Misplaced Pages

#929070

118-468: The earliest work on a non-trivial Grassmannian is due to Julius Plücker , who studied the set of projective lines in real projective 3-space, which is equivalent to G r 2 ( R 4 ) {\displaystyle \mathbf {Gr} _{2}(\mathbf {R} ^{4})} , parameterizing them by what are now called Plücker coordinates . (See § Plücker coordinates and Plücker relations below.) Hermann Grassmann later introduced

236-1244: A C ∞ {\displaystyle C^{\infty }} manifold M {\displaystyle M} , if a chart φ = ( x 1 , … , x n ) : U → R n {\displaystyle \varphi =(x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}} is given with p ∈ U {\displaystyle p\in U} , then one can define an ordered basis { ∂ ∂ x 1 | p , … , ∂ ∂ x n | p } {\textstyle \left\{\left.{\frac {\partial }{\partial x^{1}}}\right|_{p},\dots ,\left.{\frac {\partial }{\partial x^{n}}}\right|_{p}\right\}} of T p M {\displaystyle T_{p}M} by Then for every tangent vector v ∈ T p M {\displaystyle v\in T_{p}M} , one has This formula therefore expresses v {\displaystyle v} as

354-570: A i j ) {\displaystyle (a_{ij})} determines w {\displaystyle w} . In general, the first k {\displaystyle k} rows need not be independent, but since W {\displaystyle W} has maximal rank k {\displaystyle k} , there exists an ordered set of integers 1 ≤ i 1 < ⋯ < i k ≤ n {\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n} such that

472-1090: A coordinate chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} , where U {\displaystyle U} is an open subset of M {\displaystyle M} containing x {\displaystyle x} . Suppose further that two curves γ 1 , γ 2 : ( − 1 , 1 ) → M {\displaystyle \gamma _{1},\gamma _{2}:(-1,1)\to M} with γ 1 ( 0 ) = x = γ 2 ( 0 ) {\displaystyle {\gamma _{1}}(0)=x={\gamma _{2}}(0)} are given such that both φ ∘ γ 1 , φ ∘ γ 2 : ( − 1 , 1 ) → R n {\displaystyle \varphi \circ \gamma _{1},\varphi \circ \gamma _{2}:(-1,1)\to \mathbb {R} ^{n}} are differentiable in

590-403: A differentiable manifold a tangent space —a real vector space that intuitively contains the possible directions in which one can tangentially pass through x {\displaystyle x} . The elements of the tangent space at x {\displaystyle x} are called the tangent vectors at x {\displaystyle x} . This is a generalization of

708-941: A map d φ x : T x M → R n {\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} ^{n}} by d φ x ( γ ′ ( 0 ) ) := d d t [ ( φ ∘ γ ) ( t ) ] | t = 0 , {\textstyle {\mathrm {d} {\varphi }_{x}}(\gamma '(0)):=\left.{\frac {\mathrm {d} }{\mathrm {d} {t}}}[(\varphi \circ \gamma )(t)]\right|_{t=0},} where γ ∈ γ ′ ( 0 ) {\displaystyle \gamma \in \gamma '(0)} . The map d φ x {\displaystyle \mathrm {d} {\varphi }_{x}} turns out to be bijective and may be used to transfer

826-511: A smooth manifold under the quotient structure. More generally, over a ground field K {\displaystyle K} , the group G L ( V ) {\displaystyle \mathrm {GL} (V)} is an algebraic group , and this construction shows that the Grassmannian is a non-singular algebraic variety . It follows from the existence of the Plücker embedding that

944-426: A Euclidean inner product q {\displaystyle q} on V {\displaystyle V} . The real orthogonal group O ( V , q ) {\displaystyle O(V,q)} acts transitively on the set of k {\displaystyle k} -dimensional subspaces G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} and

1062-417: A basis consisting of k {\displaystyle k} linearly independent column vectors ( W 1 , … , W k ) {\displaystyle (W_{1},\dots ,W_{k})} . The homogeneous coordinates of the element w ∈ G r k ( V ) {\displaystyle w\in \mathbf {Gr} _{k}(V)} consist of

1180-495: A canonical isomorphism In particular, for any point s {\displaystyle s} of S {\displaystyle S} , the canonical morphism { s } = Spec K ( s ) → S {\displaystyle \{s\}={\text{Spec}}K(s)\rightarrow S} induces an isomorphism from the fiber G r ( k , E ) s {\displaystyle \mathbf {Gr} (k,{\mathcal {E}})_{s}} to

1298-465: A closed immersion from the projective bundle: For any morphism of S -schemes: this closed immersion induces a closed immersion Conversely, any such closed immersion comes from a surjective homomorphism of O T {\displaystyle O_{T}} -modules from E T {\displaystyle {\mathcal {E}}_{T}} to a locally free module of rank k {\displaystyle k} . Therefore,

