Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 26 https://internationalpubls.com On the Geometry of Grassmannian Manifold [ T. Venkatesh 1, Shruti Kamalakar Govekar 2 1Department of Mathematics, and Director, Mathematical Sciences Institute Belagavi , Karnataka,India. e-mail: tmathvenky@yahoo.co.in 2Department of Mathematics, Rani Channamma University, Belagavi, Karnataka, India e-mail: shru16_shruti@rediffmail.com Article History: Received: 26-02-2024 Revised: 28-04-2024 Accepted: 12-05-2024 Abstract: We start with basics in Differential manifold such as complex manifolds, tangent space to a manifold, complex sub manifolds and sub varieties and specified their generalizations to projective spaces with its rich topological and smooth manifold structure. Keywords: Complex manifolds, tangent space, Sub manifold, Sub varieties. 1. Introduction The core group of compact complex manifolds is known as the Grassmannians. "They could be viewed as an extension of projective space as well. We'll define grassmannians formally here. Understanding their individual structures is crucial for both geometric and topological analyses of grassmannians. 1.1 Definition : For this purpose, we may write 𝐺(π‘˜, 𝑛) for 𝐺(π‘˜, 𝐢𝑛) and define the Grassmannians 𝐺(π‘˜, 𝑉) as the set of π‘˜-dimensional linear subspaces of 𝑉. Let 𝑉 be a complex vector space of size n. To represent Ξ› in 𝐢𝑛, one may use a collection of π‘˜-row vectors in 𝐢𝑛that span a particular k-plane 𝐴. ie by a π‘˜π‘₯𝑛 matrix ( 𝑣11………………………𝑣1𝑛 π‘£π‘˜π‘›β€¦β€¦β€¦β€¦β€¦β€¦β€¦β€¦β€¦π‘£π‘˜π‘› ) of rank k. If ∧ =gβˆ§β€² for some g ∈ πΊπΏπ‘˜ then 𝐴 and 𝐴', two such matrices, may both represent for the same π‘˜ βˆ’ 𝑛 element in 𝐺. Any matrix of this kind clearly represents a point or an element of 𝐺(π‘˜, 𝑛). Let 𝐼 = {𝑖1, ……… π‘–π‘˜} βŠ‚ {1, …… . . , 𝑛} of cardinality π‘˜, then for 𝑉IβŠ‚ 𝐢n , a (𝑛 βˆ’ π‘˜) – plane in 𝐢n be spanned by the vectors {𝑒𝑗: 𝑗 βˆ‰ 𝐼}. 2. Complex manifolds and examples: We define a complex manifold and provide some important examples of Complex manifolds. 2.1 Definition : A differentiable manifold is a multilayered manifold. 𝑀 allows coordinate mappings πœ‘π›Ό:π‘ˆπ›ΌβŸΆπΆ 𝑛 and an open cover {π‘ˆπ›Ό: π›Όπœ–π›¬}, 𝐢 n such that, for all 𝛼, 𝛽 such that πœ‘π›Όπ‘œπœ‘π›½ -1 is holomorphic on πœ‘π›½(π‘ˆπ›Ό ∩ π‘ˆπ›½) βŠ‚ 𝐢 n Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 27 https://internationalpubls.com βˆͺ π‘ˆπ›Ό ∩ π‘ˆπ›½ βŠ‚ π‘ˆπ›Ό πœ‘π›Ό β†’ 𝐢nβŠƒ πœ‘π›Ό(π‘ˆπ›Ό βˆ©π‘ˆπ›½) πœ‘π›½ ↓ ↓ πœ‘π›Όπ‘œπœ‘π›½ -1 𝐢nβŠƒ πœ‘π›½(π‘ˆπ›Ό ∩ π‘ˆπ›½) Our intension, to define holomorphic maps on 𝑀 into 𝑁, where 𝑀 and 𝑁 smooth complex manifolds. 