Neifeld’s Connection Inducedon the Grassmann Manifold

Olga Belova

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Displaying similar documents to “Neifeld’s Connection Inducedon the Grassmann Manifold”

The generic dimension of the first derived system

Robert P. Buemi (1978)

Annales de l'institut Fourier

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Any $r$ -dimensional subbundle of the cotangent bundle on an $n$ -dimensional manifold $M$ partitions $M$ into subsets $M_{0}, ..., M_{m}$ ( $m$ being the minimum of $r$ and $C (n - r, 2)$ , the combinations of $n - r$ things taken 2 at a time). $M_{i}$ is the set on which the first derived systems of the subbundle has codimension $i$ . In this paper we prove the following: Theorem. Let $s \geq 2$ and let $Q$ be a generic $C^{s}$ $r$ -dimensional subbundle of the cotangent bundle of an $n$ -dimensional manifold $M$ . The codimension...

Liftings of 1-forms to $(J^{r} T ) $

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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Let $J^{r} T * M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r} T * M) *$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r} T * M) *$ is given.

Constructions on second order connections

J. Kurek, W. M. Mikulski (2007)

Annales Polonici Mathematici

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We classify all $ℱ ℳ_{m, n}$ -natural operators $: J ² ⟿ J ² V^{A}$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $(Γ) : V^{A} Y \to J ² V^{A} Y$ on the vertical Weil bundle $V^{A} Y \to M$ corresponding to a Weil algebra A.

Grassmann manifold $V_{3}^{4}$ in the projective space $P_{7}$ with characteristics consisting of a quadric and two planes

Josef Vala (1993)

Mathematica Bohemica

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Some results in the geometry of four-parametric manifolds of three-dimensional spaces in the projective space $P_{7}$ are found. The properties of such a manifold $V_{3}^{4}$ with characteristics consisting of a quadric and two planes are studied. The properties of the manifold dual to $V_{3}^{4}$ are found. Some results in the geometry of linear spaces from [1],[2],[3],[4] are used. The notation of the quantities is the same as in [4].

Natural affinors on the (r,s,q)-cotangent bundle of a fibered manifold

J. Kurek, W. M. Mikulski (2004)

Annales Polonici Mathematici

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For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle $T^{r, s, q *} Y$ over an (m,n)-dimensional fibered manifold Y.

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

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We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A} M$ , where $T^{A}$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic...

Bundle functors with the point property which admit prolongation of connections

W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let F:ℳ f →ℱℳ be a bundle functor with the point property F(pt) = pt, where pt is a one-point manifold. We prove that F is product preserving if and only if for any m and n there is an $ℱ ℳ_{m, n}$ -canonical construction D of general connections D(Γ) on Fp:FY → FM from general connections Γ on fibred manifolds p:Y → M.

Gauge natural constructions on higher order principal prolongations

Miroslav Doupovec, Włodzimierz M. Mikulski (2007)

Annales Polonici Mathematici

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Let $W_{m}^{r} P$ be a principal prolongation of a principal bundle P → M. We classify all gauge natural operators transforming principal connections on P → M and rth order linear connections on M into general connections on $W_{m}^{r} P \to M$ . We also describe all geometric constructions of classical linear connections on $W_{m}^{r} P$ from principal connections on P → M and rth order linear connections on M.

Lifting adapted connections from foliated manifolds to higher order adapted frame bundles

J. Kurek, W. M. Mikulski (2008)

Annales Polonici Mathematici

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Let (M,ℱ) be a foliated manifold. We describe all natural operators lifting ℱ-adapted (i.e. projectable in adapted coordinates) classical linear connections ∇ on (M,ℱ) into classical linear connections (∇) on the rth order adapted frame bundle $P^{r} (M, ℱ)$ .

On the Helmholtz operator of variational calculus in fibered-fibered manifolds

W. M. Mikulski (2007)

Annales Polonici Mathematici

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A fibered-fibered manifold is a surjective fibered submersion π: Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q an (r,s,q)th order Lagrangian on a fibered-fibered manifold π: Y → X is a base-preserving morphism $λ : J^{r, s, q} Y \to ⋀^{d i m X} T * X$ . For p= max(q,s) there exists a canonical Euler morphism $(λ) : J^{r + s, 2 s, r + p} Y \to * Y \otimes ⋀^{d i m X} T * X$ satisfying a decomposition property similar to the one in the fibered manifold case, and the critical fibered sections σ of Y are exactly the solutions of the Euler-Lagrange equation $(λ) \circ j^{r + s, 2 s, r + p} σ = 0$ . In the present...

The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

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Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators $T_{p r o j | ℱ ℳ_{m, n}} ⇝ T^{(0, 0)} (F_{| ℱ ℳ_{m, n}}) *$ transforming projectable vector fields on Y to functions on the dual bundle (FY)* for any $ℱ ℳ_{m, n}$ -object Y. Next, under some assumption on F we study natural operators $T *_{h o r | ℱ ℳ_{m, n}} ⇝ T * (F_{| ℱ ℳ_{m, n}}) *$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T *_{h o r | ℱ ℳ_{m, n}} ⇝ T * (F_{| ℱ ℳ_{m, n}}) *$ for some vector bundle functors F on fibered manifolds.

On canonical constructions on connections

Jan Kurek, Włodzimierz Mikulski (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We study how a projectable general connection $Γ$ in a 2-fibred manifold $Y^{2} \to Y^{1} \to Y^{0}$ and a general vertical connection $Θ$ in $Y^{2} \to Y^{1} \to Y^{0}$ induce a general connection $A (Γ, Θ)$ in $Y^{2} \to Y^{1}$ .