SECTION 10

#1733085819930

1416-481: A derivation at x {\displaystyle x} . This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces. If M {\displaystyle M} is an open subset of R n {\displaystyle \mathbb {R} ^{n}} , then M {\displaystyle M} is a C ∞ {\displaystyle C^{\infty }} manifold in

1534-483: A different inner product will give an equivalent norm on V {\displaystyle V} , and hence an equivalent metric. For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices. Let M ( n , R ) {\displaystyle M(n,\mathbf {R} )} denote the space of real n × n {\displaystyle n\times n} matrices and

1652-547: A differentiable manifold and also as an algebraic variety. An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators ( Milnor & Stasheff (1974) problem 5-C). For this, choose a positive definite real or Hermitian inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } on V {\displaystyle V} , depending on whether V {\displaystyle V}

1770-414: A differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold. The informal description above relies on

1888-401: A firm and independent basis projective duality . In 1836, Plücker was made professor of physics at University of Bonn . In 1858, after a year of working with vacuum tubes of his Bonn colleague Heinrich Geißler , he published his first classical researches on the action of the magnet on the electric discharge in rarefied gases. He found that the discharge caused a fluorescent glow to form on

2006-687: A linear combination of the basis tangent vectors ∂ ∂ x i | p ∈ T p M {\textstyle \left.{\frac {\partial }{\partial x^{i}}}\right|_{p}\in T_{p}M} defined by the coordinate chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} . Every smooth (or differentiable) map φ : M → N {\displaystyle \varphi :M\to N} between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces: If

2124-498: A linear map I / I 2 → R {\displaystyle I/I^{2}\to \mathbb {R} } . Conversely, if r : I / I 2 → R {\displaystyle r:I/I^{2}\to \mathbb {R} } is a linear map, then D ( f ) := r ( ( f − f ( x ) ) + I 2 ) {\displaystyle D(f):=r\left((f-f(x))+I^{2}\right)} defines

2242-454: A manifold M {\displaystyle M} , so that every vector bundle generates a continuous map from M {\displaystyle M} to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing homotopic maps to

2360-405: A manifold's ability to be embedded into an ambient vector space R m {\displaystyle \mathbb {R} ^{m}} so that the tangent vectors can "stick out" of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself. There are various equivalent ways of defining the tangent spaces of

2478-443: A manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below. In the embedded-manifold picture, a tangent vector at a point x {\displaystyle x} is thought of as the velocity of a curve passing through the point x {\displaystyle x} . We can therefore define

SECTION 20

#1733085819930

2596-530: A natural manner (take coordinate charts to be identity maps on open subsets of R n {\displaystyle \mathbb {R} ^{n}} ), and the tangent spaces are all naturally identified with R n {\displaystyle \mathbb {R} ^{n}} . Another way to think about tangent vectors is as directional derivatives . Given a vector v {\displaystyle v} in R n {\displaystyle \mathbb {R} ^{n}} , one defines

2714-425: A point) and derivations at a point. As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if v {\displaystyle v} is a tangent vector to M {\displaystyle M} at a point x {\displaystyle x} (thought of as a derivation), then define

2832-600: A sequence of elements g ∈ G L ( k , K ) {\displaystyle g\in GL(k,K)} ) to obtain its reduced column echelon form . If the first k {\displaystyle k} rows of W {\displaystyle W} are linearly independent, the result will have the form and the ( n − k ) × k {\displaystyle (n-k)\times k} affine coordinate matrix A {\displaystyle A} with entries (

2950-490: A subspace w 0 ⊂ V {\displaystyle w_{0}\subset V} of dimension k {\displaystyle k} , any element w ∈ G r ( k , V ) {\displaystyle w\in \mathbf {Gr} (k,V)} can be expressed as for some group element g ∈ G L ( V ) {\displaystyle g\in \mathrm {GL} (V)} , where g {\displaystyle g}

3068-569: A tangent vector as an equivalence class of curves passing through x {\displaystyle x} while being tangent to each other at x {\displaystyle x} . Suppose that M {\displaystyle M} is a C k {\displaystyle C^{k}} differentiable manifold (with smoothness k ≥ 1 {\displaystyle k\geq 1} ) and that x ∈ M {\displaystyle x\in M} . Pick

3186-415: Is The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space . First, recall that the general linear group G L ( V ) {\displaystyle \mathrm {GL} (V)} acts transitively on the k {\displaystyle k} -dimensional subspaces of V {\displaystyle V} . Therefore, if we choose

3304-588: Is nonsingular . We may apply column operations to reduce this submatrix to the identity matrix , and the remaining entries uniquely determine w {\displaystyle w} . Hence we have the following definition: For each ordered set of integers 1 ≤ i 1 < ⋯ < i k ≤ n {\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n} , let U i 1 , … , i k {\displaystyle U_{i_{1},\dots ,i_{k}}} be

3422-431: Is perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space , then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport . Many authors in differential geometry and general relativity use it. More strictly, this defines an affine tangent space, which