2.2 Definition: On an open set π‘ˆ βŠ‚ 𝑀, a function 𝑓 is holomorphic if and only if π‘“π‘œπœ‘π›Ό -1 is holomorphic on πœ‘π›Ό (π‘ˆβ‹‚π‘ˆπ›Ό)βŠ‚ 𝐢n for all 𝛼. Similarly, a collection 𝑍 = (𝑧1, …… . . , 𝑧𝑛) of functions on π‘ˆ βŠ‚ 𝑀 is called a holomorphic coordinate system if and only if πœ‘π›Όπ‘œπ‘§ βˆ’1 and are π‘§π‘œπœ‘π›Ό -1 holomorphic on 𝑧(π‘ˆβ‹‚π‘ˆπ›Ό) and πœ‘π›Ό (π‘ˆβ‹‚π‘ˆπ›Ό) respectively, for 𝛼. 2.3 Definition : A complex manifold map 𝑓:𝑀 ⟢ 𝑁 is holomorphic if and only if holomorphic functions provide local holomorphic coordinates on 𝑁. (i) A complex manifold that is one-dimensional is referred to as a Riemann surface. (ii) The set of all lines in 𝐢𝑛+1 that intersect at the origin is denoted as 𝑃𝑛. For any 𝑧 β‰  0 ∈ 𝑙 we can write, 𝑝n ={[𝑧] β‰  0 ∈ 𝐢𝑛+1 }/ [𝑧]~[πœ†π‘§] determines a line 𝑙 in 𝐢𝑛+1. A bijective map πœ‘π‘– to 𝐢n is given by πœ‘π‘–([𝑧0, … . , 𝑧𝑛]) = ( 𝑧0 𝑧1 , …… , 𝑧�̂� 𝑧𝑖 , … . 𝑧𝑛 𝑧𝑖 ) and a subset π‘ˆπ‘– of 𝑝n looks like, π‘ˆπ‘– = {[𝑧]: 𝑧𝑖 β‰  0} βŠ‚ 𝑝n of lines not contained in the hyper plane (𝑧𝑖 = 0). On (𝑧𝑗 β‰  0) = πœ‘π‘–(π‘ˆπ‘–β‹‚π‘ˆπ‘—)βŠ‚ 𝐢 n, πœ‘π‘—π‘œπœ‘π‘– -1(𝑧1, … . , 𝑧𝑛) = ( 𝑧1 𝑧𝑗 , …… , 𝑧�̂� 𝑧𝑗 , … 1 𝑧𝑗 , … , 𝑧𝑛 𝑧𝑗 ) is clearly holomorphic. As a consequence of this, n is a complicated manifold that has the structure of a complex projective space. The coordinates that are supplied by the mappings πœ‘π‘– are referred to as Euclidean coordinates, whereas the ''coordinates'' 𝑧 = [𝑧1, … . , 𝑧𝑛] are known as homogeneous coordinates on 𝑝n. Given that there is a continuous surjective map from the unit sphere in 𝐢𝑛+1 to 𝑝n, it is necessary to conclude that pn is compact. It should be noted that β„™1 is simply the Riemann surface 𝐢⋃{𝛼 }. Any reference to an inclusion is caused by πΆπ‘˜+1⟢ 𝐢𝑛+1. The picture of a map such as β„™π‘˜ ⟢ ℙ𝑛 is a linear subspace of 𝑝n. In general, a π‘˜-plane is the image of πΆπ‘˜+1οƒŒ 𝐢𝑛+1, a line is the image of a 2- plane 𝐢2οƒŒ Cn+1, and a hyperplane is the image of a hyperplane in 𝐢𝑛+1again. Now, we can discuss linear relations between points in 𝑝n in this particular situation. 2.4 Example : If the span of a line in 𝑝n is a (π‘˜ βˆ’ 1) -plane, then the line in 𝐢𝑛+1 is said to be linearly independent of k points. Assuming that the image of the subspace in 𝐢𝑛+1 that the lines πœ‹βˆ’1(𝑝𝑖) cover in ℙ𝑛 is the span of a collection of points 𝑝𝑖 in ℙ𝑛 , it is assumed that the span of ℙ𝑛 is the image of the subspace. 2.5 Remark : As a result, the set of hyper planes in ℙ𝑛 is a projective space in and of itself; it is known as the dual projective space and is represented by β„™π‘›βˆ—. It corresponds to the set 𝐢𝑛+1 βˆ’ {0}, of non zero linear functionals on 𝐢𝑛+1 modulo scalar multiplication. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 28 https://internationalpubls.com Sometimes it is advantageous to visualize ℙ𝑛 on the compactifization of 𝐢𝑛 that results from appending the hyper plane H at infinity. Coordinates-wise, the inclusion 𝐢n ⟢ ℙ𝑛 is [𝑧1, … . , 𝑧𝑛] ⟢ [1, 𝑧1, … . , 𝑧𝑛]; here H has identification (z0=0) and the identification 𝐻 ⋍ ℙ𝑛-1 is obtained by taking the hyper plane at infinity as the directions that go to infinity in 𝐢n. Let be a lattice 𝛬 = 𝑍kβŠ‚ 𝐢n after that, the projective map πœ‹: 𝐢n⟢ 𝐢n /𝛬 induces a complex manifold structure in the quotient group 𝐢𝑛/𝛬. Only if π‘˜ = 2𝑛 does it qualify as compact; in this instance, πΆβˆ—/𝛬 is referred to as a complex torus. When πœ‹: 𝑀 ⟢ 𝑁 is a complex manifold, it is common for 𝑁 to inherit the structure of 𝑀. It is possible for 𝑁 to acquire the structure of a complex manifold due to the fact that it is both a complex manifold and a topological covering space. However, this is only the case if 𝑀 is also a complex manifold and its deck transformations are holomorphic. 