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

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∞ 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 | 푓 훾 푡 = | 푓표휑−1 ( 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 | 푓 훾 푡 = | 푓표휑−1 (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.

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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. It is denoted by (푑휑)푎 , 푑휑 푎 휉 푔 = 휉 휑 푔 , 푔 ∈ 풴 푁 , 휉 ∈ 푇푎 푀 .

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If 휓: 푁 → 푄 is a second smooth map, then

(푑(휓 ∘ 휑))푎 = (푑휓)휑(푎) ∘ 푑휑 푎 , 푎 ∈ 푀. Moreover, for the identity map 휄: 푀 → 푀, we have

(푑휄)푎 = 휄푇푎 푀 , 푎 ∈ 푀 In particular, if 휑: 푀 → 푁 is a diffeomorphism, then

−1 푑휑 푎 : 푇푎 푀 → 푇휑(푎) 푁 and (푑휑 )휑(푎): 푇휑(푎) 푁 → 푇푎 푀 . (4.3) are inverse linear isomorphisms.

Example 4.1 Let 휑: 푀 → 푁 be a smooth map which sends a neighbourhood 푈 of a point 푎 ∈ 푀 diffeomorphically onto a neighbourhood 푉 of 휑(푎) in 푁. Then

푑휑 푎 : 푇푎 푀 → 푇휑(푎) 푁 (4.4) is a linear isomorphisms.

5. Tangent bundle In this section we shall discuss properties of smooth map using the concept of tangent bundle. Finally we state our theorem for determining the characteristics of subjective map using the definition of derivative of smooth map. We also give the definition of quotient manifold and sub manifolds by applying the definition of smooth map on manifold. Definition 5.1 (Brickell et al., 1970) : Let 푀 be a 푛-manifold. Consider the disjoint union 푇푀 = 푎∈푀 푇푎 푀 , and let 휋푀 ∶ 푇푀 → 푀 be the projection defined by

휋푀 휉 = 푎, 휉 ∈ 푇푎 푀 . 푛 Then we define a manifold structure on 푇푀 so that 휏푀 = (푇푀, 휋푀, 푀, ℝ ) is a vector bundle over 푀, whose fibre at a point 푎 ∈ 푀 is the tangent space 푇푎 푀 . Then 휏푀 is called the tangent bundle of 푀.

Let (푈훼 , 푢훼 , 푈 훼 ) be a chart for 푀 and let 푗훼 : 푈훼 → 푀 be the inclusion map. For each 푥 ∈ 푈훼 there are linear isomorphisms ≅ 푛 휆푢훼 (푥): ℝ → 푇푢훼 푥 (푈훼 ) ≅ −1 (푑푢훼 )푥 : 푇푢훼 푥 푈훼 → 푇푥 (푈훼 ) (5.1) and ≅ (푑푗훼 )푥 : 푇푥 푈훼 → 푇푥 (푀). (5.2)

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≅ 푛 Composing them we obtain a linear isomorphism 휓훼,푥 : ℝ → 푇푥 (푀).

Finally, let {(푈훼 , 푢훼 )} be an atlas for 푀. Define maps 휓훼 : 푈훼 × ℝ → 푇푀 by 푛 휓훼 푥, 푕 = 휓훼,푥 푕 , 푥 ∈ 푈훼 , 푕 ∈ ℝ . −1 If 푈훼 ∩ 푈훽 ≠ ∅ and 푢훽훼 = 푢훽 ∘ 푢훼 , the map −1 푛 푛 휓훽훼 = 휓훽 ∘ 휓훼 : 푈훼 ∩ 푈훽 × ℝ → 푈훼 ∩ 푈훽 × ℝ ′ given by 휓훽훼 푥, 푕 = (푥, 푢 훽훼 (푢훼 푥 ; 푕)). Hence it is smooth. According to the construction of vector bundles, there is a unique vector bundle

푛 휏푀 = (푇푀, 휋푀, 푀, ℝ ) for which {(푈훼 , 휓훼 )} is a coordinate representation. The fibre of this bundle at 푥 ∈ 푀 is the tangent space 푇푥 (푀). Evidently this bundle structure is independent of the choice of atlas for 푀.