3540-403: Is projective if E {\displaystyle {\mathcal {E}}} is finitely generated. When S {\displaystyle S} is the spectrum of a field K {\displaystyle K} , then the sheaf E {\displaystyle {\mathcal {E}}} is given by a vector space V {\displaystyle V} and we recover

3658-480: Is a local diffeomorphism at x {\displaystyle x} in M {\displaystyle M} , then d φ x : T x M → T φ ( x ) N {\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to T_{\varphi (x)}N} is a linear isomorphism . Conversely, if φ : M → N {\displaystyle \varphi :M\to N}

Grassmannian - Misplaced Pages Continue

3776-528: Is a derivation at the point x , {\displaystyle x,} and that equivalent curves yield the same derivation. Thus, for an equivalence class γ ′ ( 0 ) , {\displaystyle \gamma '(0),} we can define D γ ′ ( 0 ) ( f ) := ( f ∘ γ ) ′ ( 0 ) , {\displaystyle {D_{\gamma '(0)}}(f):=(f\circ \gamma )'(0),} where

3894-405: Is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication. A derivation at x ∈ M {\displaystyle x\in M} is defined as a linear map D : C ∞ ( M ) → R {\displaystyle D:{C^{\infty }}(M)\to \mathbb {R} } that satisfies

4012-732: Is an object of G r ( k , E G r ( k , E ) ) , {\displaystyle \mathbf {Gr} \left(k,{\mathcal {E}}_{\mathbf {Gr} (k,{\mathcal {E}})}\right),} and therefore a quotient module G {\displaystyle {\mathcal {G}}} of E G r ( k , E ) {\displaystyle {\mathcal {E}}_{\mathbf {Gr} (k,{\mathcal {E}})}} , locally free of rank k {\displaystyle k} over G r ( k , E ) {\displaystyle \mathbf {Gr} (k,{\mathcal {E}})} . The quotient homomorphism induces

4130-447: Is any orthonormal basis for w ⊂ R n {\displaystyle w\subset \mathbf {R} ^{n}} , viewed as real n {\displaystyle n} component column vectors. An analogous construction applies to the complex Grassmannian G r ( k , C n ) {\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{n})} , identifying it bijectively with

4248-453: Is continuously differentiable and d φ x {\displaystyle \mathrm {d} {\varphi }_{x}} is an isomorphism, then there is an open neighborhood U {\displaystyle U} of x {\displaystyle x} such that φ {\displaystyle \varphi } maps U {\displaystyle U} diffeomorphically onto its image. This

4366-506: Is determined only up to right multiplication by elements { h ∈ H } {\displaystyle \{h\in H\}} of the stabilizer of w 0 {\displaystyle w_{0}} : under the G L ( V ) {\displaystyle \mathrm {GL} (V)} -action. We may therefore identify G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} with

4484-435: Is distinct from the space of tangent vectors described by modern terminology. In algebraic geometry , in contrast, there is an intrinsic definition of the tangent space at a point of an algebraic variety V {\displaystyle V} that gives a vector space with dimension at least that of V {\displaystyle V} itself. The points p {\displaystyle p} at which

4602-532: Is frequently expressed using a variety of other notations: In a sense, the derivative is the best linear approximation to φ {\displaystyle \varphi } near x {\displaystyle x} . Note that when N = R {\displaystyle N=\mathbb {R} } , then the map d φ x : T x M → R {\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} } coincides with

4720-427: Is not a projective space is Gr (2, 4) . To endow G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} with the structure of a differentiable manifold, choose a basis for V {\displaystyle V} . This is equivalent to identifying V {\displaystyle V} with K n {\displaystyle K^{n}} , with

4838-445: Is real or complex. A k {\displaystyle k} -dimensional subspace w {\displaystyle w} determines a unique orthogonal projection operator P w : V → V {\displaystyle P_{w}:V\rightarrow V} whose image is w ⊂ V {\displaystyle w\subset V} by splitting V {\displaystyle V} into

Grassmannian - Misplaced Pages Continue

4956-620: Is the i j {\displaystyle i_{j}} -th row of W {\displaystyle W} is nonsingular. The affine coordinate functions on U i 1 , … , i k {\displaystyle U_{i_{1},\dots ,i_{k}}} are then defined as the entries of the ( n − k ) × k {\displaystyle (n-k)\times k} matrix A i 1 , … , i k {\displaystyle A^{i_{1},\dots ,i_{k}}} whose rows are those of

5074-707: Is the stalk of O X {\displaystyle {\mathcal {O}}_{X}} at p {\displaystyle p} . For x ∈ M {\displaystyle x\in M} and a differentiable curve γ : ( − 1 , 1 ) → M {\displaystyle \gamma :(-1,1)\to M} such that γ ( 0 ) = x , {\displaystyle \gamma (0)=x,} define D γ ( f ) := ( f ∘ γ ) ′ ( 0 ) {\displaystyle {D_{\gamma }}(f):=(f\circ \gamma )'(0)} (where

