Lobachevskii Journal of Mathematics Vol. 7, 2000, 31 – 42 \begin{matrix}c \\ ◯ \end{matrix}Branko Sarić

Branko Sarić

A FUNCTIONAL EXPRESSION FOR THE CURVATURE OF HYPER-DIMENSIONAL RIEMANNIAN SPACES

(submitted by B.N.Shapukov)

Abstract. Analogously to a notion of curvature of a curve and a surface, in the differential geometry, in the main part of this paper the notion of curvature of hyper-dimensional vector spaces of Riemannian metric is generally defined. The defined notion of curvature of Riemannian spaces of higher dimensions M: M ≥ 2, in the further text of the paper, is functional related to the fundamental parameters of internal geometry of a space, more exactly, to components of Riemann-Christoffel’s curvature tensor. At the end, analogously to a notion of lines of a curvature in the differential geometry, the notion of sub-spaces of curvature of Riemannian hyper-dimensional vector spaces is also generally defined.

1. Introduction

The well-known Riemannian mathematical model of defining a curvature of hyper-dimensional curvilinear metric spaces, via Gaussian concept of the two-dimensional surface curvature, [1, 2, 4], is an imperfect in spite of the fact that it is functionally related to internal geometry of a space:

1. Firstly, for a reason that the concept of the curvature of hyper-dimensional metric spaces reduced to the concept of Gaussian curvature of two-dimensional geodesic surface, it cannot be said that the Riemannian concept of the curvature of hyper-dimensional spaces is general one because it is an inapplicable to one-dimensional space.

2. As secondly, since there are more than one geodesic surface in vector spaces of higher dimensions, it is obvious that Riemannian concept of the curvature of hyper-dimensional spaces, in the general case, is not uniquely defined one.

3. Finally, at any an individual, concrete case, from the practical point of view, it is not simple to come to the quantitatively usable functional expression for the Riemannian curvature of an analyzed hyper-dimensional metric space.

1991 Mathematical Subject Classification. Primary 53A35; Secondary 53A45.

Key words and phrases. Space, curvature of space, sub-space of curvature.

Hence, the one other mathematical model of defining the curvature of the hyper-dimensional metric spaces, which essentially differs from Riemannian model, is presented in this paper. Concretely, the mathematical model is being discussed, which can be said to be generalization of the well-known model of defining the curvature of a curve and a surface in the differential geometry.

1.1. Basic characteristics of spatial continuum. A notion of geometrical point is one of the fundamental notions. Closely related to the notion of geometrical points is the system of values (a_{1}, a_{2}, ... a_{N}) of some arbitrary N variables (x_{1}, x_{2}, ... x_{N}) such that a set of all geometrical points, and for all real values of the variables, is a real N - dimensional space of a spatial continuum, [1]. The geometrical point O, defined by system of zero values (0, 0, ... 0), is the origin of system of reference (of co-ordinate system) of a space. The vector r⃗ (x^{i}), defined with respect to the origin O, is a position vector. Note that a notion of vector, in the vector hyper-dimensional spaces (N > 3), should be conditionally comprehended in the sense of its geometrical presentation in a form of segments, hence it bears a name linear tensor, [4].

Covariant vectors e⃗_{i}: e⃗_{i} = ∂_{x_{i}}r⃗ (x^{j}), where ∂_{x_{i}} denotes \frac{∂}{∂x^{i}}, form the covariant vector basis \left\{{ \atop e_{i}}\right\}_{i=1}^{N} of the tangent space of a spatial continuum. The vectors e^{i}, such that at each point of a space: e⃗_{i} · e^{k} = δ_{i}^{k}, where the second order system of the unit values δ_{i}^{k} - is an unit N × N matrix (Kronecker’s delta-symbol), [1, 2, 4], form a basis {e^{i}}_{i=1}^{N}, which is called the dual basis of the covariant vector basis \left\{{ \atop e_{i}}\right\}_{i=1}^{N}. This is so-called natural isomorphism from \left\{{ \atop e_{i}}\right\}_{i=1}^{N} onto \{e^{i}\}_{i=1}^{N}. The infinitesimal value dr⃗ of the position vector r⃗ of a representative point is defined by dr⃗ = dx^{i}e⃗_{i} = dx_{i}e^{i}, where the well-known Einstein’s convention is applied to summation with respect to the repetitive indexes (uppers and lowers), [1, 4], herein as well as in the further text of the paper.

