The Manifold of Euclidean Inner Products of Sphere

Dana Mihaela Petrosanu

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

We give an example for the manifold of all Euclidean inner products on various tangent spaces of a manifold, building the manifold S(S^{n}) of all Euclidean inner products for the sphere S^{n}. We obtain a representation of points of this manifold by matrices classes.

Mathematics Subject Classification: 58D17 Key words: Linear frame, fibre bundle, manifold of metrics

1 Introduction

Let S(M) be the manifold of all Euclidean inner products given on various tangent spaces of a n−dimensional real smooth manifold M [5 ]. This is the n + n(n + 1)/2−dimensional real smooth manifold S(M) = L(M) ×_{GL(n;R)} S(R^{n}) -the total space of the fibre bundle over the base M, with standard fibre the homogeneous space S(R^{n}) = GL(n; R)/O(n) which is associated with the principal bundle L(M) of linear frames of M. The homogeneous space S(R^{n}) is exactly the manifold of all Euclidean inner products on R^{n}. The manifold S(M) is different from the manifold M(M) of all Riemannian metrics of M which is an infinite smooth manifold [2], but M(M) is exactly the manifold of all global sections of the fibre bundle S(M).

We denote by π : S(M) → M the canonical projection. The fibre π^{−1}(x) over x ∈ M may be considered as S(T_{x}M)- the space of Euclidean inner products on the tangent space T_{x}M. If {x^{1}, ..., x^{n}} are local coordinates in a neighborhood U

of the point x ∈ M, let \{θ_{1}, ..., θ_{n}\} be the canonical frame, θ_{i} = \frac{∂}{∂x^{i}} , i = 1, ..., n. The vertical space in the point g ∈ S(M), π(g) = x is the space of all symmetric endomorphisms of T_{x}M related with g. If we denote by V_{g}S(M) the vertical space at the point g ∈ S(M),then, locally,

V_{g}S(M) = {Â(g) | A ∈ gl(n; R), Â(g) = Ag + g^{t}A}, π(g) = x ∈ U,

where gl(n; R) is the Lie algebra of endomorphisms of R^{n} and ^{t}A is the transposition of A ∈ gl(n; R) related with the standard inner product e on R^{n}.

Balkan Journal of Geometry and Its Applications, Vol.5, No.1, 2000, pp. 129-132 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

If ∇ is a linear connection on M, the horizontal space H_{g} at the point g ∈ S(M) is locally spanned by {θ̂_{1}, ..., θ̂_{n}}, where for i = 1, ..., n, θ̂_{i} is the horizontal lift of θ_{i} and is given by θ̂_{i} = θ_{i} + Γ̂_{i}(x)(g), Γ_{i} = (Γ_{ij}^{k})_{jk} ∈ gl(n; R), with ∇_{θ_{i}}θ_{j} = Γ_{ij}^{k}θ_{k}, (∀) i, j = 1, ..., n..., n.

2 The principal bundle of linear frames of the sphere

Let M be a manifold of dimension n. A linear frame at a point x ∈ M is an ordered basis (X_{1}, ..., X_{n}) of a tangent space T_{x}M. We denote by L(M) the set of all linear frames u at all points of M and let π′ be the mapping of L(M) onto M witch maps a linear frame u at x into x.The general linear group GL(n; R) acts on L(M) on the right as follows. If a = (a_{ij})_{ij} ∈ GL(n; R) and u = (X_{1}, ..., X_{n}) is a linear frame of T_{x}M, x ∈ M,then ua is, by definition, the linear frame (Y_{1}, ...., Y_{n}) of T_{x}M defined by Y_{i} = ∑_{j} a_{i}^{j}X_{j}. In order to introduce a differentiable structure in L(M), let (x^{1}, ..., x^{n}) be a local coordinate neighborhood U in M. Every frame u at T_{x}M, x ∈ M, can be expressed uniquely in the form u = (X_{1}, ..., X_{n}), with X_{i} = ∑_{k} X_{i}^{k}∂/∂x^{k}, i = 1, ..., k . Then, (x_{1}, ..., x_{n}, X_{i}^{k})_{ik} is a local coordinate system in π^{′−1}(U). It is easy to verify that L(M) is the total space of a principal fibre bundle over the base M with structure group GL(n; R), denoted by L(M) to [1].