5192-425: Is the collection of all k {\displaystyle \mathbb {k} } -derivations D : O X , p → k {\displaystyle D:{\mathcal {O}}_{X,p}\to \mathbb {k} } , where k {\displaystyle \mathbb {k} } is the ground field and O X , p {\displaystyle {\mathcal {O}}_{X,p}}

5310-453: Is the invertible k × k {\displaystyle k\times k} matrix whose l {\displaystyle l} th row is the j l {\displaystyle j_{l}} th row of A ^ i 1 , … , i k {\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}} . The transition functions are therefore rational in

5428-580: Is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the varieties considered in algebraic geometry . If D {\displaystyle D} is a derivation at x {\displaystyle x} , then D ( f ) = 0 {\displaystyle D(f)=0} for every f ∈ I 2 {\displaystyle f\in I^{2}} , which means that D {\displaystyle D} gives rise to

5546-544: Is then defined as the set of all tangent vectors at x {\displaystyle x} ; it does not depend on the choice of coordinate chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} . To define vector-space operations on T x M {\displaystyle T_{x}M} , we use a chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} and define

5664-518: The k {\displaystyle k} -dimensional subspace of R n {\displaystyle \mathbf {R} ^{n}} spanned by its columns and, conversely, sending any element w ∈ G r ( k , R n ) {\displaystyle w\in \mathbf {Gr} (k,\mathbf {R} ^{n})} to the projection matrix where ( w 1 , ⋯ , w k ) {\displaystyle (w_{1},\cdots ,w_{k})}

5782-418: The k × k {\displaystyle k\times k} submatrix W i 1 , … , i k {\displaystyle W_{i_{1},\dots ,i_{k}}} whose rows are the ( i 1 , … , i k ) {\displaystyle (i_{1},\ldots ,i_{k})} -th rows of W {\displaystyle W}

5900-423: The general linear group of invertible k × k {\displaystyle k\times k} matrices with entries in K {\displaystyle K} . This defines an equivalence relation between n × k {\displaystyle n\times k} matrices W {\displaystyle W} of rank k {\displaystyle k} , for which

6018-533: The ideal I {\displaystyle I} of C ∞ ( M ) {\displaystyle C^{\infty }(M)} that consists of all smooth functions f {\displaystyle f} vanishing at x {\displaystyle x} , i.e., f ( x ) = 0 {\displaystyle f(x)=0} . Then I {\displaystyle I} and I 2 {\displaystyle I^{2}} are both real vector spaces, and

SECTION 50

#1733085819930

6136-520: The quotient space I / I 2 {\displaystyle I/I^{2}} can be shown to be isomorphic to the cotangent space T x ∗ M {\displaystyle T_{x}^{*}M} through the use of Taylor's theorem . The tangent space T x M {\displaystyle T_{x}M} may then be defined as the dual space of I / I 2 {\displaystyle I/I^{2}} . While this definition

6254-453: The self-adjointness is with respect to the Hermitian inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } in which the standard basis vectors ( e 1 , ⋯ , e n ) {\displaystyle (e_{1},\cdots ,e_{n})} are orthonomal. The formula for

6372-470: The tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. In differential geometry , one can attach to every point x {\displaystyle x} of

6490-592: The tangent space to M {\displaystyle M} can be considered as a subspace of the tangent space of R n {\displaystyle \mathbf {R} ^{n}} , which is also just R n {\displaystyle \mathbf {R} ^{n}} . The map assigning to x {\displaystyle x} its tangent space defines a map from M to G r k ( R n ) {\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})} . (In order to do this, we have to translate

6608-442: The unitary group U ( V , h ) {\displaystyle U(V,h)} acts transitively, and we find analogously or, for V = C n {\displaystyle V=\mathbf {C} ^{n}} and w 0 = C k ⊂ C n {\displaystyle w_{0}=\mathbf {C} ^{k}\subset \mathbf {C} ^{n}} , In particular, this shows that

6726-489: The Grassmannian G r ( k , R n ) {\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{n})} of k {\displaystyle k} -dimensional subspaces of R n {\displaystyle \mathbf {R} ^{n}} given by sending P ∈ P ( k , n , R ) {\displaystyle P\in P(k,n,\mathbf {R} )} to

6844-448: The Grassmannian G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} into a metric space with metric for any pair w , w ′ ⊂ V {\displaystyle w,w'\subset V} of k {\displaystyle k} -dimensional subspaces, where ‖ ⋅ ‖ denotes the operator norm . The exact inner product used does not matter, because

6962-467: The Grassmannian are isomorphic . Here the definition of homotopy relies on a notion of continuity, and hence a topology. For k = 1 , the Grassmannian Gr (1, n ) is the space of lines through the origin in n -space, so it is the same as the projective space P n − 1 {\displaystyle \mathbf {P} ^{n-1}} of n − 1 dimensions. For k = 2 ,

7080-399: The Grassmannian as a closed subset of the sphere { X ∈ E n d ( V ) ∣ t r ( X X † ) = k } {\displaystyle \{X\in \mathrm {End} (V)\mid \mathrm {tr} (XX^{\dagger })=k\}} this is one way to see that the Grassmannian is a compact Hausdorff space. This construction also turns