Example 5.1 If 푂 is an open subset of vector space 퐸, then the tangent bundle 휏푂 is isomorphic to the product bundle 푂 × 퐸. In fact, define a map 휆: 푂 × 퐸 → 푇푂 by setting

휆 푎, 푕 = 휆푎 푕 , 푎 ∈ 푂, 푕 ∈ 퐸, where 휆푎 is the canonical linear map. Then 휆 is a strong bundle isomorphism.

Next, suppose 휑: 푀 → 푁 is a smooth map. Then a set map 푑휑: 푇푀 → 푇푀 is defined by

푑휑 휉 = (푑휑)푥 휉, 휉 ∈ 푇푥 푀 , 푥 ∈ 푀. It is called the derivative of 휑.

Proposition 5.1 (Ali et al., 2012) : The derivative of a smooth map 휑: 푀 → 푁 is a bundle map 푑휑: 휏푀 → 휏푁. Proof. It follows from the definition that 푑휑 is fibre preserving and that the restriction of 푑휑 to each fibre is linear. To show that 푑휑 is smooth we use atlases on 푀 and 푁 to reduce the case 푀 = ℝ푛 , 푁 = ℝ푝. In this case definition 2.5 shows that 푑휑: ℝ푛 × ℝ푛 → ℝ푝 × ℝ푝 is given by

푑휑 푥; 푕 = (휑 푥 ; 휑′(푥; 푕)). (5.3) Hence it is smooth.

Now let 휓: 푁 → 푄 be a smooth map into a third manifold. Then

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푑 휓 ∘ 휑 = 푑휓 ∘ 푑휑 as follows from the definition. Moreover, the derivative of the identity map 휄: 푀 → 푀 is the identity map of 푇푀,

푑휄푀 = 휄푇푀 . (5.4) It follows that if 휑: 푀 → 푁 and 휓: 푀 ← 푁 are inverse diffeomorphisms, then 푑휑 and 푑휓 are inverse bundle isomorphisms.

6. Properties of smooth maps on tangent bundle Let 휑: 푀 → 푁 be a smooth map. Then 휑 is called a local diffeomorphism (respectively an immersion, a submersion) at a point 푎 ∈ 푀 if the map

(푑휑)푎 : 푇푎 푀 → 푇휑(푎) 푁 (6.1) is a linear isomorphism (respectively injective, subjective). If 휑 is a local diffeomorphism (respectively an immersion, a submersion) for all points 푎 ∈ 푀, it is called a local diffeomorphism (respectively an immersion, a submersion ) of 푀 into 푁. Theorem 6.1 (Brickell et al., 1970) : Let 휑: 푀 → 푁 be a smooth map where dim 푀 = 푛 and dim 푁 = 푟. Let 푎 ∈ 푀 be given a point. Then (a) If 휑 is a local diffeomorphism at 푎, there are neighbourhoods 푈 of 푎 and 푉 of 푏 such that 휑 maps 푈 diffeomorphically onto 푉.

(b) If (푑휑) is injective, there are neighbourhoods 푈 of 푎 and 푉 of 푏, and 푊 of 푂 in 푎 ≅ ℚ푛−푟 , and a diffeomorphism 휓: 푈 × 푊 → 푉 such that 휑 푥 = 휓 푥, 0 , 푥 ∈ 푈.

(c) If (푑휑) is surjective, there are neighbourhoods 푈 of 푎 and 푉 of 푏, and 푊 of 푂 in 푎 ≅ 푛−푟 ℚ , and a diffeomorphism 휓: 푈 → 푉 × 푊 such that 휑 푥 = 휋푉휓 푥 , 푥 ∈ 푈, where 휋푉: 푉 × 푊 → 푉 is the projection. Proof. By using charts we may reduce to the case 푀 = ℚ푛 , 푁 = ℚ푟 in part (a). Then, we are assuming that 휑′ 푎 : ℚ푛 → ℚ푛 is an isomorphism, and the conclusion in the inverse function theorem.

