Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 136 https://internationalpubls.com Group Actions on Manifolds Prof. T.Venkatesh1 and Smt.Ashma F Ganachari2 1Department of Mathematics, Rani Channamma University, Belagavi and Director, Mathematical Sciences Institute Belagavi, Karnataka,India e-mail: tmathvenky@yahoo.co.in 2Department of Mathematics, (Research Scholar) ,RCU Belagavi And Assistant Professor , P.C.Jabin Science College Vidyanagar, Hubballi. Karnataka, India g-mail: ashmagmath786@gmail.com Article History: Received: 28-09-2023 Revised: 07-11-2023 Accepted: 29-11-2023 Abstract: In this paper some Riemann Geometry aspects will be gathered with classical time, the study naturally concentrate to PDE’s from the relation outside geometry. The classical frame work of differential geometry (intending smooth manifolds) and later it endured with Riemannian metric is briefly described in one of the sections that were formed to be essential for our understanding of the inner structure of the space. Keywords: Diffeomorphism, Locally integrable structures and Integral curves. 1. Introduction In this paper we discuss the inner structure of the space, we mean a smooth manifold of dimension , where .Naturally, it sets a pace with geometry ever since the human civilization started confronting the enchanting beauty of the nature in which geometry manifests. If one were to specifically mention its historical anecdotes, then it is certainly of Greek times. Euclid’s axiomatic approach to mathematics and in particular his flat geometry model. Space is curved is altogether another revelation and that guides us to move forward with the agenda. But this picture underwent radical changes during 19th century, we notice these changes in the significant work of Gauss and Riemann, he was a student of Gauss. Infact, Riemann revolutionized our notions of space and liberating mathematics from its Euclidean sholders. That laed to believe formally that objects no longer lead to be confined to the flat, linear space of Euclidean geometry. Riemann proposed a much more abstract conception of space , of dimension for any in which we could describe distance and curvature and a form of calculus that intend to this idea of abstract space, Riemann geometry base the stump, as non- Euclidean geometry, and get in a standard Euclidean spirit. Any investigation /research in geometry in the middle of 20th century lead to forever, Einstein realized that this kind of geometry , which involved curved spaces with exactly what was needed by him to unity geometry (Newton’s) with special relativity and that later lead to the famous theory of general relativity. Under classification theme we distinguish all 2-dimensional oriented closed manifolds. Indeed this has been a classical problem of topology and was successfully done by the people in 1940’s later, this study lead to higher analogies. Here, we shall give a simple treatment of Riemann’s idea of presenting a surface, from topological view point, to begin with a sphere S2 and torus T2 both are oriented closed 2-dimensional manifolds and are least homoeomorphic to each other. Other part Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 137 https://internationalpubls.com of discussion is vector fields and some inerrability related issues; this is an important and deeply studied theme of geometry under Riemannian metric. The ambient space being either or over or (as the case may be ) a word about low – dimensional topology and finite space (infinite dimensional space) is not out of place, rather it is relevant we come across linear groups, to address the stability issues relating to their dynamics (Dynamical systems, perturbations , etc seen from classical mechanics). 2.Differentiable manifolds and some examples Let M be a smooth manifold of dimension , .Then locally we have , coordinate chatrs , which are smooth and on thus ( pairs as compertable , when it comes to for different and the transformation maps being smooth the following diagram clarifies these issues. : is smooth.Similarly, we can take the other one as well i.e being smooth.To make things simple we shall use notation instead , and , , is open in some times we keep switching from one notion to the other , without any compassion . 2.1 Definition:Diffeomorphisms:Diff is a group under composition of maps and if G is any group then is an important group of homomorphisms. i.e then , where the map is an automorphism (group action naturally arise in this fashion). Observe that and things go inthat way .i.e is at the map is at the map i.e as which we denote it by i.e , 2.2 Proposition:Diffeomorphism group (M) is smooth in fact it is a lie group. Proof:A lie group is a group and at the same times a topological space. For our study we confine to and its subgroups, which are smooth manifolds at the same time. S. T. Yaninitiated and came up without studying results on PDE’s and their intense connections with the geometry of the underlying manifold. Recently,there are results from the study on weak and strong unique continuationfor systems of linear and non-linear PDE’s which arise as sections of a vector sub- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 138 https://internationalpubls.com bundle of the complex tangent bundle which we take it as of and , its cotangent versions, orthogonal to and is locally generated as exact forms (intigrebilityissues,incorporated). Firstly, we shall deal the local integrability for the sub-bundle ϑ of and provide explicit expressions for the basis a local basis of ϑ over a neighbourhood for each point in . 