Local properties and differential forms of smooth map and tangent bundle
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Local properties and differential forms of smooth map and tangent bundle Local properties and differential forms of smooth map and tangent bundle Document Transcript

  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org Local Properties and Differential Forms of Smooth Map and Tangent Bundle Md. Shafiul Alam 1 , Khondokar M. Ahmed 2 & Bijan Krishna Saha 3 1 Department of Mathematics, University of Barisal, Barisal, Bangladesh 2 3 Department of Mathematics, University of Dhaka, Dhaka, Bangladesh Department of Mathematics, University of Barisal, Barisal, Bangladesh 1 Corresponding author: Email: shafiulmt@gmail.com, Tel: +88-01819357123 Abstract In this paper, some basic properties of differential forms of smooth map and tangent bundle are d e fi n ed b y is t he d eri va tive of at a, developed. The li n ear m ap where is a smooth map and . If is a smooth bijective ma p and if the maps are all injective, then is a diffeomorphism. Finally, it is shown that if X is a vector field on a manifold M then there is a radial neighbourhood gh of in and a smooth map such that and , where is given by . Keywords: Smooth map, manifold, tangent bundle, isomorphism, diffeomorphism, vector field. 1. Introduction S. Kobayashi1 and K. Nomizu1 first worked on differential forms of smooth map and tangent bundle. Later H. Flanders2, F. Warner3, M. Spivak4 and J. W. Milnor5 also worked on differential forms of smooth map and tangent bundle. A number of significant results were obtained by S. Cairns 6, W. Greub9, E. Stamm9, R. G. Swan10, R. L. Bishop12, R. J. Crittenden12 and others. Let M be a smooth manifold and let S (M) be the ring of smooth functions on M. A tangent vector of M at a point is a linear map S (M) such that S (M). under the linear operations The tangent vectors form a real vector space S (M). is called the tangent space of M at a. Let E be an n-dimensional real vector space. Let be an open subset of E and let define a linear isomorphism . If is a smooth map of sp ace F, then the classic al d erivative o f at a is the linear map Moreover, in the special case . We shall into a second vector given by we have the product formula S (O). S (O) given by This shows that the linear map of O at a. Hence, we have a canonical linear map is a tangent vector given by . 2. The Derivative of a Smooth Map Let be a smooth map and let . The li nea r map is ca ll ed t h e d er i va tive of at a. It will be denoted by , 101 d efi ned b y
  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org S (N), If map is a second smooth map, then we have and Lemma 1 . Let to map from Proof. Let . . Moreover, for the identity . In particular, if is a diffeomorphism, then are inverse linear isomorphisms. , then . and the corre spondence be a smooth map. S (N), . defines a linear S (N) induces a homomorphism S (M) given by is a linear map from S (N) to . Moreover, S (N) and so . Clearly is linear. Therefore, the lemma is proved. be a basis for Proposition 1. Let (a) the functions the functions S ( ). Then S ( ) are given by (b) and let S ( ) satisfy Proof. By the fundamental theorem of calculus we have Thus the theorem follows, with Proposition 2. The canonical linear map . is an isomorphism. Proof. First we consider the case . We show first that is surjective. Let and let S where the satisfy conditions (a) and (b) of Proposition 1. ( ). Write Since maps the constant function into zero, so Since the functions are independent of f, we can write surjective. To show that is injective, let be any linear function in . Then for Now suppose . Then . Hence and so is injective. Finally, let be any open subset of E and let identity map and we have the commutative diagram be the inclusion map. Then 102 . Thus , is . is the
  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org is a linear isomorphism, the canonical linear map Since completes the proof. . Then dim Corollary 1. Let M be a smooth manifold and let Proof. Let we find dim is an isomorphism, which be a chart for M such that = dim = dim = dim . = dim m . Using the result of Lemma 1 and Proposition 2 be a smooth map Proposition 3. The derivative of a constant map is zero. Conversely, let Then, if M is connected, is a constant map. such that Proof. Assume that function given by follows that each is the constant map where . Hence, for . S ( ) is the constant . It . Then, for , is a smooth map satisfying and let be Conversely, assume that connected. Then, given two points and , there exists a smooth curve such and . Consider the map . We have that . Now using an atlas for N we see that must be constant. In particular . Thus is a constant map and the proposition is proved. and so 