3. Tangent space to a manifold M We offer a complex manifold 𝑀 as well as a holomorphic coordinate system 𝑧 = (𝑧1, … . , 𝑧𝑛) that revolves around 𝑝. A tangent space to 𝑀 at 𝑝 is defined in this article in three distinct ways. a) 𝑇𝑅,𝑝(𝑀) We define 𝑀 to be a real manifold of size 2𝑛. (𝑀) is the normal real tangent space at 𝑀 at 𝑝. If we write 𝑧𝑖 = π‘₯𝑖 + 𝑖𝑦𝑖 , 𝑇𝑅,𝑝 (𝑀) =𝑅 { πœ• πœ•π‘₯𝑖 , πœ• πœ•π‘¦π‘– } then 𝑇𝑅,𝑝 (𝑀) may be expressed as the set of all 𝑅-linear derivations on the set of all C^∞ functions with real values that are close to p. b) 𝑇𝑐,𝑝(𝑀) =𝑇𝑅,𝑝(𝑀)⨂𝑅𝐢 We refer to the complexified tangent space to 𝑀 as at 𝑝. The space of 𝐢 βˆ’linear derivations in the ring of Complex-valued 𝐢∞ functions on 𝑀 around 𝑝 is one way to realise it. We may write 𝑇𝑐,𝑝(𝑀)=𝐢 { πœ• πœ•π‘₯𝑖 , πœ• πœ•π‘¦π‘– } =𝐢 { πœ• πœ•π‘§π‘– , πœ• πœ•οΏ½Μ…οΏ½π‘– } c) The space of holomorphic tangents to 𝑀 at point 𝑝 is represented by an 𝑇𝑝(M) = 𝐢 { πœ• πœ•π‘§π‘– } βŠ‚ 𝑇𝑐,𝑝(M). Due to the fact that the subspace of 𝑇𝐢,𝑝(𝑀) is composed of derivations that vanish on anti- holomorphic functions or functions such that f is holomorphic, the coordinate system that was Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 29 https://internationalpubls.com selected (𝑧1, … . , 𝑧𝑛) is not valid. The subspace is shown by the anti holomorphic tangent space to 𝑀 at p is 𝑇′′𝑝(M) =𝐢 { πœ• πœ•οΏ½Μ…οΏ½π‘– }. Clearly, Actually, a holomorphic map 𝑓:𝑀 ⟢ 𝑁 exists for each p in M only if πΉβˆ—(𝑇 β€² 𝑝(M)) βŠ‚ 𝑇′𝑓(𝑝)(N) exists. Observe that, because, 𝑇𝐢,𝑝(𝑀) is provided naturally as the real vector space , 𝑇𝑅 ,𝑝(𝑀) tensored with, the statement conjugation sending πœ• πœ•π‘§π‘– π‘‘π‘œ πœ• πœ•οΏ½Μ…οΏ½π‘– is well defined and 𝑇′′𝑝(𝑀) =𝑇′𝑝(𝑀) . The projection 𝑇𝑅,𝑝(M) ⟢ 𝑇𝐢,𝑝(M)⟢ 𝑇′𝑝(M) is therefore an R-linear isomorphism of this type. Therefore, this meant that the only place where we could ''do geometry" was in the holomorphic tangent space. 3.1 Example: Assume that 𝑧(𝑑), where 0 ≀ 𝑑 ≀ 1, is a smooth arc in the complex z-plane. Then 𝑧(𝑑) = π‘₯(𝑑) + βˆšβˆ’1𝑦(𝑑) and The tangent to the arc may be calculated as follows: π‘₯β€²(t) πœ• πœ•π‘₯ + 𝑦′(t) πœ• πœ•π‘¦ in 𝑇𝑅(𝐢) Or 𝑧′(t) πœ• πœ•π‘§ in 𝑇′(𝐢) and these two coincide under the projection 𝑇𝑅(𝐢)⟢ 𝑇′(𝐢). Suppose that 𝑀 and 𝑁 are Complex Manifolds. The set of holomorphic coordinates centred at p∈M is denoted as 𝑧 = (𝑧1, … . , 𝑧𝑛). A set of holomorphic coordinates centred at π‘ž ∈ 𝑁 is denoted as 𝑀 =(𝑀1, … . , 𝑀𝑛). We have different thoughts on the Jacobian of 𝑓 ,we refer to [𝐺, 𝐻]. With 𝑓(𝑝) = π‘ž, the holomorphic map 𝑓:𝑀 ⟢ 𝑁 corresponds to the different tangent spaces to 𝑀 and 𝑁 at 𝑝 and π‘ž, respectively. 