A coordinate-free survey on pseudo-connections

              Rev.Acad.Canar.Cienc., V (Núm. 1), 125-137 (1993)
A COORDINA TE -FREE SURVEY ON PSEUDO-CONNECTIONS
Fernando Etayo
Departamento de Matemáticas, Estadística y Computación
Facultad de Ciencias. Universidad de Cantabria
Avda. de los Castros, s/n, 39071 SANTANDER
ABSTRACT. In thls paper we present the general framework of dlffer-ent
notlons of pseudo- and quasl-connectlons. We show the relatlons amone
them. Local coordlnates are avolded, looklng for a more readable style.
A.M.S. Math. Sub. Class.: 53815, 53C05.
Key words: Pseudo-Connectlon, Quasl-Connectlon, Generallzed Connec-tlon,
Otsukl Connectlon.
l. INTRODUCTION.
Let M be a real n-dimensional smooth manifold. We shall use the
following standard notation: :D(M) is the set of derivatives on M; ~(M) the
ring of functions; :X(M) the module of vector fields. As is well-known, a
derivative 8 is given by its action over the functions and the vector fields.
Finally, the set of ~(M)-linear endomorphisms of :X(M) can be identified with
the set of (1,1)-tensor fields. In general, the module of (r,s)-tensor fields
will be noted as '.Vr(M), and then, :X(M) = '.V 1(M).
• o
A linear connection on the manifold M (in the sense of Koszul, [KO],
[K.N]) is given by an ~(M)-linear map 17::X(M) ---? :D(M), 17: X ---? 17X , such
that 17x(fYJ = f 17X Y + (Xf)Y and 17X(Y+Zl = 17X(YJ + 17X(Zl, for ali fe~(MJ and
X, Y,Ze:X(M). Different generalizations of this concept are given by severa!
authors.
Definition 1.1. A linear pseudo-connection 17 on M with fundamental
tensor field Fe'.V:(M) is an ~(M)-linear map 17::X(M) ---? :D(M), 17: X ---? 17X ,
such that 17X(fYl = f 17X Y + (F(X)f)Y and 17X(Y+ZJ = 17X(Yl + 17X(Z), for ali
fe~(M) and X,Y,Ze:X(M).
This notion was introduced by Wong [W], Di Comite [DC.1] and
Giublesi [G]. 'In Wong's notation, this is a "quasi-connection". Moreover, this
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notion is introduced by Wong as the answer to a geometrical problem (see [W]).
In this paper we concentrate on the algebraic aspects of the definition. In
Obadeanu's notation [OB.21. it is called an "M-connection".
On the other hand, we have the "dual" definition:
Definition .lb An Otsuki connection 'il on M with fundamental tensor
field Fe~:(M) is an IR-linear map íl:'.r(M) x X(M) ~ '.r(M), íl: (X,Y) ~ 'i/XY ,
such that ílfXY = f 'i/XY and 'i/X(fY) = f 'i/XY + (Xf)F(Y), for ali fe?f(M) and
X, Y ,Ze'.r(M).
This notion was introduced by Otsuki [OT.1), [OT.2], as "general
connection". The name "Otsuki connection" is given by Abe [A.11. In [E.F.3) we
have given the name "fere-connection" for this operator.
In this paper we shall study the above topics and their natural
extensions. There exist severa! different notions of "general connections" or
"connections of higher order" which will not be showed in this work (see
[E.F.5) per bibl. ). For this reason, we shall not use the name "general
conne.ction" nor "generalized connection".
In §2 (resp. §3) we shall study linear pseudo-connections on a
manifold (resp. Otsuki connections). In §4 we shall study the extensions of
the above concepts on fibre bundles, with special emphasis on principal and
vector bundles. Finally, we s hall present a table with the different
definitions.
As we have pointed out in the abstract, local coordinates are avoid­ed,
looking for a more readable style.
2. LINEAR PSEUDO-CONNECTIONS ON A MANIFOLD.
The notion of a pseudo-connection (see definition 1.1) was intro­duced
in 1962 by Wong [W) and by Di Comite [DC.l) and Giublesi [G) in 1969. We
can define special kinds of pseudo-connections:
Definition li Let 'il be a linear pseudo-connection on M with
fundamental tensor field F.