2. The main results

2.1. A curvature of hyper-dimensional Riemannian spaces. By the following transformation low: x^{i} = x^{i} (q^{α}); i = 1, 2, ..., N, α = 1, 2, ..., M ≤ N, in the general case, an arbitrary M - dimensional metrical space is defined, [1]

ds^{2} = ∂_{x^{i}}r⃗∂_{q^{α}}x^{i} · ∂_{x^{j}} r⃗∂_{q^{β}}x^{j}dq^{α}dq^{β} = e_{ij}∂_{q^{α}}x^{i}∂_{q^{β}}x^{j}dq^{α}dq^{β} = g_{αβ}dq^{α}dq^{β},

where the positive definite symetric square matrices e_{ij}: e_{ij} = e⃗_{i} · e⃗_{j} and g_{αβ}:

g_{αβ} = ∂_{q^{α}}r⃗ · ∂_{q^{β}} r⃗ = g⃗_{α} · g⃗_{β} = ∂_{x^{i}}r⃗∂_{q^{α}}x^{i} · ∂_{x^{j}} r⃗∂_{q^{β}}x^{j} = e⃗_{i} · e⃗_{j}∂_{q^{α}}x^{i}∂_{q^{β}}x^{j},

of degree: N and M, respectively, are fundamental (metrical) tensors of an ambient N - dimensional Euclidean (more exactly Cartesian) space x^{i}, as well as, in the general case of Riemannian covering map (M < N), of an internal M - dimensional Riemannian curvilinear space q^{α} immersed in it. As it is well-known if M = N a smooth map x^{i} → q^{α} is called an isometric immersion.

The smallest possible dimensional difference C: C = N − M, is said to define a class of the Riemannian vector space q^{α}, [1]. Hence, a Riemannian space q^{α}: q^{α} = q, of a class C: C = N −1, is an arbitrary curve of the ambient N - dimensional Euclidean (Cartesian) space x^{i}.

Since the vector d_{q}g⃗, where d_{q} denotes \frac{d}{dq}, as derivative of the fundamental vector g⃗: g⃗ = d_{q}r⃗, lying in the tangent space of a Riemannian space q of a class C: C = N − 1 (on the tangent of curve), along the curve, from the point of view of the interior of an ambient space x^{i}, is a vector of the ambient space x^{i}, then at each points of the curve there exist C: C = N − 1, the linearly independent and mutually orthogonal unit vectors: n^{Λ}; Λ = 1, 2, ..., N − 1, being orthogonal on the unit vector of tangent t⃗: t⃗ = d_{s}r⃗, of the curve, such that

d_{qq}^{2}r⃗ = d_{q} (d_{q}x^{i}e⃗_{i}) = ∂_{x^{j}} e⃗_{i}d_{q}x^{j}d_{q}x^{i} + d_{qq}^{2}x^{i}e⃗_{i} =  (2.1)
= d_{q}g⃗ = (d_{q}g⃗ · n^{Λ}) n⃗_{Λ} + (d_{q}g⃗ · g⃗) g⃗.

The covariant vectors n⃗_{Σ}, as well as the vectors n^{Λ} satisfying the condition: n⃗_{Σ} · n^{Λ} = δ_{Σ}^{Λ}, form covariant \left\{{ \atop n_{Σ}}\right\}_{Σ=1}^{N−1}, as well as dual vector basis \{n^{Λ}\}_{Λ=1}^{N−1} of normal vector space of a Riemannian space q of class C: C = N − 1, respectively.

From the point of view of the internal geometry of a Riemannian space q, the vector κ⃗:

κ⃗ = (d_{q}g⃗ · n^{Λ}) n⃗_{Λ},

is a vector of curvature of curve, more exactly, from the point of view of the internal geometry of an ambient Euclidean space x^{i}, it is a vector of geodesic curvature of curve, [1]. Namely, by projection of the vector κ⃗ onto the normal vector space of a Riemannian space q and onto the tangent vector space of an ambient Euclidean space x^{i}, it is obtained that

κ⃗ · n^{Σ} = d_{q}g⃗ · n^{Σ} = (∂_{x^{j}} e⃗_{i} · e^{k}d_{q}x^{i}d_{q}x^{j} + d_{qq}^{2}x^{k}) (e⃗_{k} · n^{Σ}) =
= (Γ_{ij}^{k}d_{q}x^{i}d_{q}x^{j} + d_{qq}^{2}x^{k}) n_{k}^{Σ}

and

κ⃗ · e^{l} = (d_{q}g⃗ · n^{Λ}) n⃗_{Λ} · e^{l} = (Γ_{ij}^{k}d_{q}x^{i}d_{q}x^{j} + d_{qq}^{2}x^{k}) e⃗_{k} · n^{Λ} (n⃗_{Λ} · e^{l}) ,

where the mixed systems Γ_{ij}^{k}: Γ_{ij}^{k} = ∂_{x^{j}} e⃗_{i} · e^{k}, are coefficients of connection (Christoffel’s symbols) of the second type, and an intensity of vector

\frac{1}{g⃗ · g⃗} (Γ_{ij}^{k}d_{q}x^{i}d_{q}x^{j} + d_{qq}^{2}x^{k}) e⃗_{k} = (d_{s}t⃗ · e^{k}) e⃗_{k} = d_{s}t⃗,

being in the direction of the vector κ⃗:

