Structure (differential geometry)

In differential geometry, a structure on a manifold , a geometric quantity, or a field of geometric objects is the section of the bundle associated with the main bundle of corepers of a variety $M$ ${\ displaystyle M}$ $M$ . Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on the point $x$ ${\ displaystyle x}$ $x$ variety $M$ ${\ displaystyle M}$ $M$ , but also from the choice of core, that is, from the choice of the infinitesimal coordinate system at the point $x$ ${\ displaystyle x}$ $x$ (see also Map ).

Formal definition of a structure on a variety

To formally define structures on a manifold, we consider $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ Is a general differential group of order $k$ ${\ displaystyle k}$ (group $k$ ${\ displaystyle k}$ -jet at zero space transformations $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ preserving the origin) $M_{k}$ ${\ displaystyle M_ {k}}$ - variety of order corers $k$ ${\ displaystyle k}$ $n$ ${\ displaystyle n}$ -dimensional diversity $M$ ${\ displaystyle M}$ (i.e. variety $k$ ${\ displaystyle k}$ -jet $j_{x}^{k}(u)$ ${\ displaystyle j_ {x} ^ {k} (u)}$ local cards $u:M\supset U\to \mathbb {R} ^{n}$ ${\ displaystyle u: M \ supset U \ to \ mathbb {R} ^ {n}}$ starting at point $x=u^{-1}(0)$ ${\ displaystyle x = u ^ {- 1} (0)}$ )

Group $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ acts on the left on the manifold $M_{k}$ ${\ displaystyle M_ {k}}$ according to the formula

j_{0}^{k}(\varphi )j_{0}^{k}(u)=j_{x}^{k}(\varphi \circ u),\quad j_{0}^{k}(\varphi )\in GL^{k}(n),\quad j_{x}^{k}(u)\in M_{k}.

{\ displaystyle j_ {0} ^ {k} (\ varphi) j_ {0} ^ {k} (u) = j_ {x} ^ {k} (\ varphi \ circ u), \ quad j_ {0} ^ {k} (\ varphi) \ in GL ^ {k} (n), \ quad j_ {x} ^ {k} (u) \ in M_ {k}.}

This action defines $M_{k}$ ${\ displaystyle M_ {k}}$ structure of the main $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ -bundles $\pi _{k}:M_{k}\to M$ ${\ displaystyle \ pi _ {k}: M_ {k} \ to M}$ called a bundle of order corers $k$ ${\ displaystyle k}$ .

Let now $W$ ${\ displaystyle W}$ - arbitrary $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ -manifold, i.e., a manifold with a left action of a group $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ , a $W(M)$ ${\ displaystyle W (M)}$ - space of orbits of the left action of the group $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ at $M_{k}\times W$ ${\ displaystyle M_ {k} \ times W}$ . Bundle $\pi _{W}:W(M)\to M$ ${\ displaystyle \ pi _ {W}: W (M) \ to M}$ , which is the natural projection of the orbit space onto $M$ ${\ displaystyle M}$ and associated with $W$ ${\ displaystyle W}$ so with $M_{k}$ ${\ displaystyle M_ {k}}$ is called a bundle of geometric structures of type $W$ ${\ displaystyle W}$ order no more $k$ ${\ displaystyle k}$ , and its sections with structures of the type $W$ ${\ displaystyle W}$ . Structures of this type are in natural one-to-one correspondence with $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ -quasivariant mappings $S:M_{k}\to W$ ${\ displaystyle S: M_ {k} \ to W}$ .

So structures like $W$ ${\ displaystyle W}$ can be seen as $W$ ${\ displaystyle W}$ -value function $S$ ${\ displaystyle S}$ on variety $M_{k}$ ${\ displaystyle M_ {k}}$ $k$ ${\ displaystyle k}$ -reper satisfying the following condition of equivariance:

S(gu^{k})=gS(u^{k}),\quad g\in GL^{k}(n),\quad u^{k}\in M_{k}.