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(a) V is banal pseudo-connection if F=O
(b) V is an quasi-connection if F is an isomorphism.
(e) V is a canonically generated pseudo-connection if there exists a
linear connection V on M such that V = V•F.
Notation (b) is introduced by Vamanu [V.l], [V.2].
Remark b.b_ A banal pseudo-connection is a tensor field of type
(1,2). A linear connection is a linear pseudo-connection with identity funda­mental
tensor field.
We have the following:
Proposition li Let V be a linear pseudo-connection on M.
(a) If V is a quasi-connection, then V is a canonically generated
quasi-connection.
(b) Let V be a linear connection on M. Then, V V•F + H, He!r~(M),
i. e., a linear pseudo-cónnection can de decomposed as the sum of a
canonically generated pseudo-connection and a banal pseudo-connection.
(e) The set of linear pseudo-connections on M is an affine space and
an ~(M)-module.
Proof. Let F be the fundamental tensor field of V. (a) Observe that
V•F-1 is a linear connection and then, V = (V •F- 1)oF as we wanted. (b) Observe
that V -V•F is a banal pseudo-connection H and then V = V•F + H. (e) If V and
V' are pseudo-connections with fundamental tensor fields F and F', then, for
ali telR, tV + (1-t)V' is a linear pseudo-connection with fundamental tensor
field tF + (1-t)F' and for ali f,ge~(M), fV + gV' is a linear pseudo­connection
with fundamenta l t ensor field fF + fF' . •
Remark b±:_ The set of linear connections on l.11 is not an ~(M)­module.
So, the concept of pseudo-connection gives an algebraic completion of
that of a connection.
Let V be a linear pseudo-connection with fundamental tensor field F.
Then, the theory of t or sion and curvature tensor fields of V can be
developped. In [DC.l] the following operator:
LF(X,Yl = [F(X),Y] + [X,F(Y)] - F[X,Y]
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and the following definitions are introduced:
Definition li (a) The torsion of V is given by T(X, Y) = VX Y - VYX
- LF(X,Y). (b) The curvature of V is given by R(X,Y) = [VX,VY) - VLF(X,Y)"
Remark ~ Te:1~(M) although Re:1~(M) if and only if NF vanishes, NF
being the Nijenhuis tensor field of F.
Bianchi identities, conjugate and symmetric pseudo-connections,
etc., can be obtained (see [DC.l), [E.F.l), [F.P.2), [G], etc.). Moreover, the
theories of geodesics (see [DC.4), [F.P.2), etc), projectively equivalent
pseudo-connections ( see [F. P. 2), [P. DT], etc.), linear pseudo-connections
adapted to structures and others (see [AM.C), [CR.1), [F.l.P), [F.P.l,2,3,4),
[G), [NI, [R), etc.), and the theory of lifts of pseudo-connections to the
tangent bundle ([F. l.P), [Mil and to the tangent bundle of order r ([E.F.4,6))
have been developped.
Moreover, !et us consider the concept of a differentiation on a
manifold: a map D: :1 1(M) ~ '.D(M), which assigns to each vector field X a
o
derivation DX which X as a vector field. lf D is an !1(M)-linear map the
differentiation is covariant, i.e., D is a linear connection. In [E.J.1,2,3)
J. Etayo introduces and studies the notion of a pseudo-differentiation: it is
map D: :1~(M) ~ '.D(M), which assigns to each vector field X a derivation DX
which F(X) as a vector field, F being a (1,1)-tensor field on M. If D is an
!1(M)-linear map then D is a linear pseudo-connection on M. Algebraic
properties of pseudodifferentiations can be found in [E.J.1), [PE), etc., and
lifts to the tangent bundle in [E.J.21.
On the other hand, Vamanu gave [V.!), [V.21. a new definition of
torsion and curvature of a quasi-connection V: the torsion T' is given by
T'(X,Yl = VXY - VYX F- 1[F(X),F(Y)] and the curvature of V is given by
R'(X,Yl = [VX,VY) - VF-1[F(X),F(Y))" Then, T'e:1~(M) and R'e:1~(M). Moreover, as
we have proved in [E.F. l], T=T' and R=R' if and only if the Nijenhuis tensor
field NF vanishes. Bianchi identities, conjugate and symmetric quasi­connections,
etc, can also be obtained with these definitions of torsion and
curvature (see [E.F.l ), [NAV.Hl , [PE), [V.1), [V.2), etc. ).