\frac{1}{g⃗ · g⃗} (Γ_{ij}^{k}d_{q}x^{i}d_{q}x^{j} + d_{qq}^{2}x^{k}) (e⃗_{k} · n^{Λ}) n⃗_{Λ} = \frac{1}{g⃗ · g⃗} (κ⃗ · e^{l}) e⃗_{l} = \frac{1}{g⃗ · g⃗} κ⃗,

defines both the curvature κ of the curve, from the point of view of the internal geometry of a Riemannian space q of a class C: C = N − 1 and geodesic curvature κ_{g} of the curve, from the point of view of the internal geometry of an ambient Euclidean space x^{i}, [1, 4].

Analogously to the vector of curvature: κ⃗ = (d_{q}g⃗ · n^{Λ}) n⃗_{Λ}, of a Riemannian space q of class C: C = N − 1, in the general case of a Riemannian space q^{α} of class C: C = N − M, immersed in the N - dimensional ambient Euclidean space x^{i}, from the viewpoint of its internal geometry, the matrix scheme of vectors

K⃗_{αβ} = (∂_{q^{β}} g⃗_{α} · n^{Λ}) n⃗_{Λ} = τ_{αβ}^{Λ}n⃗_{Λ},  (2.2)

is a matrix scheme of the sectional curvature vectors, of a Riemannian space q^{α} of class C: C = N − M. The determinant |τ_{αβ}^{Λ}τ_{γδ}^{Σ}g^{δβ}n_{ΛΣ}| of the matrix obtained by the matrix multiplication (rows × columns) of the matrix schemes of curvature vectors: τ_{αβ}^{Λ}n⃗_{Λ} and τ_{γδ}^{Σ}g^{δβ}n⃗_{Σ}, is the proportionate to the square of the curvature κ of a Riemannian space q^{α} of class C: C = N −M, and with the coefficient of proportionality that is equal to the determinant of matrix of fundamental tensor g_{αβ}

|K⃗_{αβ} · K⃗_{γ}^{β}| = |τ_{αβ}^{Λ}τ_{γδ}^{Σ}g^{δβ}n_{ΛΣ}| = |g_{αβ}| κ^{2},

more exactly,

κ^{2} = \frac{|τ_{αβ}^{Λ}τ_{γδ}^{Σ}g^{δβ}n_{ΛΣ}|}{|g_{αβ}|}.  (2.3)

Comment: If N - dimensional ambient space x^{i} is either a Riemannian curvilinear space of a class C or an Euclidean curvilinear space of a class C: C ≥ 1, both with normal vector space w⃗_{P}; P = 1, 2, ..., C, then the matrix scheme of vectors

K⃗_{αβ} = (∂_{q^{β}} g⃗_{α} · n^{Λ}) n⃗_{Λ} + (∂_{q^{β}} g⃗_{α} · w^{P} ) w⃗_{P} ,

is a matrix scheme of vectors of the curvature of a Riemannian space q^{α} of class Ĉ: Ĉ = C + N − M.

From the point of view of internal geometry of the ambient space x^{i}, the partial matrix schemes of vectors of curvature: N⃗_{αβ} = (∂_{q^{β}} g⃗_{α} · w^{P} ) w⃗_{P} = κ_{αβ}^{P}w⃗_{P} and G⃗_{αβ} = (∂_{q^{β}} g⃗_{α} · n^{Λ}) n⃗_{Λ} = τ_{αβ}^{Λ}n⃗_{Λ}, are matrix schemes of vectors of normal curvature, as well as of geodesic curvature, of a Riemannian space q^{α} of class Ĉ, respectively.

The square of intensity of curvature κ of a Riemannian space q^{α} of class Ĉ, in this case is equal to the sum of square of the normal curvature κ_{n}:

κ_{n} = \frac{|κ_{αβ}^{P}κ_{γδ}^{L}g^{βδ}w_{P L}|}{|g^{αβ}|}

and of the geodesic curvature κ_{g}:

κ_{g} = \frac{|τ_{αβ}^{Λ}τ_{γδ}^{Σ}g^{βδ}n_{ΛΣ}|}{|g^{αβ}|},

of a Riemannian space q^{α} of class Ĉ:

κ^{2} = κ_{n}^{2} + κ_{g}^{2}.