7198-577: The Grassmannian functor associates the set of quotient modules of locally free of rank k {\displaystyle k} on T {\displaystyle T} . We denote this set by G r ( k , E T ) {\displaystyle \mathbf {Gr} (k,{\mathcal {E}}_{T})} . This functor is representable by a separated S {\displaystyle S} -scheme G r ( k , E ) {\displaystyle \mathbf {Gr} (k,{\mathcal {E}})} . The latter

SECTION 60

#1733085819930

7316-563: The Grassmannian is compact , and of (real or complex) dimension k ( n − k ) . In the realm of algebraic geometry , the Grassmannian can be constructed as a scheme by expressing it as a representable functor . Let E {\displaystyle {\mathcal {E}}} be a quasi-coherent sheaf on a scheme S {\displaystyle S} . Fix a positive integer k {\displaystyle k} . Then to each S {\displaystyle S} -scheme T {\displaystyle T} ,

7434-514: The Grassmannian is complete as an algebraic variety. In particular, H {\displaystyle H} is a parabolic subgroup of G L ( V ) {\displaystyle \mathrm {GL} (V)} . Over R {\displaystyle \mathbf {R} } or C {\displaystyle \mathbf {C} } it also becomes possible to use smaller groups in this construction. To do this over R {\displaystyle \mathbf {R} } , fix

7552-412: The Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces Gr (2, 3) , Gr (1, 3) , and P (the projective plane ) may all be identified with each other. The simplest Grassmannian that

7670-419: The Grassmannian of k {\displaystyle k} -dimensional subspaces of an n {\displaystyle n} -dimensional vector space V {\displaystyle V} . By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them

7788-692: The Grassmannians G r ( k , R N ) {\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{N})} , G r ( k , C N ) {\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{N})} in the space of real or complex n × n {\displaystyle n\times n} matrices R n × n {\displaystyle \mathbf {R} ^{n\times n}} , C n × n {\displaystyle \mathbf {C} ^{n\times n}} , respectively. Since this defines

7906-767: The Leibniz identity ∀ f , g ∈ C ∞ ( M ) : D ( f g ) = D ( f ) ⋅ g ( x ) + f ( x ) ⋅ D ( g ) , {\displaystyle \forall f,g\in {C^{\infty }}(M):\qquad D(fg)=D(f)\cdot g(x)+f(x)\cdot D(g),} which is modeled on the product rule of calculus. (For every identically constant function f = const , {\displaystyle f={\text{const}},} it follows that D ( f ) = 0 {\displaystyle D(f)=0} ). Denote T x M {\displaystyle T_{x}M}

8024-450: The capillary part now called a Geissler tube , by means of which the luminous intensity of feeble electric discharges was raised sufficiently to allow of spectroscopic investigation. He anticipated Robert Wilhelm Bunsen and Gustav Kirchhoff in announcing that the lines of the spectrum were characteristic of the chemical substance which emitted them, and in indicating the value of this discovery in chemical analysis. According to Hittorf, he

8142-503: The choice of basis is arbitrary, two such maximal rank rectangular matrices W {\displaystyle W} and W ~ {\displaystyle {\tilde {W}}} represent the same element w ∈ G r k ( V ) {\displaystyle w\in \mathbf {Gr} _{k}(V)} if and only if for some element g ∈ G L ( k , K ) {\displaystyle g\in GL(k,K)} of

8260-498: The concept in general. Notations for Grassmannians vary between authors; they include G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} , G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} , G r k ( n ) {\displaystyle \mathbf {Gr} _{k}(n)} , G r ( k , n ) {\displaystyle \mathbf {Gr} (k,n)} to denote

8378-666: The consecutive complementary rows. On the overlap U i 1 , … , i k ∩ U j 1 , … , j k {\displaystyle U_{i_{1},\dots ,i_{k}}\cap U_{j_{1},\dots ,j_{k}}} between any two such coordinate neighborhoods, the affine coordinate matrix values A i 1 , … , i k {\displaystyle A^{i_{1},\dots ,i_{k}}} and A j 1 , … , j k {\displaystyle A^{j_{1},\dots ,j_{k}}} are related by

8496-551: The coordinate matrices A i 1 , … , i k {\displaystyle A^{i_{1},\dots ,i_{k}}} may take arbitrary values, and they define a diffeomorphism from U i 1 , … , i k {\displaystyle U_{i_{1},\dots ,i_{k}}} to the space of K {\displaystyle K} -valued ( n − k ) × k {\displaystyle (n-k)\times k} matrices. Denote by

8614-475: The corresponding directional derivative at a point x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} by This map is naturally a derivation at x {\displaystyle x} . Furthermore, every derivation at a point in R n {\displaystyle \mathbb {R} ^{n}} is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at