For part (b), we choose a subspace 퐸 of ℚ푟 such that Im 휑′ 푎 ⨁ 퐸 = ℚ푟 . (6.2) and consider the map 휓: ℝ푛 × 퐸 → ℝ푟 given by 휓 푥, 푦 = 휑 푥 + 푦, 푥 ∈ ℚ푛 , 푦 ∈ 퐸.

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Then

휓′ 푎, 0; 푕, 푘 = 휑′ 푎; 푕 + 푘, 푕 ∈ ℚ푛 , 푘 ∈ 퐸 (6.3)

It follows that 휓′ 푎, 0 is injective and thus an isomorphism (푟 = dim Im 휑′ 푎 + dim 퐸 = 푛 + dim 퐸).

Thus part (a) implies the existence of neighborhoods 푈 of 푎 and 푉 of 푏, and 푊 of 푂 in 퐸 such that 휓: 푈 × 푊 → 푉 is a diffeomorphism. Clearly, 휓 푥, 0 = 휑 푥 .

Finally, for part (c), we choose a subspace 퐸 of ℚ푟 such that ker 휑′ 푎 ⨁ 퐸 = ℚ푛 .

Let 휌: ℚ푛 → 퐸 be the projection induced by this decomposition, and define 휓: ℚ푛 → ℚ푟 ⨁ 퐸 by 휓 푥 = 휑 푥 , 휌 푥 , 푎 ∈ ℚ푛 , 푕 ∈ ℚ푛 . (6.4)

It follows easily that 휓′(푎) is a linear isomorphism. Hence there are neighbourhoods 푈 of 푎 and 푉 of 푏, and 푊 of 0 ∈ 퐸 such that 휓: 푈 → 푉 × 푊 is a diffeomorphism.

Proposition 6.1 : If 휑: 푀 → 푁 be a smooth bijective map and if the maps

(푑휑)푥 : 푇푥 푀 → 푇휑(푥) 푁 are all surjective, then 휑 is a diffeomorphism.

Proof. Let dim 푀 = 푛 and dim 푁 = 푟. Since (푑휑)푥 is surjective, we have 푟 ≥ 푛. Now we show that 푟 = 푛. For every 푎 ∈ 푀 there exist a neighborhoods 푈 of 푎 and 푉 of 푏, and ≅ 푛−푟 푊 of 0 ∈ ℝ together with a diffeomorphism 휓푎 : 푈 푎 × 푊 → 푉 such that the diagram 1

푈 푎 × 푊 휓푎 푉

푖 휑

푈 푎 Diagram 1: Composite map commutes (푖 denotes the inclusion map opposite 0)

Choose a countable open covering 푈푖 (푖 = 1, . . , 푛) of 푀 such that each 푈 푖 is compact and contained in some 푈(푎푖). Since 휑 is surjective, it follows that 푈푖 휑( 푈 푖) ⊃ 푁.

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Now assume that 푟 > 푛. Then the diagram implies that no 휑( 푈 푖) contains an open set. Thus, by the category theorem 푁 could not be Hausdorff. This contradiction shows that 푛 = 푟. Since 푛 = 푟, 휑 is a local diffeomorphism. On the other hand, 휑 is bijective. Since it is a local diffeomorphism and its inverse is smooth. This implies that 휑 is a diffeomorphism. 7. Quotient manifold A quotient manifold of a manifold 푀 is a manifold 푁 together with a smooth map 휋: 푀 → 푁 such that 휋 and each linear map (푑휋)푥 : 푇푥 푀 → 푇휋(푥) 푁 is surjective and thus dim 푀 ≥ dim 푁. Proposition 7.1 (Narasimhan. 1968) : Let 휋: 푀 → 푁 make 푁 into a quotient manifold of 푀. Assume that 휑: 푀 → 푄, 휓: 푁 → 푄 are maps into a third manifold 푄 such that the diagram 2 푀 휑 푄