The space q^{α} of a class Ĉ, as well as of geodesic curvature zero κ_{g}: κ_{g} = 0, with respect to the ambient space x^{i} of a class C, is a geodesic sub-space of that ambient space.▼ 2.1.1. Functional expression for the curvature of hyper-dimensional Riemannian spaces, from the point of view of internal geometry of a space. If the vector functional equality (2.1)

∂_{q^{β}} g⃗_{α} = (∂_{q^{β}} g⃗_{α} · g^{γ}) g⃗_{γ} + (∂_{q^{β}} g⃗_{α} · n^{Λ}) n⃗_{Λ},

in the general case of an ambient Riemannian space q^{α} of class C: C = N −M, is partially differentiated with respect to dual co-ordinates q^{δ}, it is obtained that

∂_{q^{β}q^{δ}}^{2}g⃗_{α} = ∂_{q^{δ}}Γ_{αβ}^{γ}g⃗_{γ} + Γ_{αβ}^{γ}∂_{q^{δ}}g⃗_{γ} + ∂_{q^{δ}}τ_{αβ}^{Λ}n⃗_{Λ} + τ_{αβ}^{Λ}∂_{q^{δ}} n⃗_{Λ},  (2.4)

that is,

∂_{q^{δ}q^{β}}^{2}g⃗_{α} = ∂_{q^{β}}Γ_{αδ}^{γ}g⃗_{γ} + Γ_{αδ}^{γ}∂_{q^{β}} g⃗_{γ} + ∂_{q^{β}}τ_{αδ}^{Λ}n⃗_{Λ} + τ_{αδ}^{Λ}∂_{q^{β}} n⃗_{Λ}.  (2.5)

Since differentials dg⃗_{α} = ∂_{q^{β}} g⃗_{α}dq^{β} are absolute ones, more exactly, the covariant vector basis \left\{{ \atop g_{α}}\right\} of a Riemannian space q^{α} of class C: C = N − M, is uniquely defined at each point of the space, and in accordance with that the condition of integrability: ∂_{q^{β}q^{δ}}^{2}g⃗_{α} −∂_{q^{δ}q^{β}}^{2}g⃗_{α} = 0, is satisfied, then by projection of difference of the respectable vector functional expressions on both sides of the previous equations: (2.4) and (2.5), onto covariant vector basis \left\{{ \atop g_{ρ}}\right\}, the functional relation is obtained

∂_{q^{δ}}Γ_{αβ}^{γ}g_{γρ} − ∂_{q^{β}}Γ_{αδ}^{γ}g_{γρ} + Γ_{αβ}^{λ}Γ_{λδ}^{γ}g_{γρ} − Γ_{αδ}^{λ}Γ_{λβ}^{γ}g_{γρ} = τ_{αβ}^{Λ}τ_{δρ,Λ} − τ_{αδ}^{Λ}τ_{βρ,Λ},

having in mind both the equation (2.2) and orthogonality of vectors: g⃗_{α} and n⃗_{Λ}. On account of the fact that the functional expression on the left hand side of preceding relation represents Riemann-Christoffel’s curvature tensor of a Riemannian space q^{α} of class C: C = N − M, [1, 2, 4], finally it follows that

R_{ραδβ} = τ_{αβ}^{Λ}τ_{δρ,Λ} − τ_{αδ}^{Λ}τ_{βρ,Λ}.  (2.6)

By an application of Gauss-Chiò’s procedure for condensation of determinants, [3], and on the basis of the tensorial relation (2.6), as well as of the functional relation (2.3), it is possible the curvature of a Riemannian space q^{α} of class C: C = N − M, to come into functional relation to internal geometry of a space, more exactly to the components of Rimann-Christoffel’s curvature tensor R_{ραδβ}.

Namely, if in addition to the matrix scheme of vectors: K⃗_{αβ} = τ_{αβ}^{Λ}n⃗_{Λ}, the matrix schemes of vectors:

Φ⃗_{αβ} = [ \begin{matrix}ϕ^{Λ}n⃗_{Λ} & 0 & ... & 0 & 0 \\ τ_{12}^{Λ}n⃗_{Λ} & τ_{11}^{Λ}n⃗_{Λ} & ... & 0 & 0 \\ ... & ... & ... & ... & ... \\ τ_{1(M−1)}^{Λ}n⃗_{Λ} & 0 & ... & τ_{11}^{Λ}n⃗_{Λ} & 0 \\ τ_{1M}^{Λ}n⃗_{Λ} & 0 & ... & 0 & τ_{11}^{Λ}n⃗_{Λ} \end{matrix} ];

Z⃗_{αβ} = [ \begin{matrix}τ_{12}^{Λ}n⃗_{Λ} & 0 & ... & 0 & 0 \\ τ_{11}^{Λ}n⃗_{Λ} & ζ^{Λ}n⃗_{Λ} & ... & τ_{1(M−1)}^{Λ}n⃗_{Λ} & τ_{1M}^{Λ}n⃗_{Λ} \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & τ_{12}^{Λ}n⃗_{Λ} & 0 \\ 0 & 0 & ... & 0 & τ_{12}^{Λ}n⃗_{Λ} \end{matrix} ];