Let (R^{n+1}, <, >) be the standard Euclidean n + 1−dimensional space and let S^{n} = {x ∈ R^{n+1} |< x, x >= 1} be the n−dimensional sphere. A frame at the point x ∈ S^{n} is given by n independents vectors of R^{n+1}, orthogonals on the vector x.If u = (X_{1}, X_{2}, ..., X_{n}) is a frame in x ∈ S^{n}, x = (x_{1}, ..., x_{n+1}),where X_{i} = (x_{1i}, ..., x_{n+1i}), i = 1, ..., n, then we consider the matrix

(1)  X = ( \begin{matrix}x_{11} & . & . & . & x_{1n} & x_{1} \\ x_{21} & . & . & . & x_{2n} & x_{2} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ x_{n+11} & . & . & . & x_{n+1n} & x_{n+1} \end{matrix} ) ∈ GL(n + 1; R).

We observe initially that ^{t}XX ∈ G̃L(n; R), where G̃L(n; R) is the Lie subgroup of the Lie group GL(n + 1; R) with the elements the non–degenerated matrices

(2)  M̃ = ( \begin{matrix}M & 0 \\ 0 & 1 \end{matrix} ) ,  M ∈ GL(n; R)

and ^{t}X is the transposed of the matrix X.

It is easy to check that the manifold of linear frames of the n-dimensional sphere L(S^{n}), is just: L(S^{n}) = {X ∈ GL(n+1; R) |^{t} XX ∈ G̃L(n; R)}, and, so, we give a representation of L(S^{n}) as matrices. The Lie group GL(n; R), identified with G̃L(n; R), acts naturally, freely, on the manifold L(S^{n}) through matriceal multiplication. Let π′ : L(S^{n}) → S^{n} be the projection π′(X) = x, x = (x_{1}, ...., x_{n+1}) ∈ S^{n} where X ∈ L(S^{n}) is give by (1). We observe that for every M̃ ∈ G̃L(n; R), we have π′(XM̃) = π′(X) and, moreover, for all X, Y ∈ L(S^{n}) we have π′(X) = π′(Y) if and only if there is a matrix M̃ ∈ GL(n; R) so that Y = XM̃. Consequently, L(S^{n}) is the total space of the fibre bundle of linear frames of S^{n}.

Let M be the set of all matrices X ∈ GL(n + 1, R) with the property ^{t}XX = λI_{n+1} , λ ∈ R, I_{n+1} being the identity matrix. Then, M is a Lie subgroup of GL(n+ 1; R) if and only if λ = 1, and so M = O(n + 1)− the group of orthogonal matrices of order n + 1.

3 Construction of the manifold of Euclidean inner products on the sphere

According with general theory [1], the fibre bundle over the base S^{n}, with standard fibre S(R^{n}) and structure group GL(n; R) which is associated with the principal bundle L(S^{n}) is obtained as follows. On the product manifold L(S^{n})XS(R^{n}), GL(n; R) ≃ G̃L(n; R) acts on the right as follows: an element M̃ ∈ G̃L(n; R) maps (X, g) ∈ L(S^{n})XS(R^{n}) into (X, g)M̃ = (XM̃, M̃^{−1}g̃M̃^{−1}) ∈ L(S^{n})XS(R^{n}), with

g̃ = ( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ), where we are denoted by g the associated matrix with the inner prod-

uct g on R^{n} related to the canonical frame of R^{n}. The quotient space of L(S^{n})XS(R^{n}) by this group action is the set S(S^{n}) of all Euclidean inner products on the sphere S^{n}.The mapping L(S^{n})XS(R^{n}) → S^{n} which maps (X, g) in π′(X) induces a mapping π, called the projection, of S(S^{n}) onto S^{n}. We can introduce a differentiable structure in S(S^{n}) by the requirement that for every coordinates neighborhood U of the manifold S^{n}, the set π^{−1}(U) is diffeomorphic with UXS(R^{n}). The projection π is then a differentiable mapping of S(S^{n}) onto S^{n}.