Natural operators lifting linear vector fields from a vector bundle to its r-jet prolongations

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

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All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r} E$ of E are given. Similar results are deduced for the r-jet prolongations $J_{v}^{r} E$ and $J^{[r]} E$ in place of $J^{r} E$ .

The natural operators lifting 1-forms to some vector bundle functors

J. Kurek, W. M. Mikulski (2002)

Colloquium Mathematicae

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Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{| ℳ f ₙ} ⇝ T^{(0, 0)} (F_{| ℳ f ₙ}) *$ transforming vector fields to functions on the dual bundle functor $(F_{| ℳ f ₙ}) *$ . Next, we study the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ lifting 1-forms to $(F_{| ℳ f ₙ}) *$ . As an application we classify the natural operators $T *_{| ℳ f ₙ} ⇝ T * (F_{| ℳ f ₙ}) *$ for some well known vector bundle functors F.

Remarks on the behaviour of higher-order derivations on the gluing of differential spaces

Krzysztof Drachal (2015)

Czechoslovak Mathematical Journal

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This paper is about some geometric properties of the gluing of order $k$ in the category of Sikorski differential spaces, where $k$ is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour...

Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles

Vadim V. Shurygin, Svetlana K. Zubkova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The second order transverse bundle $T_{}^{2} M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra $𝔻^{2}$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $𝔻^{2}$ -smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $𝔻^{2}$ -smooth foliated diffeomorphism between two second order transverse...

On prolongations of projectable connections

Jan Kurek, Włodzimierz M. Mikulski (2011)

Annales Polonici Mathematici

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We extend the concept of r-order connections on fibred manifolds to the one of (r,s,q)-order projectable connections on fibred-fibred manifolds, where r,s,q are arbitrary non-negative integers with s ≥ r ≤ q. Similarly to the fibred manifold case, given a bundle functor F of order r on (m₁,m₂,n₁,n₂)-dimensional fibred-fibred manifolds Y → M, we construct a general connection ℱ(Γ,Λ):FY → J¹FY on FY → M from a projectable general (i.e. (1,1,1)-order) connection $Γ : Y \to J^{1, 1, 1} Y$ on Y → M by means of an...

Some Classes of Lorentzian $α$ -Sasakian Manifolds Admitting a Quarter-symmetric Metric Connection

Santu DEY, Buddhadev Pal, Arindam BHATTACHARYYA (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The object of the present paper is to study a quarter-symmetric metric connection in an Lorentzian $α$ -Sasakian manifold. We study some curvature properties of an Lorentzian $α$ -Sasakian manifold with respect to the quarter-symmetric metric connection. We study locally $φ$ -symmetric, $φ$ -symmetric, locally projective $φ$ -symmetric, $ξ$ -projectively flat Lorentzian $α$ -Sasakian manifold with respect to the quarter-symmetric metric connection.

On natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$

A. Ntyam, A. Mba (2009)

Annales Polonici Mathematici

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Some properties and applications of natural vector bundle morphisms $T_{A} \circ ⨂_{s}^{q} \to ⨂_{s}^{q} \circ T_{A}$ over $i d_{T_{A}}$ are presented.

The Killing Tensors on an $n$ -dimensional Manifold with $S L (n,)$ -structure

Sergey E. Stepanov, Irina I. Tsyganok, Marina B. Khripunova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $S L (n,)$ -structure and discuss possible applications of obtained results in Riemannian geometry.

Structure of a neighbourhood of a complex compact submanifold in a complex manifold

Roman Dwilewicz

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CONTENTSIntroduction...................................................................................................................................................................51. Notation and some definitions...................................................................................................................................6 (a) Analytic families.....................................................................................................................................................6 (b)...

On prolongation of connections

Włodzimierz M. Mikulski (2010)

Annales Polonici Mathematici

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Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. Let r, m,n be positive integers. We present a construction $B^{r}$ of rth order holonomic connections $B^{r} (Γ, \nabla) : Y \to J^{r} Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M. Then we prove that any construction B of rth order holonomic connections $B (Γ, \nabla) : Y \to J^{r} Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M is equal...

Lifting to the r-frame bundle by means of connections

J. Kurek, W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let m and r be natural numbers and let $P^{r} : ℳ f_{m} \to ℱ ℳ$ be the rth order frame bundle functor. Let $F : ℳ f_{m} \to ℱ ℳ$ and $G : ℳ f_{k} \to ℱ ℳ$ be natural bundles, where $k = d i m (P^{r} ℝ^{m})$ . We describe all $ℳ f_{m}$ -natural operators A transforming sections σ of $F M \to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G (P^{r} M) \to P^{r} M$ . We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

The vertical prolongation of the projectable connections

Anna Bednarska (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We prove that any first order $ℱ_{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operator transforming projectable general connections on an $(m_{1}, m_{2}, n_{1}, n_{2})$ -dimensional fibred-fibred manifold $p = (p, p) : (p_{Y} : Y \to Y) \to (p_{M} : M \to M)$ into general connections on the vertical prolongation $V Y \to M$ of $p : Y \to M$ is the restriction of the (rather well-known) vertical prolongation operator $𝒱$ lifting general connections $\bar{Γ}$ on a fibred manifold $Y \to M$ into $𝒱 \bar{Γ}$ (the vertical prolongation of $\bar{Γ}$ ) on $V Y \to M$ .