Ξ⃗_{αβ} = [ \begin{matrix}ξ^{Λ}n⃗_{Λ} & 0 & ... & 0 & 0 \\ τ_{22}^{Λ}n⃗_{Λ} & τ_{21}^{Λ}n⃗_{Λ} & ... & 0 & 0 \\ ... & ... & ... & ... & ... \\ τ_{2(M−1)}^{Λ}n⃗_{Λ} & 0 & ... & τ_{21}^{Λ}n⃗_{Λ} & 0 \\ τ_{2M}^{Λ}n⃗_{Λ} & 0 & ... & 0 & τ_{21}^{Λ}n⃗_{Λ} \end{matrix} ];

Ψ⃗_{αβ} = [ \begin{matrix}τ_{22}^{Λ}n⃗_{Λ} & 0 & ... & 0 & 0 \\ τ_{21}^{Λ}n⃗_{Λ} & ψ^{Λ}n⃗_{Λ} & ... & τ_{2(M−1)}^{Λ}n⃗_{Λ} & τ_{2M}^{Λ}n⃗_{Λ} \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & τ_{22}^{Λ}n⃗_{Λ} & 0 \\ 0 & 0 & ... & 0 & τ_{22}^{Λ}n⃗_{Λ} \end{matrix} ] ,

are also introduced into analysis, where: ϕ^{Λ}n⃗_{Λ};ζ^{Λ}n⃗_{Λ};ξ^{Λ}n⃗_{Λ} and ψ^{Λ}n⃗_{Λ}, are arbitrary vector functions of the normal vector space n⃗_{Λ} of a Riemannian space q^{α} of class C: C = N − M, then having in view the fact that determinant of the product of matrices, as it is well-known, is equal to the product of determinants of any matrix separately, [1, 3], it follows that

(Φ⃗_{γη} · K⃗_{δ}^{η}) (K⃗_{αλ} · Ψ⃗_{β}^{λ}) = \begin{matrix}(τ_{11}^{Λ}ϕ_{Λ}) & (τ_{22}^{Λ}ψ_{Λ}) \\ |g_{αβ}| & 2 \end{matrix} ×  (2.7)

× | \begin{matrix}R_{1212} & R_{1213} & ... & R_{121M} \\ R_{1312} & R_{1313} & ... & R_{131M} \\ ... & ... & ... & ... \\ R_{1M12} & R_{1M13} & ... & R_{1M1M} \end{matrix} | | \begin{matrix}R_{1212} & R_{1232} & ... & R_{12M2} \\ R_{3212} & R_{3232} & ... & R_{32M2} \\ ... & ... & ... & ... \\ R_{M212} & R_{M232} & ... & R_{M2M2} \end{matrix} | =

= \begin{matrix}(τ_{11}^{Λ}ϕ_{Λ}) & (τ_{22}^{Λ}ψ_{Λ}) \\ |g_{αβ}| & 2 \end{matrix}Δ_{τ_{11}^{Λ}}^{R}Δ_{τ_{22}^{Λ}}^{R},

more exactly^{1}

|K⃗_{αδ} · K⃗_{β}^{δ}| (τ_{11}^{Λ}τ_{22,Λ})^{M−2} = \frac{1}{|g_{αβ}|} | \begin{matrix}R_{1212} & R_{1213} & ... & R_{121M} \\ R_{1312} & R_{1313} & ... & R_{131M} \\ ... & ... & ... & ... \\ R_{1M12} & R_{1M13} & ... & R_{1M1M} \end{matrix} | ×

× | \begin{matrix}R_{1212} & R_{1232} & ... & R_{12M2} \\ R_{3212} & R_{3232} & ... & R_{32M2} \\ ... & ... & ... & ... \\ R_{M212} & R_{M232} & ... & R_{M2M2} \end{matrix} | = \frac{1}{|g_{αβ}|}Δ_{τ_{11}^{Λ}}^{R}Δ_{τ_{22}^{Λ}}^{R},

|Φ⃗_{αδ} · Ψ⃗_{β}^{δ}| = \frac{1}{|g_{αβ}|} | \begin{matrix}ϕ^{Λ}τ_{22,Λ} & 0 & ... & 0 \\ − (τ_{22}^{Λ}τ_{12,Λ} + τ_{11}^{Λ}τ_{21,Λ}) & ψ^{Λ}τ_{11,Λ} & ... & −τ_{11}^{Λ}τ_{2M,Λ} \\ ... & ... & ... & ... \\ −τ_{22}^{Λ}τ_{1M,Λ} & 0 & ... & τ_{11}^{Λ}τ_{22,Λ} \end{matrix} | =

= \frac{1}{|g_{αβ}|} (ϕ^{Λ}τ_{22,Λ}) (ψ^{Λ}τ_{11,Λ}) (τ_{11}^{Λ}τ_{22,Λ})^{M−2} .