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3. OTSUKI CONNECTIONS ON A MANIFOLD.
In 1958 Otsuki introduced the notion of "general connection" (we
shall call it Otsuki connection, see definition 1.2) looking for a concept
which includes those of linear connection and (1,2)-tensor field. As we have
pointed out in Remark 2.2, the notion of a linear pseudo-connection also
includes those of linear connection and (1,2)-tensor field. Following our
introduction of pseudo-connections, we have:
Definition ~ Let 'í1 be an Otsuki connection on M with fundamental
tensor field F.
(a) 'í1 is a banal Otsuki connection if F=O
(b) 'í1 is a regular Otsuki connection if F is an isomorphism.
(c) 'í1 is a canonically generated Otsuki connection if there exists a
linear connection V on M such that 'í1 = v•r' where 'í1 X y (V*F)x y = V x(F(Y)).
Remark lb. A banal Otsuki connection is a tensor field of type
0,2). A linear connection is an Otsuki-connection with identity fundamental
tensor field.
The following proposition corresponds to 2.3 and has a similar
proof:
Proposition 2.l_ Let 'í1 be an Otsuki connection on M.
(a) If 'í1 is a regular Otsuki connection, then 'í1 is a canonically
generated Otsuki connection.
- - 1 (b) Let 'í1 be an Otsuki connection on M. Then, 'í1 = 'íl*F + H, He~ 2 (M),
i. e., an Otsuki connection can de decomposed as the sum of a canonically
generated and a banal Otsuki connection.
(e) The set of Otsuki connections on M is an affine space and an
~(M)-module.
Using propositions 2.3.(b) and 3.3. (b) we can prove the following
Theorem 2..1.,_ There exists a bijection from the set of linear pseudo­connections
on M onto the set of Otsuki connections on M.
Proof. Let V be a linear connection on M. Each pseudo-connection V
on M can be written as V = VoF + H in a unique manner. The associated Otsuki
connection is V*F + H.•
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As iq the case of linear pseudo-connections, the theories of
torsion, curvature, etc. of an Otsuki connection have been developped (see
[OT.2,3,4,5,6), etc. Specially indicated is [OT.6), which is a survey of the
theory, with emphasjs on geodesics and curvature. Moreover, this theory has
been used to a new presentation of General Relativity (see [NAG)).
4. PSEUDO- AND OTSUKI CONNECTIONS ON A FIBRE BUNDLE.
In this section, we show the different extensions of the above
concepts to fibre bundles. These extensions must verify the fellowing two
conditions: (1) connections on fibre bundles are pseudo- and Otsuki connec­tions
on fibre bundles; (2) pseudo- (resp. Otsuki) connections on a manifold
are pseudo- (resp. Otsuki) connections on the tangent bundle of the manifold.
Then, it is clear that we need the definition of a connection on a
fibre bundle (which it is often called a "general connection"!). We present
the results in three steps:
4.1. Connections and pseudo-connections on Vector Bundles.
Let n::E ----7 M be a vector bundle. Let r(E) denotes the :1(M)-module
of sections of n:. A linear connection on n: is defined as an IR-linear map:
V: X(M) X r(E) ~ r(E)
(X,s) ----? 'íJxs
such that VX(fs) = f
ser(E). (See [PO)).
'íJ Xs + (Xf)s and 'íJ fX s = f 'íJ Xs' fer ali fe:1(M), XeX(M) and
Then, it is easy to obtain the "natural" extensions of
pseudo- and Otsuki connections:
Definition ~ A linear pseudo-connection 'íJ on the vector bundle
n:, with fundamental tensor field Fe~ 1 (M) is an IR-linear map 'íJ: X(M) x r(E)
1
~ r(E), such that VX(fs) = f 'íJXs + (F(X)f)s and 'íJfX s = f 'íJXs, fer all
fe:1(M), XeX(M) and ser(E).
