∘ 휑 ∈ 푇휑(푎) 푁 . Clearly 휉 ⟼ 휉 ∘ 휑 is linear. Definition 4.1 (Brickell et al., 1970) : Let 휑: 푀 → 푁 be a smooth map and let 푎 ∈ 푀. ∗ The linear map 푇푎 푀 → 푇휑(푎) 푁 defined by 휉 ⟼ 휉 ∘ 휑 is called the derivative of 휑 ∗ at 푎.
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