1

considering the fact that the functional expression in the relation (2.7), which is independent of the choice of arbitrary vector functions: ϕ^{Λ}n⃗_{Λ} and ψ^{Λ}n⃗_{Λ} (the specially interesting case is the one in which ϕ^{Λ} = ψ^{Λ}), defines the determinant |K⃗_{αδ} · K⃗_{β}^{δ}|.

Similarly

|K⃗_{αδ} · K⃗_{β}^{δ}| (τ_{12}^{Λ}τ_{21,Λ})^{M−2} = \frac{1}{|g_{αβ}|} | \begin{matrix}R_{1212} & R_{1232} & ... & R_{12M2} \\ R_{1213} & R_{1233} & ... & R_{12M3} \\ ... & ... & ... & ... \\ R_{121M} & R_{123M} & ... & R_{12MM} \end{matrix} | ×

× | \begin{matrix}R_{2121} & R_{2123} & ... & R_{212M} \\ R_{2131} & R_{2133} & ... & R_{213M} \\ ... & ... & ... & ... \\ R_{21M1} & R_{21M3} & ... & R_{21MM} \end{matrix} | = \frac{1}{|g_{αβ}|}Δ_{τ_{12}^{Λ}}^{R}Δ_{τ_{21}^{Λ}}^{R}.

On the basis of the functional formulation of the curvature of a Riemannian space q^{α} of class C: C = N − M, the relation (2.3): κ^{2} = \frac{|K⃗_{αδ}·K⃗_{β}^{δ}|}{|g_{αβ}|}, as well as

of the two previously derived relations, and for (M > 2), finally it follows that

κ^{2} = \frac{1}{|g_{αβ}|^{2} (R_{1212})^{M−2}} [(Δ_{τ_{11}^{Λ}}^{R}Δ_{τ_{22}^{Λ}}^{R})^{\frac{1}{M−2}} − (Δ_{τ_{12}^{Λ}}^{R}Δ_{τ_{21}^{Λ}}^{R})^{\frac{1}{M−2}} ]^{M−2} . (2.8)

Clearly, in the case of a Riemannian space q^{α} of class C: C = N − M (M = 2), functional expression (2.3) for the curvature of a surface is reduced to the well-known Gaussian curvature of a surface in the theory of surfaces, [1, 5]

κ = \frac{R_{1212}}{|g_{αβ}|} .

Comment: In the case when the component of Riemann-Christoffel’s curvature tensor is equal to zero, it is possible, in the functional expression (2.8) for the curvature of Riemannian spaces, to take any another combination of the components of the matrix scheme of the curvature vectors of a space: K⃗_{αβ} = τ_{αβ}^{Λ}n⃗_{Λ}, as the support elements of Gauss-Chiò’s procedure for condensation of determinants, and for which some of the components of Riemann-Christoffel’s curvature tensor are not equal to zero. Clearly, if all components of Riemann-Christoffel’s curvature tensor of a space are identically equal to zero, then the space is a Riemannian space of curvature zero (an Euclidean space).▼

2.2. Sub-spaces of curvature of hyper-dimensional spaces. Let N - dimensional ambient space of space continuum x^{i}, just as in the first Comment of preceding Section 2.1 of this paper, be either a Riemannian space of class C or an Euclidean curvilinear space of class C: C ≥ 1, both with the normal vector space w⃗_{P}; P = 1, 2, ..., C.

If and only if the vector components:

(∂_{q^{δ}} N⃗_{αβ}g^{αδ} · e⃗_{i}) e^{i} = κ_{αβ}^{P}g^{αδ} (∂_{q^{δ}} w⃗_{P} · e⃗_{i}) e^{i},

of vectors obtained by a partial differentiation of the matrix scheme of vector of normal curvature: N⃗_{αβ} = (∂_{q^{δ}}g⃗_{α} · w^{P} ) w⃗_{P} = κ_{αβ}^{P}w⃗_{P}, of a Riemannian space q^{α} of class Ĉ: Ĉ = C + N − M, immersed in an ambient space x^{i} of class C, are vectors of the tangent vector space g⃗_{α} of a Riemannian space q^{α} of class Ĉ, in other words if and only if the mutually equivalent conditions:

− (∂_{q^{δ}} N⃗_{αβ}g^{αδ} · e⃗_{i}) e^{i} = κ̂^{2}e_{ij}∂_{q^{β}}x^{j}e^{i} = κ̂^{2} (g⃗_{β} · e⃗_{i}) e^{i} = κ̂^{2}g⃗_{β}  (2.9)

and

−κ_{αβ}^{P}g^{αδ} (∂_{q^{δ}} w⃗_{P} · e⃗_{i}) = κ_{αβ}^{P}g^{αδ}t_{ij,P} ∂_{q^{δ}}x^{j} = κ̂^{2}e_{ij}∂_{q^{β}}x^{j},  (2.10)

are satisfied^{2}, then the Riemannian sub-space q^{α} of class Ĉ, is a sub-space of the curvature of the ambient space x^{i} of class C. In the case in which the Riemannian space q^{α} of class Ĉ: Ĉ = C + N − M, is an arbitrary curve (M = 1), the preceding conditions are reduced to the conditions: − (∂_{q}N⃗ \frac{1}{g⃗·g⃗} · e⃗_{i}) e^{i} =