휋 휓

푁 Diagram 2: Smooth map commutes. Then 휑 is smooth if and only if 휓 is smooth. 8. Sub manifold Let 푀 be a manifold. An embedded manifold is a fair (푁, 휑), where 푁 is a second manifold and 휑: 푁 → 푀 is a smooth map such that the derivative 푑휑 = 푇푁 → 푇푀 is injective. In particular, since the maps (푑휑)푥 : 푇푥 푀 → 푇휑(푥) 푁 are injective, it follows that dim 푁 ≤ dim 푀.

Given an embedded manifold (푁, 휑), consider the subset 푀1 = 휑(푁). 휑 may be considered as a bijective map

휑1: 푁 → 푀1.

This bisection defines a smooth structure on 푀1, such that 휑1 becomes a diffeomorphism.

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Definition 8.1 (Narasimhan. 1968) : A submanifold of a manifold 푀 is an embedded manifold (푁, 휑) such that 휑1: 푁 → 휑(푁) is a homomorphism, when 휑(푁) is given the topology induced by the topology of 푀. If 푁 is a subset of 푀 and 휑 is the inclusion map, we say simply that 푁 is a submanifold of 푀. Not every embedded manifold is a submanifold, as the following example shows: Let 푀 be the 2-torus 푇2 and let 푁 = ℝ. Define a map 휑: ℝ → 푇2 by 휑 푡 = 휋 푡, 휆푡 , 푡 ∈ ℝ, where 휆 is an irrational number and 휋: ℝ2 → 푇2 denotes the projection. Then

푑휑: 푇ℝ → 푇푇2 is injective and so (ℝ, 휑) is an embedded manifold. Since 휆 is an + 2 irrational, 휑(ℝ ) is dense in 푇 . In particular there are real numbers 푎푖 > 0 such that 2 휑(푎푖 ) → 휑(−1). Thus 푇 does not induce the standard topology in 휑(ℝ). Proposition 8.1 (Kobayashi et al., 1963) : Let (푁, 푖) be a submanifold of 푀. Assume that 푄 is a smooth manifold and

푄 휑 푁

휓 푖

푀 Diagram 3: Commutative map is a commutative map. Then 휑 is smooth if and only if 휓 is.

Proof. If 휑 is smooth then clearly so is 휓 is. Conversely, assume that 휓 is smooth. Fix a point 푎 ∈ 푄 and set 푏 = 휓(푎). Since 푑푖 is injective, there are neighborhoods 푈, 푉 of 푏 in

푁 and 푀, respectively, and there is a smooth map 휒: 푉 → 푈 such that 휒 ∘ 푖푈 = 휄. Since 푁 is a submanifold of 푀, the map 휑 is continuous. Hence there is neighborhood 푊 of 푎 such that 휑(푊) ⊂ 푈. Then 푖푈 ∘ 휑푊 = 휓푊, where 휑푊, 휓푊 denote the restrictions of 휑, 휓 to 푊. It follows that 휒 ∘ 휓푊 = 휒 ∘ 푖푈 ∘ 휑푊 = 휑푊 and so 휑 is smooth in 푊; thus 휑 is a smooth map.

56 Barishal University Journal Part 1, 5(1&2): 47-58 (2018) Ahmed et al.

9. Conclusion

The main objective of this study is finding existence of the diffeomorphism smooth map on manifold. In order to achieve this result we described some propositions and related lemmas to illustrate the theorem. Our result discussed here on the basis of derivative of smooth map on manifolds. Finally we can say that the generalization of the theorem 6.1 can be used in further study on tangent bundle of manifolds for finding the existence of diffeomorphism smooth map.

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57 Barishal University Journal Part 1, 5(1&2): 47-58 (2018) A Study of smooth map on manifolds

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