Definition 4.1. 2. An Otsuki linear connection 'íJ on the vector bundle
n:, with fundamental endomorphism Fe End(r(E)) is an IR-linear map V: X(M) x
r(E) ~ f(E) , such that 'íJX(fs) = f 'íJXs + (Xf)F(s) and 'íJfX s = f VXs' fer
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ali fE:J(Ml, XEX(Ml and ser(E), End(r(E)) being the :J(M)-module of endomorphims
of sectlons.
one,
These definitions are given in [CR.2]. Observe that in the first
FE~1 (M) and in the second one Fe End(r(E)).
1
Obviously, if F is the
identity map, in both cases we have linear connections. Moreover, if E=TM,
then we recover the pseudo-connection and Otsuki connection on a manifold.
The importance of these notions in the study qf subfibre bundles,
distributions and Finsler structures is pointed out in . [CR.2). Moreover,
Otsuki linear connections have been used in Paracomplex Geometry (see [B)) and
Relativity (see [NAG), [NE)).
Remark 4.1.3. The above definitions show a possible extension of the
corresponding concepts on manifolds. Neverthless, there are sorne other
possible extensions. For example, Candela [CA.1,2) defines a pseudoconnection
on a vector bundle n by means of a pair (A, IJ), where A:E ~ TM is an M­homomorphism
and IJ is an IR-linear map IJ: r(E) x f(E) ~ f(E), such that
IJs(ft) = f IJst + (Aos){flt and 'ii'fs t = f IJst, for ali fe'.1(M), s,ter(E). If
E=TM, we recover the definition of a pseudo-connection on a manifold. But if A
is the identity, the above concept does not coincide with that of a linear
connection on a vector bundle. Far this reason, we do not consider this notion
in the work.
4.2. Connections and pseudo-connections on Fibre Bundles.
Ehresmann [El gave the first definition of a connection on a locally
trivial fibre bundle n:E ~ M. Following [L) and [K.M.S) we can give three
equivalent definitions. First of ali, let us int' róduce sorne notations:
If n:E ~ M is a locally trivial fibre bundel (a fibre bundle in
the ·future), VE represents the vertical vector bundle VE = ker n. ~ E and
n -'l(TM) the pull-back of TM. Then, we obtain the following exact sequence of
vector bundles over E:
O ~ VE ~ TE ~ n-1(TM) ~ O
.pnally, J(E) denotes the !-jet bundle of local sections of n:E ~ M. Obser­ve
that il.:J(E) ~ E is an affine bundle.
A connection on n:E ~ M is given by:
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(1) an E-homomorphism V: TE ~ VE, such that Voi=id.
(2) an E-homomorphism W:rr- 1(TM) ~ TE, such that rr' o W=id.
(3) an M-homorphism 0:E ~ JE, such that i\o 0=id.
If rr :E ~ M is a vector bundle, definitions (1) and (2) have been
studied by Vilms in [VII. Giving sorne linear conditions one recovers the
definition obtained in §4.1. Moreover, in the case of a vector bundle one can
give the following exact sequence of vector bundles over M:
O --7T • M®E ~J1 (E) ----+E ---.+O
Definitions of pseudo- and Otsuki connections can be obtained as
follows:
Definition 4.2.1. A pseudo-connection on rr:E ~ M is an E-homomor­phism
W:rr - 1(TM) ~ TE.
Definition 4.2.2. An Otsuki connection on rr:E ~ M is an E-homo­morphism
V: TE ~VE.
These definitions were given by Spesivykh in [S. l,2,3], with
different notation. In fact, Spesivykh gave a more general concept (which was
called a "general connection" once again) which included almost every
definitions of connections, pseudo-connections, etc. It is obvious that
connections on fibre bundles are pseudo- and Otsuki connections. It is more
difficult to see that we can recover pseudo- and Otsuki connections on a
manifold when E=TM (see [E.F.21. [S.1,2,3]). Curvature, covariant derivative,
etc., have been studied by Spesivykh. Lifts of pseudo- and Otsuki connections
have been studied in [E.F.61. The above definitions had been pointed out in
[E.J.3].