3. Formulation of local integrable structure Since, PDE’s arise as sections of vector bundles rather sub-bundles of vector bundles where is a connected smooth manifold. The bundle satisfies the involutionly condition ϑ ………… (1) The usual [ , ] lie bracket of sections in the sub-bundle ϑ of the complexified tangent bundle of .If are sections of ϑ, the lie bracket [ is also section of ϑ. We always assume that ϑ is locally integrable.Then there exists sections which are solutions of in U ……….(2) And { are linearly independent over at each part of areterm { a complete set of first integrals on . Since ϑ satisfies local integrability condition hence it satisfies the involutionly conditionalso. Thus we refer to the pair as a locally integrable structure. In such a structure, given any point these are local co-ordinates , (i.e , for the sections which are satisfies of (2), vanishing at such that ϑ is generated locally by basis of the form ( ………….(3). 3.1 Examples: For the non-linear systems ), there exists local co-ordinates . In which the equations take the form ……..(4) Following are some examples of locally integrable structures, first one is for and the later one . (a) Let be smoothlinearly independent vector fields on a domain such that the lie bracket [ is in the linear span of Let ϑ denote the sub bundles of enerated by these vector fields. By Frobenius theorem, each is centre of local coordinates in which the bundles is locally generatedby . Hence is locally integrable and in these coordinates any solution has the form, . For the case, , we have the following description. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 139 https://internationalpubls.com (b) We know that ≃ hence, the basis in this case will be 2-times the linearly independent set of vector field (as noticed in the real case) .Then ϑ is the associated bundle generated by . The co-ordinate functions are complete set of first integrals and so ( ϑ) is locally integrable. Here the solutions are holomorphic functions. 3.2Proposition:Letm , as earlier and then give rise to a locally integrable system ( ϑ), where the basis is generated by linearly independent real analytic and complex valued coefficient on which is complex valued. Proof: On the same lines of example (a) and holomorphicversion of Frobeniustheorem, the properties follow. 4. The locally integrable structures and integral curves Here the setting is and sections are interms of integral curves, generating them. Let be a smooth curve, such that (i) M (ii) For and , is locally defined smooth vector field for each is an integral curve as the case may be we assume that M is orientable). We shall be interested in thus for in is an non singular matrix with real entries. For an open set of as neighborhood for each in , we shall consider its , where and (( ϑ) a locally integrable structure. For connected we shall, simply deal with its analogue i.e i.e and corresponding locally integrable structure on (( ϑ). 4.1 Definition: The locally integrable structure is said to satisfies the weak unique continuation property if any solution that vanishes on a non-empty open subset vanishes on . 4.2 Definition:The locally integrable structure satisfies the strong unique continuation property if any solution that is flat at a point vanishes on . The validity of weak unique continuation property both for linear and non-linear systems is connected with the orbits of the system, which is a very useful geometric objects associated with the given family of real vector fields .The above setting of linear local vector field are the formal setting for this explanation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 140 https://internationalpubls.com The following technical deposits provide the construction of such orbits, for more clarity and clear understanding of the motions .This is one side of the study, the real objective in global theory i.e Globally integrable systems. Before we take up them we will present the following theorem on local integrability structures. From the standard analogues the following theorem in an important result. 4.3 Theorem:Let be a locally integrable structure and set, is a smooth section of ϑ}. If is a solution on that vanishes in a neighborhood of then it vanishes in a neighbourhood of the Sussman orbit through .Thus, the support of is a union of orbits of .In particular, if is an orbit, then satisfies the weak unique continuationproperty. 4.4 Example: Let be locally integrable and suppose at each , the linearspace of all of the repeated brackets of sections of equals Then is the only orbit of and so by the above theorem, the weak unique continuation property holds for These are explains of locally integrable structures, where only orbit of although the hypothesis in the above example is violated. 5. Conclusion P.Cohen gives an example for a smooth vector field in with a smooth solution on , for whose support is for ,this is a consequence of certain group acting on the upper local of place of . That is locally integrable structure and maximally real sub manifolds of provide some interesting results characterizing holomorphic maps. And a useful application of locally integrable structure on unique weak continuation. References [1] Eckhard Meinrenken – Group actions on manifolds –university of Toronto, spring 2003.. [2] S. Berhanu, P.D. Cordaro, J. Hounie, An introduction to involutive structures, Cambridge University Press (2008). [3] S. Berhanu, J. Cordaro, J. Hounie, Uniqueness for locally integrable solutions of overdetermined systems, Duke Math. Journal, 105 no.3 (2000), 387-410. [4] M. S. Baouendi , F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 113 (1981), 387-421. [5] E.M. Chirka, Introduction to the geometry of CR manifolds, Russian Math. Surveys 46:1 (1991), 95-197. [6] J. Jost, Riemannian Geometry and Geometric Analysis, Springer 2011. [7] H. Sussmann, Orbits of families of vector fields and integrability of systems with singularities, Bull. Am.Math.Soc. 79, No. 1 (1973), 197-199. [8] S. 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