3. Local Properties of Smooth Maps be a smooth map. Then is called a local diffeomorphism at a point is a linear isomorphism. If is a local diffeomorphism for all points called a local diffeomorphism of M into N. if the map , it is Theorem 1. Let point. Then be a given Let be a smoo th map where d im M = n and dim N = r. Let (a) If is a local diffeomorphism at a, there are neighbourhoods U of a and V of b such that maps U diffeomorphically onto V. (b) If is injective, there are neighbourhoods U of a, V of b, and such that a diffeomorphism (c) If of 0 in . is surjective, there are neighbourhoods U of a, V of b, and W of 0 in a diffeomorphism the projection. and such that where , and is . In part (a), then, we are Proof. By using charts we may reduce to the case assuming that is an isomorphism, and the conclusion is the inverse function theorem. For part (b), we choose a subspace E of given by such that , and consider the map . Then 103
  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org . It follows that i s i n j e c t i v e a n d t h u s a n i s o m o r p h i s m (r = dim Im + dim E = n + dim E ). Thus part (a) implies the existence of neighborhoods U of a, V of b, and of 0 in E such that is a diffeomorphism. Clearly, such that Finally, for part (c), we choose a subspace E of projection induced by this decomposition and define . Let . It follows easily that Then Hence there are neighborhoods U of a, V of b and W of diffeomorphism. Hence the proof of the theorem is complete. be the by is a linear isomorphism. such that is a smooth bijective map and if the maps Proposition 4. If all injective, then is a diffeomorphism. is a are is inj ective, we have . Now we show that r Proof. Let d im M = n , dim N = r. Since = n. In fact, according to Theorem 1, part (b), for every there are neighborhoods of a, V of and of together with a diffeomorphism such that the diagram commutes (i denotes the inclusion map opposite 0). of such that each is compact and contained Choose a countable open covering in some . Since is surjective, it follows that . Now assume that r > n. Then the diagram implies that no contains an open set. Thus, N could not be Hausdorff. This contradiction shows that . Since , is a local diffeomorphism. On the other hand, is bijective. Since it is a local diffeomorphism, Theorem 1 implies that its inverse is smooth. Thus is a diffeomorphism and the theorem is proved. Definition 1. A q uotient manifold o f a manifold M is a manifold N together with a smooth map such that and each linear map is surjective and thus dim M dim N. Lemma 2. Let make N into a quotient manifold of M. Then the map is open. , there is a neighbourhood U of a such that the restriction Proof. It is sufficient to show that, for each of to U is open. This follows at once from part (c) of Theorem 1. 4. Submanifolds Let M be a manifold. An embedded manifold is a pair , where N is a second manifold and is a smooth map such that the derivative is injective. In particular, since the maps are injective, it follows that dim N dim M. Given an embedded manifold , the subset may be considered as a bijective map . A submanifold of a manifold M is an embedded manifold such that is a homeomorphism, when is given the topology induced b y the topology of M. If N is a subset o f M and is the inclusion map, we say simply that N is a submanifold of M. 104
  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 Proposition 5. Let www.iiste.org be a submanifold of M. Assume that Q is a smooth manifold and is a commutative diagram of maps. Then is smooth if and only if is smooth. Proof. If is smooth then clearly is smooth. Conversely, assume that is smooth. Fix a point a Q and set . Since is injective, there are neighbourhoods U, V of b in N and M respectively, such that . and there is a smooth map Since N is a submanifold of , the map is continuous. Hence there is a neighbourhood W of a such that . Then , where denote the restrictions of to W. It follows that and so is smooth in W; thus is a smooth map. Hence the proposition is proved. 5. Vector Fields A vector field X on a manifold M is a cross-section in the tangent bundle . Thus a vector field X a tangent vector such that the map so obtained is smooth. assigns to every point The vector fields on M form a module over the ring S ( ) which will be denoted by . onto the S ( )-module of derivations in the Theorem 2. There is a canonical isomorphism of algebra S ( ), i.e., Der S ( ). Proof. Let be a vector field. For each S ( ), we define a function . is smooth. To see this we may assume that is smooth. Hence every vector field X on M determines a map S( ) Obviously, S ( ) given by . is a derivation in the algebra S ( ). The assignment Der S ( ). We show now that is an isomorphism. Suppose on M by . But then , for