Remark 4.2.3. Let W be a pseudo-connection. Then, rr 'o W: rr-1(TM) ~
rr - 1(TM). If «=rr' o W is an isomorphism, then a.o W is a connection, i. e., a
"quasi-connection" is a "canonically generated connection" (see proposition
2.3. (a)). The same idea for an Otsuki connection.
On the other hand, Abe [A.1,2] has defined an Otsuki connection (a
"general connection" in his notation!) on a vector bundle rr:E ~ M as an M­homomorphism
;r:E ~ JE. Obviously, this notion is a generalization of those
of a connection on a vector bundle and of an Otsuki connection on a manifold.
Moreover, it coincides with definition 4.2.2. when rr:E ~ M is a vector
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bundle, because J(E) ~ E is afl affine bundle modelled on the vector bundle
Hom(n-1(TM), VE) ~ E (see [L]). Covariant derivative, curvature, etc. · have
been studied by Abe [A.1,2], [A.N.Y].
4.3. Connections and pseudo-connections on Principal Bundles.
Pseudo-connections on principal bundles, specially the linear frame
bundle of a manifold, have been studied by the Italian School, developping the
theory together the theory of pseudo-connections on a manifold. Di Comite's
works [DC.2,3] began this topic.
Our starting-point will be the work [DC.3]: Let P(M,G) be a
principal bundle over M, with projection n:P ~ M and group G. Let R denotes
the action of G over P.
Definition 4.3.1. A pseudo-connection on P(M,G) is a tensor field re
~:(P), such that r(X) is a vertical vector field and í•(RªJ.=(Ra)••r, for ali
XeX(P) and aeG.
Obviously, connections on principal bundles (in the sense of [K.Nll
are pseudo-connections. The relationship between pseudo-connections on
principal bundles and pseudo-connections on manifolds is similar to that of
connections on principal bundles and connections on manifolds. So, a pseudo­connection
on the linear frame bundle of a manifold is called a linear pseudo­connection;
each linear pseudo-connection on a manifold induces a pseudo­connection
on its linear frame bundle; pseudo-connections on associated vector
bundles can be defined; pseudo-connection form can be studied; etc. These
results have been obtained by Di Comite and other Italian researches (see
[AM.F], [DC.2,3], [F], [FA.L], [M]. etc.).
On the other hand, pseudo-connections on a principal bundle are
related with "generalized connections" (once again!) of Verona [VE], as Di
Comite [DC.3] has pointed out (see also [MO]).
Remark 4. 3. 2. Observe that pseudo-connections on principal bundles
are Otsuki connections! in the sense of definition 4.2.2. For this reason
Spesivykh introduces a nota tion which differs from the ours. But it is not
possible to find a global coherent notation (see the Appendix).
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APPENDIX: RELATION AMONG THE CONCEPTS ANO DIFFERENT NAMING.
Pseudo-, quasi- and Otsuki connections on a manifold
17X( fY) = f 17XY + (F(X)flY 17X(fYl = f 17XY + (Xf )F(Y)
Pseudo-connection (.) Pseudo-connection
Di Comite, 1ta1 i an school Spesi v ykh
Quasi-connection General connection
Wong, Spesivykh Ot suk i
M-connection Otsuki connection (.)
Obadeanu Abe
Wong quasiconnection Otsuki quasiconnection
Cruceanu Crucean u
1 f F is an i somorph i sm 1 f F is an isomorphism
Quasi-connection (.) Regular general connection
Vamanu, Et ayo J .• Peralta Otsuki
Regular Otsuki connection (•)
(•) denotes the notation used in this paper.
Pseudo-, quasi- and Otsuki connections on a fibre bundle
O --~VE -~TE -~n - 1 ( TM) ---¿O O -~VE --~TE -~n - 1 (TM l ---¿O (!)
~- f-y--
0-~T • M®E -----¿J 1 (El -~E ___...¿O (2)
(------ r
Quasi-connection Pseudo- e onne et ion ( 1 )
Spesivykh Spesivykh
Pseudo-connection (•) General connection (2)
Abe
Otsuki connection (2) (.)
(•) denotes the notation used in this paper.
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