4. Sub manifolds and sub varieties 4.1 Definition: Specified sub manifold S of a complicated manifold M with complex characteristics A finite collection of holomorphic functions 𝑓1, 𝑓2, . . . , π‘“π‘˜ with rank 𝑔(𝑓) = π‘˜ is represented locally by the subset 𝑀, which is known as the zeros. The sign π‘‰βˆ— is used to denote the location of smooth points in the curve 𝑉. On the other hand, a singular point of 𝑉 is defined as 𝑝 ∈ 𝑉 - π‘‰βˆ—, and the singular locus of 𝑉 is denoted by 𝑉𝑠. The only conditions under which 𝑉 is deemed smooth or non- singular are those in which it is a sub-manifold of 𝑀 or if 𝑉 is equal to π‘‰βˆ—. Specifically, for every point 𝑝 on an analytic hyper surface 𝑉 βŠ‚ 𝑀 defined in terms of local coordinates 𝑍 by the function 𝑓, we say that the multiplicity π‘šπ‘’π‘™π‘(𝑉) is the degree to which 𝑓 eliminates at 𝑝, or the greatest number m for which all partial derivatives are identical πœ•π‘˜π‘“ πœ•π‘§π‘–1,β€¦β€¦β€¦πœ•π‘§π‘–π‘˜ (𝑃) = 0 , π‘˜ ≀ π‘š βˆ’ 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 30 https://internationalpubls.com When dealing with a family of objects that are parametrized locally by a complex manifold or an analytic subvariety of a complex manifold, the statement ''a generic relative has a certain property'' indicates that the set of objects in the family that do not possess the property is contained in a sub variety of strictly smaller dimension. This is the case when dealing with a complex manifold. In most cases, the proper way to parametrize items in our family will be obvious. In ℙ𝑛, the generic π‘˜-plane is an exception( this one will be again revisited in the sections on Grassmannians in the work that follows later). 4.2 Remark: Grassmannians are a family of compact complex manifolds. To be specific are generalizations to Projective spaces. They have rich topological and smooth manifold structure. Before we define them formally some background of affine and projective vector spaces are defined. The setting is complex. 5. Vector spaces: These are rich algebraic structures under additive binary operation they form abelian group and multiplication/ scalar distribution operation ensures distribution ensures distribution properties. One main thing about them is that they appear as pairs of fields. Thus every field is a vector space over itself. The best examples are the real field 𝑅 is a vector space projected onto itself. 𝐢, the self- describing complex field vector space. Next, is their higher analogues 𝑅𝑛 over 𝑅, 𝐢𝑛 over 𝐢. If we choose a vector space with n dimensions that is isomorphic to 𝑅2𝑛, then n is its dimension. 5.1 Proposition: 𝐹𝑛 is isomorphic to any finite dimensional vector space of size 𝑛, where 𝐹 is a scalar field. The real field 𝐹 = 𝑅 is defined as a real vector space 𝑉 ′𝑠of size n over 𝑅. Proof: Each vector space has a basis, which is not unique; two bases of V can be equivalent, but the dimension of n is unique. We take these vital facts for granted, despite their importance. An instance of an inner product on 𝑅𝑛 is the bilinear map 𝑓: 𝑅𝑋𝑅 β†’ 𝑅, which is positive definite, linear in the slot, and compatible with scalar multiplication. In reality, the only differences are that the inner product is Hermitian and complex numbers have conjugates. A metric induces an inner product space, which in turn is a metric