κ̂^{2}e_{ij}∂_{q}x^{j}e^{i} and (d_{q}g⃗\frac{1}{g⃗·g⃗} · w^{P} ) t_{ij,P} ∂_{q}x^{j}e^{i} = κ̂^{2}e_{ij}∂_{q}x^{j}e^{i}, as well as (d_{s}t⃗ · w^{P} ) t_{ij,P} ∂_{s}x^{j}e^{i} = k_{ij}∂_{s}x^{j}e^{i} = κ̂^{2}e_{ij}∂_{s}x^{j}e^{i}.

The last of them is the well-known condition for the curve to be a line of the curvature of an ambient space x^{i} of class C, [5]. It is obvious from this that conditions: (2.9) and (2.10), are a generalization of the preceding conditions.

By projection of the condition (2.9), onto covariant vector basis \left\{{ \atop g_{γ}}\right\}, it is obtained that

κ_{αβ}^{P}g^{αδ}t_{ij,P} ∂_{q^{δ}}x^{j}∂_{q^{γ}}x^{i} = κ_{αβ}^{P}κ_{γ,P}^{α} = κ̂^{2}e_{ij}∂_{q^{β}}x^{j}∂_{q^{γ}}x^{i} = κ̂^{2}g_{βγ},

more exactly,

(κ_{n})^{2} = \frac{|κ_{αβ}^{P}κ_{γ,P}^{α} |}{|g_{βγ}|} = κ̂^{2M},

with regard to the relation (2.3).

Comment: A Riemannian curvature space q^{α} of class Ĉ, as a sub-space of an ambient space x^{i} of class C, is said to be the principal, if and only if

t_{j,P}^{k} t_{ik}^{P}∂_{q^{β}}x^{j} = κ̂^{2}e_{ij}∂_{q^{β}}x^{j},  (2.11)

in other words, if all vectors of the vector components: (∂_{q^{δ}} w⃗_{P} · e^{i}) e⃗_{i}, are vectors of the tangent vector space (see the Footnote 2 of the paper).

2The second of the two relations of equality is reduced to: t_{k,P}^{i} t_{ij}^{P}∂_{q^{β}}x^{k} = κ̂^{2}e_{ij}∂_{q^{β}}x^{i}, clearly on the condition that all vectors of vector components (∂_{q^{δ}}w^{P} · e^{i}) e⃗_{i} are also vectors of the tangent vector space g⃗_{α}. Namely, since: t_{k,P}^{i} ∂_{q^{β}}x^{k} = ∂_{q^{β}} w⃗_{P} · e^{i} and κ_{αβ,P} g^{αδ}∂_{q^{δ}}x^{i} = (∂_{q^{β}} w⃗_{P} · g⃗_{α}) g^{α} · g^{δ} (e^{i} · g⃗_{δ}), then κ_{αβ,P} g^{αδ}∂_{q^{δ}}x^{i} = t_{k,P}^{i} ∂_{q^{β}}x^{k}.

In order to exist nontrivial solutions of the previous homogeneous linear system (2.11) of ordinary differential equations with respect to the unknowns ∂_{q^{β}}x^{j}, the following condition

|t_{j,P}^{k} t_{ik}^{P} − κ̂^{2}e_{ij}| = 0,  (2.12)

must be satisfied, [1, 3].

On the basis of the developed form of the preceding condition (2.12), expressed by polynomial of N-th degree with respect to the unknown κ̂^{2}, as well as of Viète’s formulas [3], and taking the relation (2.3) into consideration, finally it follows that

k^{2} = \frac{|t_{j,P}^{k} t_{ik}^{P}|}{|e_{ij}|} = ∏^{N} ; i=1κ̂_{i}^{2}.

If the varied of all eigevalues κ̂_{i} of the matrix: t_{j,P}^{k} t_{ik}^{P}, is unit, then the principal Riemannian curvature sub-spaces q^{α} of class Ĉ, are onedimensional ones (M = 1), in other words there exist N principal directions of curvature with the normal curvatures: (κ_{n})_{i} = κ̂_{i}, of the ambient space x_{i} of class C. Under the condition that there exists at least one of all eigevalues κ̃ with the varied M, then there exists at least one principal M - dimensional Riemannian curvature sub-spaces q^{α} of class Ĉ, with the normal curvature: κ_{n} = κ̃^{M}, as well as N − M mutually orthogonal principal directions of curvature with the normal curvatures: (κ_{n})_{i} = κ̂_{i} , such that they are orthogonal onto the principal M - dimensional Riemannian curvature sub-spaces q^{α} of class Ĉ. In this emphasized case

k = κ̃^{M} ∏^{N−M} ; i=1κ̂_{i}.