some defines a homomorphism . Then S ( ). This implies that Then, for every point ; i.e. X = 0. To prove that , determines the vector is surjective, let , given by be any derivation in S ( ). S ( ). . T o sho w that this map is s mo o th, fix a point . Using a construct vector fields and smooth functions on M such that , (V is some neighbourhood of a). Then the vectors form a basis for . Hence, for each , there is a unique system of numbers such that . Applying to we obtain . Hence . Since the are smooth functions on M, this equation shows that X is smooth in V; i.e. X is a vector field. Finally, it follows from the definition that . Thus is surjective, which completes the proof. We d efine chart, it is easy by to 6. Differential Equations Let X be a vector field on a manifold M. An orbit for X is a smooth map 105 where is some
  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 open interval such that www.iiste.org . Co n sid er t h e p ro d u ct m an i fo ld . W e cal l a s ub se t or where is an open interval on union and finite intersection of radial sets is again radial. and Proposition 6. Let X be a vector field on M. Fix of X such that . Moreover, if an orbit some then . Proof. For the first statement of the proposition we may assume stand ard P icard existenc e theo rem. radial if for each , containing the point 0. The . Then there is an interval and are orbits for X which agree at . In this case it is the for which is both closed and To prove the second part we show that the set of open, and hence all of . It is obviously closed. To show that it is open we may assume and then we apply the Picard uniqueness theorem, which completes the proof. Theorem 3. Let X be a vector field on a manifold M. Then there is a radial neighbourhood in and a smooth map such that of and where is given by . Moreover, is uniquely determined by X. Proof. Let be an atlas for M. The Picard existence theorem implies our theorem for each . Hence there are radial neighbourhoods of in and there are smooth maps and . such that . Then is a radial neighbourhood of in . Moreover, is a Now set ; if , then is an interval radial neighbourhood of containing . Clearly, are orbits of X agreeing at 0, and so by Proposition 6 they agree in . It follows that they agree in . Thus the defines a smooth map which follows from Proposition 6. Thus the theorem is proved. has the desired properties. The uniqueness of 7. Conclusions Differential forms are among the fundamental analytic objects treated in this paper. The derivative of a smooth map appears as a bundle map between the corresponding tangent bundles. They are the cross-sections in the exterior algebra bundle of the dual of the tangent bundle and they form a graded anticommutative algebra. Vector fields on a manifold has been introduced as cross-sections in the tangent bundle. The module of vector fields is canonically isomorphic to the module of derivations in the ring of smooth functions. References 1. Kobayashi, S. and K. Nomizu, 1963. Foundations of Differential Geometry, Vol. I, Wiley, New York, 51-57. 2. Flanders, H., 1963. Differential Forms, Academic Press, New York, 44-49. 3. Warner, F., 1971. Foundations of Differentiable Manifolds and Lie Groups, ScottForesman, Glenview, Illinois, 29-38. 4. Spivak, M., 1970. A Comprehensive Introduction to Differential Geometry, Brandeis Univ., Waltham, Massachusetts, 37-42. 5. Milnor, J. W., 1965. Topology from the Differentiable Viewpoint, Univ. Press of Virginia, Charlottesville, 87-95. 6. Cairns, S., 1965. Differential and Combinatory Topology, Princeton Univ. Press, Princeton, New Jersey, 76-82. 106
  • Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org 7. Alam, M. S., 2009. Manifolds with Fibre, Vector, Tangent Bundles and De Rham Cohomology, MS thesis, Department of Mathematics, University of Dhaka, 33-44. 8. Hermann, R., 1960. A sufficient condition that a mapping of Riemannian spaces be fibre bundle, Proc. Amer. Math. Soc. 11, 226-232. 9. Greub, W. and E. Stamm, 1966. On the multiplication of tensor fields, Proc. Amer. Math. Soc. 17, 1102-1109. 10. Swan, R. G., 1962. Vector bundles and projective modules, Trans. Amer. Math. Soc. 105, 223-230. 11. Narasimhan, R., 1968. Analysis on Real and Complex Manifolds, North-Holland Publ., Amsterdam, 110-115. 12. Bishop, R. L. and R. J. Crittenden, 1964. Geometry of Manifolds, Academic Press, New York, 212-220. 13. Husemoller, D., 1966. Fibre Bundles, McGraw-Hill, New York, 39-44. 14. Ahmed, K. M. and M. S. Rahman, 2006. On the Principal Fiber Bundles and Connections, Bangladesh Journal of Scientific and Industrial Research. 15. Ahmed, K. M., 2003. A note on Affine Connection on Differential Manifold, The Dhaka University Journal of Science 107
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