8732-675: The curve γ {\displaystyle \gamma } being used, and in fact it does not. Suppose now that M {\displaystyle M} is a C ∞ {\displaystyle C^{\infty }} manifold. A real-valued function f : M → R {\displaystyle f:M\to \mathbb {R} } is said to belong to C ∞ ( M ) {\displaystyle {C^{\infty }}(M)} if and only if for every coordinate chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} ,

8850-404: The curve γ ∈ γ ′ ( 0 ) {\displaystyle \gamma \in \gamma '(0)} has been chosen arbitrarily. The map γ ′ ( 0 ) ↦ D γ ′ ( 0 ) {\displaystyle \gamma '(0)\mapsto D_{\gamma '(0)}} is a vector space isomorphism between

8968-417: The derivative is taken in the ordinary sense because f ∘ γ {\displaystyle f\circ \gamma } is a function from ( − 1 , 1 ) {\displaystyle (-1,1)} to R {\displaystyle \mathbb {R} } ). One can ascertain that D γ ( f ) {\displaystyle D_{\gamma }(f)}

9086-422: The dimension of the tangent space is exactly that of V {\displaystyle V} are called non-singular points; the others are called singular points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of V {\displaystyle V} are those where the "test to be a manifold" fails. See Zariski tangent space . Once

9204-597: The directional derivative D v {\displaystyle D_{v}} in the direction v {\displaystyle v} by If we think of v {\displaystyle v} as the initial velocity of a differentiable curve γ {\displaystyle \gamma } initialized at x {\displaystyle x} , i.e., v = γ ′ ( 0 ) {\displaystyle v=\gamma '(0)} , then instead, define D v {\displaystyle D_{v}} by For

9322-524: The element w ∈ G r k ( V ) {\displaystyle w\in \mathbf {Gr} _{k}(V)} does not affect the values of the affine coordinate matrix A i 1 , … , i k {\displaystyle A^{i_{1},\dots ,i_{k}}} representing w on the coordinate neighbourhood U i 1 , … , i k {\displaystyle U_{i_{1},\dots ,i_{k}}} . Moreover,

9440-473: The elements of G r ( k , E ) ( T ) {\displaystyle \mathbf {Gr} (k,{\mathcal {E}})(T)} are exactly the projective subbundles of rank k {\displaystyle k} in P ( E ) × S T . {\displaystyle \mathbf {P} ({\mathcal {E}})\times _{S}T.} Julius Pl%C3%BCcker Julius Plücker (16 June 1801 – 22 May 1868)

9558-411: The elements of the n × k {\displaystyle n\times k} maximal rank rectangular matrix W {\displaystyle W} whose i {\displaystyle i} -th column vector is W i {\displaystyle W_{i}} , i = 1 , … , k {\displaystyle i=1,\dots ,k} . Since

9676-411: The equivalence classes are denoted [ W ] {\displaystyle [W]} . We now define a coordinate atlas. For any n × k {\displaystyle n\times k} homogeneous coordinate matrix W {\displaystyle W} , we can apply elementary column operations (which amounts to multiplying W {\displaystyle W} by

9794-510: The further structure of a differentiable manifold , one can talk about smooth choices of subspace. A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space . Suppose we have a manifold M {\displaystyle M} of dimension k {\displaystyle k} embedded in R n {\displaystyle \mathbf {R} ^{n}} . At each point x ∈ M {\displaystyle x\in M} ,

9912-417: The generalization of these co-ordinates to k × k {\displaystyle k\times k} minors of the n × k {\displaystyle n\times k} matrix of homogeneous coordinates, also known as Plücker coordinates , apply. The embedding of the Grassmannian G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} into

10030-402: The glass walls of the vacuum tube, and that the glow could be made to shift by applying an electromagnet to the tube, thus creating a magnetic field. It was later shown that the glow was produced by cathode rays. Plücker, first by himself and afterwards in conjunction with Johann Hittorf , made many important discoveries in the spectroscopy of gases. He was the first to use the vacuum tube with

10148-458: The homogeneous coordinate matrix having the identity matrix as the k × k {\displaystyle k\times k} submatrix with rows ( i 1 , … , i k ) {\displaystyle (i_{1},\dots ,i_{k})} and the affine coordinate matrix A i 1 , … , i k {\displaystyle A^{i_{1},\dots ,i_{k}}} in

10266-487: The homogeneous space If we take V = R n {\displaystyle V=\mathbf {R} ^{n}} and w 0 = R k ⊂ R n {\displaystyle w_{0}=\mathbf {R} ^{k}\subset \mathbf {R} ^{n}} (the first k {\displaystyle k} components) we get the isomorphism Over C , if we choose an Hermitian inner product h {\displaystyle h} ,

10384-415: The influence of the great school of French geometers, whose founder, Gaspard Monge , had only recently died. In 1825 he returned to Bonn, and in 1828 was made professor of mathematics. In the same year he published the first volume of his Analytisch-geometrische Entwicklungen , which introduced the method of "abridged notation". In 1831 he published the second volume, in which he clearly established on