{\ displaystyle S (gu ^ {k}) = gS (u ^ {k}), \ quad g \ in GL ^ {k} (n), \ quad u ^ {k} \ in M_ {k}.}

The bundle of geometric objects is a natural bundle in the sense that the group of diffeomorphisms of a manifold $M$ ${\ displaystyle M}$ acts as a group of automorphisms $\pi _{W}$ ${\ displaystyle \ pi _ {W}}$ .

If a $W$ ${\ displaystyle W}$ is a vector space with linear (respectively, affine) action of the group $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ , then structures like $W$ ${\ displaystyle W}$ are called linear (respectively, affine ).

The main examples of linear structures of the first order are tensor structures , or tensor fields . Let be $V=\mathbb {R} ^{n}$ ${\ displaystyle V = \ mathbb {R} ^ {n}}$ , $V^{*}=\mathrm {Hom} \,(V,\;\mathbb {R} )$ ${\ displaystyle V ^ {*} = \ mathrm {Hom} \, (V, \; \ mathbb {R})}$ and $V_{g}^{p}=((\otimes ^{p}V))\otimes ((\otimes ^{q}V^{*}))$ ${\ displaystyle V_ {g} ^ {p} = ((\ otimes ^ {p} V)) \ otimes ((\ otimes ^ {q} V ^ {*}))}$ Is the space of tensors of type $(p,\;q)$ ${\ displaystyle (p, \; q)}$ with the natural tensor representation of the group $GL^{1}(n)=GL(n)$ ${\ displaystyle GL ^ {1} (n) = GL (n)}$ . Type structure $V_{q}^{p}$ ${\ displaystyle V_ {q} ^ {p}}$ called a tensor field of type $(p,\;q)$ ${\ displaystyle (p, \; q)}$ . It can be considered as a vector function on the variety of corepers $M_{1}$ ${\ displaystyle M_ {1}}$ matching coreper $\theta =j_{1}^{1}(u)=(du^{1},\;du^{2},\;\ldots ,\;du^{n})$ ${\ displaystyle \ theta = j_ {1} ^ {1} (u) = (du ^ {1}, \; du ^ {2}, \; \ ldots, \; du ^ {n})}$ set of coordinates $S(\theta )_{j_{1}j_{2}\ldots j_{q}}^{i_{1}i_{2}\ldots i_{p}}$ ${\ displaystyle S (\ theta) _ {j_ {1} j_ {2} \ ldots j_ {q}} ^ {i_ {1} i_ {2} \ ldots i_ {p}}}$ tensor $S(\theta )\in V_{q}^{p}$ ${\ displaystyle S (\ theta) \ in V_ {q} ^ {p}}$ relative to the standard basis

\{e_{i_{1}}\otimes e_{i_{2}}\otimes \ldots \otimes e_{i_{p}}\otimes e^{*j_{1}}\otimes e^{*j_{2}}\otimes \ldots \otimes e^{*j_{q}}\}

{\ displaystyle \ {e_ {i_ {1}} \ otimes e_ {i_ {2}} \ otimes \ ldots \ otimes e_ {i_ {p}} \ otimes e ^ {* j_ {1}} \ otimes e ^ { * j_ {2}} \ otimes \ ldots \ otimes e ^ {* j_ {q}} \}}

of space $V_{q}^{p}$ ${\ displaystyle V_ {q} ^ {p}}$ . With linear conversion of the coroner $\theta \to g\theta =(g_{a}^{i}\,du^{a})$ ${\ displaystyle \ theta \ to g \ theta = (g_ {a} ^ {i} \, du ^ {a})}$ coordinates $S_{j_{1}j_{2}\ldots j_{p}}^{i_{1}i_{2}\ldots i_{p}}$ ${\ displaystyle S_ {j_ {1} j_ {2} \ ldots j_ {p}} ^ {i_ {1} i_ {2} \ ldots i_ {p}}}$ are transformed according to the tensor representation:

S_{j_{1}j_{2}\ldots j_{q}}^{i_{1}i_{2}\ldots i_{p}}(q\theta )=g_{a_{1}}^{i_{1}}g_{a_{2}}^{i_{2}}\ldots g_{a_{p}}^{i_{p}}(g^{-1})_{j_{1}}^{b_{1}}(g^{-1})_{j_{2}}^{b_{2}}\ldots (g^{-1})_{j_{q}}^{b_{q}}S(\theta )_{b_{1}b_{2}\ldots b_{q}}^{a_{1}a_{2}\ldots a_{p}}.

