A tensor is symmetric in a given set of contravariant or covariant indices if it is equal to its symmetrized part on these indices, and is antisymmetric if it is equal to its antisymmetrized part. Thus, for example, a tensor T of type (0,2) is symmetric if T^ = + (which we can also express in the form: i[a6) =0).

A particularly important subset of tensors is the set of tensors of type (0, q) which are antisymmetric on all q positions (so q ^ n) \ such a tensor is called a q-form. If A and B arejp- and g-forms respectively, one can define a (jp + gJ-form A a B from them, where a is the skew-symmetrized tensor product ®; that is, A a B is the tensor of type (0, p + q) with components determined by

This rule implies (AaB) = (-)M(BaA). With this product, the space of forms (i.e. the space of all jp-forms for all p, including one-forms and defining scalars as zero-forms) constitutes the Grassmann algebra of forms. If {E°} is a basis of one-forms, then the forms E°i a ... a E°p (ai run from 1 to n) are a basis of jp-forms, as any jp-form A can be written A = Aa bE° a ... a E6, where Aa b =

So far, we have considered the set of tensors defined at a point on the manifold. A set of local coordinates {a;1} on an open set tfl in defines a basis {(d/dx1)]^ of vectors and a basis {(da;1)!,,} of one-forms at each point p of and so defines a basis of tensors of type (r, a) at onoh point of <ty. Sunh n. l>nmn of tonsors will bo callod n. coorflinnlo basis. A Ck tensor field T of type (r, s) on a set V c: ^ is an assignment of an element of Z%(p) to each point pe'V such that the components of T with respect to any coordinate basis defined on an open subset of y are Ck functions.

In general one need not use a coordinate basis of tensors, i.e. given any basis of vectors {Ea} and dual basis of forms {E°} on "V, there will not necessarily exist any open set in 'V on which there are local coordinates {a;°} such that E„ = d/dx" and E° = da;°. However if one does use a coordinate basis, certain specializations will result; in particular for any function/, the relations Ea(Ebf) = Eb(E,J) are satisfied, being equivalent to the relations d2f/dxadx^ = d2f/dx^dxa. If one changes from a coordinate basis E„ = 8/8xa to a coordinate basis Ea, = 8\dxa\ applying (2.2), (2.3) to a;", a;"' shows that

Clearly a general basis {E„} can be obtained from a coordinate basis

{d/ftc'} by giving the functions EJ which are the components of the En with respect to the basis {8/dx*}; then (2.2) takes the form Ea = Eaidjdxi and (2.3) takes the form E° = Eai6xi, where the matrix Eai is dual to the matrix EJ.

In this section we define, via the general concept of a Ck manifold map, the concepts of 'imbedding', 'immersion', and of associated tensor maps, the first two being useful later in the study of submanifolds, and the last playing an important role in studying the behaviour of families of curves as well as in studying symmetry properties of manifolds.

A map (¡> from a Ck n-dimensional manifold^ to a Ck' «'-dimensional manifold is said to be a Cr map (r < k, r < k') if, for any local coordinate systems in ^ and the coordinates of the image point 4>(p) in are CT functions of the coordinates of p in . As the map will in general be many-one rather than one-one (e.g. it cannot bo one-one if n > n), it will in general not have an inverse; and if a Cr map does have an inverse, this inverse will in general not be Cr (e.g. if <f> is the map Rl-+Rl given by a;-* a;3, then is not differentiable at the point x = 0).

If/is a function on the mapping 0 defines the function 0 */on as the function whose value at the point p of is the value of / at

Thus when <p maps points from to maps functions linearly from to

If A (t) is a curve through the point p then the image curve 0(A(i)) in passes through the point 4>{p). If r ^ 1, the tangent vector to this curve at 4>(p) will be denoted by one can regard it as the image, under the map <j>, of the vector (8jdt)x\p. Clearly is a linear map of TV(J() into T^{Ji'). From (2.5) and the definition (2.1) of a vector as a directional derivative, the vector map tp^ can be characterized by the relation: for each G' (r ^ 1) function / at 4>{p) and vector X at p,

Using the vector mapping from Jt to we can if r ^ 1 define a linear one-form mapping from to T*v(Jl) by the condition: vector-one-form contractions are to be preserved under the maps. Then the one-form AeT*^p) is mapped into the one-form 0*A eT*p where, for arbitrary vectors Xe5^,,

A consequence of this is that

The maps and <f>* can be extended to maps of contravariant tensors from ^ to and covariant tensors from to respectively, by the rules T e T£(p)->- eTfttfip)) where for any ri'sT*^, T(4>w ^ V)jj) = ...^rji^

and <f>*:TeT°(<f>(P))-+<f>*Te'r0s(p), where for any Xt- 6 Tp,

4>*T(Xlt...,Xs)\p - T(<j>ifX1 <&»XS)|,W.

When r > 1, the Cr map (¡> from to is said to be of rank s at p if the dimension of <j>m(Tv(*4i()) is s. It is said to be injeclive at p if« = n (and son < n') at p; then no vector in Tp is mapped to zero by $5*. It is said to be surjective if s = n' (so n > n').

A Cr map 0 {r > 0) is said to be an immersion if it and its inverse are Cr maps, i.e. if for each point p&J( there is a neighbourhood % of p in such that the inverse restricted to is also a Cr map. This implios n < n'. By l.ho implir.it- function thenrrin (Spivak (1965), p. 41), when r > 1, <p will be an immersion if and only if it is infective at every point p sJ(\ then is an isomorphism of Tp into the image <f>*(Tp) <= T^py The image is then said to be an «-dimensional immersed submanifdd in ui". This submanifold may intersect itself, i.e. <p may not be a one-one map from ^ to although it is one-one when restricted toasuffi cientlysmallneighbo ur-hood of^. An immersion is said to be an imbedding if it is a homeo-morphism onto its image in the induced topology. Thus an imbedding is a one-one immersion; however not all one-one immersions are imbeddings, cf. figure 6. A map <p is said to be a proper map if the inverse image 0-1pT) of any compact set JT <= is compact. It can be shown that a proper one-one immersion is an imbedding. The image <f>(J() oi^M under an imbedding 0 is said to be an «-dimensional imbedded submanifold of

The map 0 from ^ to is said to be a CT diffeomorphism if it is a one-one Cr map and the inverse tp'1 is a Cr map from to In this case, n = n', and <p is both injective and surjective if r > 1; conversely, the implicit function theorem shows that if is both injective and surjective atp, then there is an open neighbourhood ^ of p such that ^: % -> is a diffeomorphism. Thus 0 is a local diffeomorphism near p if is an isomorphism from Tp to

Figure 6. A one-one immersion of U1 in U! which is not an imbedding, obtained by joining smoothly part of the curve y = sin (l/z) to the curve

Figure 6. A one-one immersion of U1 in U! which is not an imbedding, obtained by joining smoothly part of the curve y = sin (l/z) to the curve

When the map 0 is a Cr (r > 1) diffeomorphism, maps to

T^M") and (0-1)*"maps T*p(J() to Thus we can define a map <f>* of Trs{p) to Trs(^(p)) for any r, s, by

for any Xt 6 Tp, tjisT*p. This map of tensors of type (r, s) on ^ to tensors of type (r, s) on preserves symmetries and relations in the tensor algebra; e.g. the contraction of <f>*T is equal to (the contraction of T).

We shall stud}' three differential operators on manifolds, the first two being defined purely by the manifold structure while the third is defined (see § 2.5) by placing extra structure on the manifold.

exterior differentiation

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