10502-516: The level of germs of functions. The reason for this is that the structure sheaf may not be fine for such structures. For example, let X {\displaystyle X} be an algebraic variety with structure sheaf O X {\displaystyle {\mathcal {O}}_{X}} . Then the Zariski tangent space at a point p ∈ X {\displaystyle p\in X}

10620-401: The map f ∘ φ − 1 : φ [ U ] ⊆ R n → R {\displaystyle f\circ \varphi ^{-1}:\varphi [U]\subseteq \mathbb {R} ^{n}\to \mathbb {R} } is infinitely differentiable. Note that C ∞ ( M ) {\displaystyle {C^{\infty }}(M)}

10738-589: The matrix W W i 1 , … , i k − 1 {\displaystyle WW_{i_{1},\dots ,i_{k}}^{-1}} complementary to ( i 1 , … , i k ) {\displaystyle (i_{1},\dots ,i_{k})} , written in the same order. The choice of homogeneous n × k {\displaystyle n\times k} coordinate matrix W {\displaystyle W} in [ W ] {\displaystyle [W]} representing

10856-533: The matrix elements of A i 1 , … , i k {\displaystyle A^{i_{1},\dots ,i_{k}}} , and { U i 1 , … , i k , A i 1 , … , i k } {\displaystyle \{U_{i_{1},\dots ,i_{k}},A^{i_{1},\dots ,i_{k}}\}} gives an atlas for G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} as

10974-408: The notion of a vector , based at a given initial point, in a Euclidean space . The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 {\displaystyle 2} - sphere , then one can picture the tangent space at a point as the plane that touches the sphere at that point and

11092-733: The ordinary sense (we call these differentiable curves initialized at x {\displaystyle x} ). Then γ 1 {\displaystyle \gamma _{1}} and γ 2 {\displaystyle \gamma _{2}} are said to be equivalent at 0 {\displaystyle 0} if and only if the derivatives of φ ∘ γ 1 {\displaystyle \varphi \circ \gamma _{1}} and φ ∘ γ 2 {\displaystyle \varphi \circ \gamma _{2}} at 0 {\displaystyle 0} coincide. This defines an equivalence relation on

11210-491: The orthogonal direct sum of w {\displaystyle w} and its orthogonal complement w ⊥ {\displaystyle w^{\perp }} and defining Conversely, every projection operator P {\displaystyle P} of rank k {\displaystyle k} defines a subspace w P := I m ( P ) {\displaystyle w_{P}:=\mathrm {Im} (P)} as its image. Since

11328-458: The orthogonal projection matrix P w {\displaystyle P_{w}} onto the complex k {\displaystyle k} -dimensional subspace w ⊂ C n {\displaystyle w\subset \mathbf {C} ^{n}} spanned by the orthonormal (unitary) basis vectors ( w 1 , ⋯ , w k ) {\displaystyle (w_{1},\cdots ,w_{k})}

11446-542: The projectivization P ( Λ k ( V ) ) {\displaystyle \mathbf {P} (\Lambda ^{k}(V))} of the k {\displaystyle k} th exterior power of V {\displaystyle V} is known as the Plücker embedding . Plücker was the recipient of the Copley Medal from the Royal Society in 1866. Tangent space In mathematics ,

11564-405: The quotient space of left cosets of H {\displaystyle H} . If the underlying field is R {\displaystyle \mathbf {R} } or C {\displaystyle \mathbf {C} } and G L ( V ) {\displaystyle \mathrm {GL} (V)} is considered as a Lie group , this construction makes the Grassmannian

11682-663: The rank of an orthogonal projection operator equals its trace , we can identify the Grassmann manifold G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} with the set of rank k {\displaystyle k} orthogonal projection operators P {\displaystyle P} : In particular, taking V = R n {\displaystyle V=\mathbf {R} ^{n}} or V = C n {\displaystyle V=\mathbf {C} ^{n}} this gives completely explicit equations for embedding

11800-451: The set of all derivations at x . {\displaystyle x.} Setting turns T x M {\displaystyle T_{x}M} into a vector space. Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties . However, instead of examining derivations D {\displaystyle D} from the full algebra of functions, one must instead work at

11918-679: The set of all differentiable curves initialized at x {\displaystyle x} , and equivalence classes of such curves are known as tangent vectors of M {\displaystyle M} at x {\displaystyle x} . The equivalence class of any such curve γ {\displaystyle \gamma } is denoted by γ ′ ( 0 ) {\displaystyle \gamma '(0)} . The tangent space of M {\displaystyle M} at x {\displaystyle x} , denoted by T x M {\displaystyle T_{x}M} ,

12036-529: The set of elements w ∈ G r k ( V ) {\displaystyle w\in \mathbf {Gr} _{k}(V)} for which, for any choice of homogeneous coordinate matrix W {\displaystyle W} , the k × k {\displaystyle k\times k} submatrix W i 1 , … , i k {\displaystyle W_{i_{1},\dots ,i_{k}}} whose j {\displaystyle j} -th row