{\ displaystyle S_ {j_ {1} j_ {2} \ ldots j_ {q}} ^ {i_ {1} i_ {2} \ ldots i_ {p}} (q \ theta) = g_ {a_ {1}} ^ {i_ {1}} g_ {a_ {2}} ^ {i_ {2}} \ ldots g_ {a_ {p}} ^ {i_ {p}} (g ^ {- 1}) _ {j_ {1 }} ^ {b_ {1}} (g ^ {- 1}) _ {j_ {2}} ^ {b_ {2}} \ ldots (g ^ {- 1}) _ {j_ {q}} ^ { b_ {q}} S (\ theta) _ {b_ {1} b_ {2} \ ldots b_ {q}} ^ {a_ {1} a_ {2} \ ldots a_ {p}}.}

The most important examples of tensor structures are:

vector field ;
differential form ;
metric tensor ;
symplectic structure ;
complex structure ;
affinor .

All linear structures (of any orders) are exhausted by Rashevsky supertensors ^[1] .

An example of a second-order affine structure is the torsion-free affine connection , which can be considered as a structure of the type $V_{(2)}^{1}$ ${\ displaystyle V _ {(2)} ^ {1}}$ where $V_{(2)}^{1}\approx V\otimes S^{2}V^{*}$ ${\ displaystyle V _ {(2)} ^ {1} \ approx V \ otimes S ^ {2} V ^ {*}}$ - the core of natural homomorphism $GL^{2}(n)\to GL^{1}(n)$ ${\ displaystyle GL ^ {2} (n) \ to GL ^ {1} (n)}$ , which can be considered as a vector space with the natural action of the group $GL^{2}(n)=GL(n)V_{(2)}^{1}$ ${\ displaystyle GL ^ {2} (n) = GL (n) V _ {(2)} ^ {1}}$ .

Another important and fairly broad class of structures is the class of infinitesimal homogeneous structures , or $G$ ${\ displaystyle G}$ structures. They can be defined as structures of type $W$ ${\ displaystyle W}$ where $W=GL^{k}(n)/G$ ${\ displaystyle W = GL ^ {k} (n) / G}$ - homogeneous group space $GL^{k}(n)$ ${\ displaystyle GL ^ {k} (n)}$ .

For further generalization, we can consider general $G$ ${\ displaystyle G}$ -structures are principal bundles that are homomorphically mapped onto $G$ ${\ displaystyle G}$ -structure, and sections of bundles associated with them. In this case, one can consider a number of important general geometric structures, such as spinor structures , symplectic spinor structures , etc.

Literature

Bourbaki, N. Set Theory / Transl. with french - M .: Mir, 1965 .-- 457 p.
Veblen, O., Whitehead, J. Foundations of differential geometry . - M .: IIL, 1949 .-- 230 p.
Sternberg, S. Lectures on differential geometry . - M .: Mir, 1970 .-- 413 p.
Vasiliev, A. M. Theory of differential geometric structures . - M .: Moscow State University, 1987 .-- 190 p.
Laptev G.F. Basic infinitesimal structures of higher orders on a smooth manifold // Transactions of a Geometric Seminar. - v. 1. - M .: VINITI , 1966, p. 139-189.

Notes

↑ Rashevsky P.K. Proceedings of the Moscow Mathematical Society. - 1957. - T. 6. - p. 337-370.

[1] Rashevsky P.K. Proceedings of the Moscow Mathematical Society. - 1957. - T. 6. - p. 337-370.

Structure (differential geometry)

Formal definition of a structure on a variety

Literature

See also

Notes

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