space. Grassmannians are obviously of importance to us. We briefly touch on affine projective geometry as a digression before moving on to the primary topic. Observe that 𝑅𝑛 can be decomposed by writing it as π‘…π‘˜(π‘›βˆ’π‘˜). If it were 𝐢 then πΆπ‘˜(π‘›βˆ’π‘˜). π‘˜ rows and (𝑛 βˆ’ π‘˜) column matrices are imaginable in the language of matrices π‘˜. (𝑛 βˆ’ π‘˜). When a metric is represented by a π‘˜-row vector and vice versa, two matrices A and B are declared to have the same mean if there is a scalar 𝑔 such that 𝐴 = 𝑔. 𝐡. This connection splits the set into equivalent classes and is an equivalent relation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 31 https://internationalpubls.com Since π‘₯ and 𝑦 in the real plane 𝑅2 are equivalent if and only if 𝑦 = π‘Žπ‘₯ for some constant a, we can assert that π‘₯ and 𝑦 are equivalent. The equivalence class determined by 𝑋 is indicated by [𝑋], allowing 𝑅2 to be divided. In other words, the line passing through the origin sets a constant slope. Thus, one can set, 𝑦 = π‘Žπ‘₯. How to realise non trivial examples where projectivization crop up. The stereographic projections of the sphere and circle are the best examples. 6. Conclusion Imagine a line passing through origin in 𝑅2 and 𝑅3 with vectors originating from the origin of unit length then we are done. Differential Geometry methods vividly capture them as one and two dimensional smooth compact manifolds. A natural inclusive tower of subspaces of 𝑅𝑛 is 𝑅1contained in 𝑅2………𝑅(π‘›βˆ’1) and contained in 𝑅𝑛. Therefore π‘…π‘˜ with π‘˜ row vectors and π‘˜ Γ— 𝑛 matrix and totality of them would be aright frame work for imagining Grassmannians. References [1] Harris, Algebraic Geometry, A first course, Springer-Verlag,1995. Lakshmibai and Gonciulea Flag Varieties,Hermann,2001. [2] Hatcher,Allen (2003).Vector Bundles & K-Theory. [3] JEAN- PIERRE SERRE, TREES,Springer Mongr.Math., Springer-Verlag,Berlin ,2003. [4] Peter Buser, Geometry and Spectra of compact Rieman surfaces,Progress in mathematics,Vol.106,Birkhouser Boston,inc.,Boston,MA,1992.mr 11833224. [5] M Arym Mirza Khani and Bram Petri,Lengths of closed jodesics on random surfaces of large genus,comment.Math.Helv. 94(2019),No.4,869-889,DOI 10.4171/CMH/477.MR4046008. [6] Thomas Bendokat , Ralf Zimmermann and P.-A. Absil -Basic Geometry and Computational Aspects. [7] P.-A. Absil, R. Mahony, and R. Sepulchre. Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica, 80(2):199–220, 2004. [8] A. A. Borisenko and Yu. A. Nikolaevski˘ı. Grassmann manifolds and Grassmann image of submanifolds. Uspekhi Mat. Nauk, 46(2(278)):41–83, 240, 1991. [9] J. M. Lee. Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer New York, 2012. [10] A. Machado and I. Salavessa. Grassmannian manifolds as subsets of Euclidean spaces. In Differential geometry (Santiago de Compostela, 1984), volume 131 of Res. Notes in Math., pages 85–102. Pitman, Boston, MA, 1985. [11] L. Qiu, Y. Zhang, and C.-K. Li. Unitarily Invariant Metrics on the Grassmann Space. SIAM Journal on Matrix Analysis and Applications, 27(2):507–25, 2005.