12154-702: The space of lines in projective space P 3 {\displaystyle \mathbf {P} ^{3}} as a quadric in P 5 {\displaystyle \mathbf {P} ^{5}} . The construction uses 2×2 minor determinants , or equivalently the second exterior power of the underlying vector space of dimension 4. It is now part of the theory of Grassmannians G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} ( k {\displaystyle k} -dimensional subspaces of an n {\displaystyle n} -dimensional vector space V {\displaystyle V} ), to which

12272-472: The space of the equivalence classes γ ′ ( 0 ) {\displaystyle \gamma '(0)} and that of the derivations at the point x . {\displaystyle x.} Again, we start with a C ∞ {\displaystyle C^{\infty }} manifold M {\displaystyle M} and a point x ∈ M {\displaystyle x\in M} . Consider

12390-440: The stabiliser of a k {\displaystyle k} -space w 0 ⊂ V {\displaystyle w_{0}\subset V} is where w 0 ⊥ {\displaystyle w_{0}^{\perp }} is the orthogonal complement of w 0 {\displaystyle w_{0}} in V {\displaystyle V} . This gives an identification as

12508-478: The standard basis denoted ( e 1 , … , e n ) {\displaystyle (e_{1},\dots ,e_{n})} , viewed as column vectors. Then for any k {\displaystyle k} -dimensional subspace w ⊂ V {\displaystyle w\subset V} , viewed as an element of G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} , we may choose

12626-457: The subset P ( k , n , C ) ⊂ M ( n , C ) {\displaystyle P(k,n,\mathbf {C} )\subset M(n,\mathbf {C} )} of complex n × n {\displaystyle n\times n} matrices P ∈ M ( n , C ) {\displaystyle P\in M(n,\mathbf {C} )} satisfying where

12744-500: The subset P ( k , n , R ) ⊂ M ( n , R ) {\displaystyle P(k,n,\mathbf {R} )\subset M(n,\mathbf {R} )} of matrices P ∈ M ( n , R ) {\displaystyle P\in M(n,\mathbf {R} )} that satisfy the three conditions: There is a bijective correspondence between P ( k , n , R ) {\displaystyle P(k,n,\mathbf {R} )} and

12862-486: The tangent space at each x ∈ M {\displaystyle x\in M} so that it passes through the origin rather than x {\displaystyle x} , and hence defines a k {\displaystyle k} -dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.) This can with some effort be extended to all vector bundles over

12980-502: The tangent space is defined via differentiable curves, then this map is defined by If, instead, the tangent space is defined via derivations, then this map is defined by The linear map d φ x {\displaystyle \mathrm {d} {\varphi }_{x}} is called variously the derivative , total derivative , differential , or pushforward of φ {\displaystyle \varphi } at x {\displaystyle x} . It

13098-401: The tangent spaces of a manifold have been introduced, one can define vector fields , which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: A solution to such

13216-614: The transition relations where both W i 1 , … , i k {\displaystyle W_{i_{1},\dots ,i_{k}}} and W j 1 , … , j k {\displaystyle W_{j_{1},\dots ,j_{k}}} are invertible. This may equivalently be written as where A ^ j 1 , … , j k i 1 , … , i k {\displaystyle {\hat {A}}_{j_{1},\dots ,j_{k}}^{i_{1},\dots ,i_{k}}}

13334-471: The usual Grassmannian G r ( k , E ⊗ O S K ( s ) ) {\displaystyle \mathbf {Gr} (k,{\mathcal {E}}\otimes _{O_{S}}K(s))} over the residue field K ( s ) {\displaystyle K(s)} . Since the Grassmannian scheme represents a functor, it comes with a universal object, G {\displaystyle {\mathcal {G}}} , which

13452-413: The usual Grassmannian variety of the dual space of V {\displaystyle V} , namely: G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} . By construction, the Grassmannian scheme is compatible with base changes: for any S {\displaystyle S} -scheme S ′ {\displaystyle S'} , we have

13570-490: The usual notion of the differential of the function φ {\displaystyle \varphi } . In local coordinates the derivative of φ {\displaystyle \varphi } is given by the Jacobian . An important result regarding the derivative map is the following: Theorem  —  If φ : M → N {\displaystyle \varphi :M\to N}

13688-520: The vector-space operations on R n {\displaystyle \mathbb {R} ^{n}} over to T x M {\displaystyle T_{x}M} , thus turning the latter set into an n {\displaystyle n} -dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} and

13806-492: Was a German mathematician and physicist . He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron . He also vastly extended the study of Lamé curves . Plücker was born at Elberfeld (now part of Wuppertal ). After being educated at Düsseldorf and at the universities of Bonn , Heidelberg and Berlin he went to Paris in 1823 , where he came under

13924-411: Was the first who saw the three lines of the hydrogen spectrum, which a few months after his death, were recognized in the spectrum of the solar protuberances. In 1865, Plücker returned to the field of geometry and invented what was known as line geometry in the nineteenth century. In projective geometry , Plücker coordinates refer to a set of homogeneous co-ordinates introduced initially to embed

#929070