We observe now that every element X ∈ L(S^{n}) is given by the conditions (1), (2) and therefore give rise at an inner product on the tangent space T_{x}S^{n}.Indeed, this inner product have his associated matrix M related to the frame (X_{1}, X_{2}, ..., X_{n}) of T_{x}S^{n} , the matrix:

(3)  M = ( \begin{matrix}< X_{1}, X_{1} > & < X_{1}, X_{2} > & . & . & . & < X_{1}, X_{n} > \\ < X_{2}, X_{1} > & < X_{2}, X_{2} > & . & . & . & < X_{2}, X_{n} > \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ < X_{n}, X_{1} > & < X_{n}, X_{2} > & . & . & . & < X_{n}, X_{n} > \end{matrix} ) .

We have immediately that

(3′)  ^{t}XX = ( \begin{matrix}M & 0 \\ 0 & 1 \end{matrix} ) ,

with the matrix M given by (3). Moreover, if we have X, Y ∈ L(S^{n}) with the same last columns x = (x_{1}, ..., x_{n+1}), then ^{t}XX =^{t} Y Y if and only if there is an element O ∈ Õ(n) so that B = OA where Õ(n) = O(n+1)∩G̃L(n; R).Therefore, the matrices X and Y of L(S^{n}), with the same last columns x, give rise at the same inner product of T_{x}S^{n} if and only if Y = XO with O ∈ Õ(n) . Following [5], we may conclude that

S(S^{n}) = {A^{t}S̃(R^{n})A | A ∈ GL(n + 1; R),^{t} AA ∈ G̃L(n; R)} =
= {A^{t}A | A ∈ GL(n + 1; R), A^{t}A ∈ G̃L(n; R)}.

This last remark shows that the vertical space V_{g} of the fibre bundle S(S^{n}), in the point g =^{t} AA ∈ S(S^{n}), which is the tangent space at the fibre through g at g, is

V_{g} = {M̂(g) | M̂(g) = M̃A^{t}A + A^{t}A^{t̃}M, M̃ ∈ G̃L(n; R)}.

In order to obtain in every point g of the manifold S(S^{n}) an horizontal space H_{g} , we introduce on the sphere S^{n} a linear connexion ∇. Then, the tangent space T_{g}S(S^{n}) at the point g ∈ S(S^{n}) splits, and we have T_{g}S(S^{n}) = V_{g} ⊕ H_{g}. We take now in consideration the property of S^{n} to be a Riemannian manifold and therefore the existence of an atlas of S^{n} with her Jacobi’s matrices orthogonals. Then, the manifold of Euclidean inner product S(S^{n}) may be endowed in a natural way with a Riemannian metric G so that the horizontal and vertical spaces at every point g are orthogonal spaces and

(4)  G(M̂(g), N̂(g)) = \frac{1}{2}T rM̂(g)N̂(g)

(4′)  G(X̂_{g}, Ŷ_{g}) = g(X, Y),

where M̂(g), N̂(g) ∈ V_{g}, X̂_{g}, Ŷ_{g} ∈ H_{g} and, for every horizontal vector X̂_{g}, we have denoted by X his projection on S^{n}.

References

[1] S. Kobayashi, K. Nomizu, Foundations of differential Geometry I, Interscience Publishers, New York, London, 1963.

[2] Olga Gil-Medrano, Peter W.Michor, The Riemannian manifold of all Riemannian metrics, Quart. J. Math. Oxford (2), 42(1991, 183-202.

[3] K. Nomizu, T. Sasaki, Affine differential geometry, Geometry of affine immersions, Cambridge University Press, 1994.

[4] S. Sternberg, Lectures on differential geometry, Prentice–Hall, Inc. Englewood Cliffs, New Jersey, 1964.

[5] A. Turtoi, The manifold of Euclidean inner products, The fourth International Workshop of Differential Geometry and its Applications, Brasov, Romania, Sept.1-22, 1999 (in print).

[6] C. Udriste, Differential Geometry, Differential Equations (in romanian), Geometry Balkan Press, Bucuresti, Romania, 1997.

University Politehnica of Bucharest Department of Mathematics I Splaiul Independenţei 313 77206 Bucharest, Romania