Furthermore, if the vector components:

(∂_{q^{δ}} N⃗_{αβ} · e⃗_{i}) e^{i} = κ_{αβ}^{P} (∂_{q^{δ}} w⃗_{P} · e⃗_{i}) e^{i},

are also vectors of the tangent vector space g⃗_{δ} of the Riemannian subspace q^{α} of class Ĉ

− (∂_{q^{δ}} N⃗_{αβ} · e⃗_{i}) e^{i} = κ̂^{2}g_{αβ}e_{ki}∂_{q^{δ}}x^{k}e^{i} = κ̂^{2}g_{αβ} (g⃗_{δ} · e⃗_{i}) e^{i} = κ̂^{2}g_{αβ}g⃗_{δ},

more exactly,

κ_{αβ}^{P}t_{ij,P} ∂_{q^{γ}}x^{i}∂_{q^{δ}}x^{j} = κ_{αβ}^{P}κ_{γδ,P} = κ̂^{2}g_{αβ}g_{γδ},

then, the normal curvature κ_{n} of curvature sub-space q^{α} of class Ĉ, of the ambient space x^{i} of class C, is determined by the functional form

(κ_{n})^{2} = \frac{|κ_{αβ}^{P}κ_{δ,P}^{α} |}{|g_{αβ}|} = κ̂^{2M},

more exactly,

M \sqrt{(κ_{n})^{2}} = κ̂^{2} = \frac{(t_{ij}^{P}t_{kl,P} − t_{il}^{P} t_{kj,P} ) ∂_{q^{α}}x^{i}∂_{q^{β}}x^{j}∂_{q^{γ}}x^{l}∂_{q^{δ}}x^{k}}{(e_{ij}e_{kl} − e_{il}e_{kj}) ∂_{q^{α}}x^{i}∂_{q^{β}}x^{j}∂_{q^{γ}}x^{l}∂_{q^{δ}}x^{k}};

M \sqrt{(κ_{n})^{2}} = κ̂^{2} = \frac{R_{ikjl}∂_{q^{α}}x^{i}∂_{q^{β}}x^{j}∂_{q^{γ}}x^{l}∂_{q^{δ}}x^{k}}{(e_{ij}e_{kl} − e_{il}e_{kj}) ∂_{q^{α}}x^{i}∂_{q^{β}}x^{j}∂_{q^{γ}}x^{l}∂_{q^{δ}}x^{k}} =

= \frac{R_{αδβγ}}{g_{αβ}g_{γδ} − g_{αγ}g_{βδ}} = −\frac{R}{M (M − 1)},

where the scalar invariant R: R = g^{αγ}g^{βδ}R_{αδβγ}, is so-called invariant of curvature (scalar curvature), [1].

It can be proved by an application of the result of the Schur theorem, [1], that the Riemannian curvature sub-spaces q^{α} of class Ĉ, of the ambient space x^{i} of class C, whether they are principals or not, in this emphasized case are isotropic spaces of the constant normal curvature κ_{n}.▼

3. Example

The curvature of Riemannian spaces:

ds^{2} = e^{μ(ρ)}dρ^{2} + ρ^{2}dθ^{2} + sin^{2} θdφ^{2} + e^{ν(ρ)}dτ^{2};

μ (ρ) = −ν (ρ) = − ln (1 − \frac{2m}{ρ}); m = const.

and

ds̄^{2} = e^{μ(ρ)} (dρ^{2} + ρ^{2}dθ^{2} + sin^{2} θdφ^{2} + dτ^{2});

μ (ρ) = −\frac{2m}{ρ}; m = const.,

with spherical symmetry.

Components of Christoffel’s symbols:

Γ_{αβ,γ} = \frac{1}{2} (∂_{q^{α}}g_{βγ} + ∂_{q^{β}}g_{αγ} − ∂_{q^{γ}}g_{αβ}) ,

as well as of Reimann-Christoffel’s curvature tensor:

R_{αβδγ} = ∂_{q^{δ}}Γ_{βγ,α} − ∂_{q^{γ}}Γ_{βδ,α} + g^{λσ} (Γ_{βδ,λ}Γ_{αγ,σ} − Γ_{βγ,λ}Γ_{αδ,σ}) ,

which are not identically zeros, for these sub-classes of the general class of Riemannian spaces with spherical symmetry, are the following forms:

Γ_{11,1} = Γ̄_{11,1} = \frac{1}{2}e^{μ}∂_{ρ}μ; Γ_{22,1} = −Γ_{12,2} = −ρ; Γ̄_{22,1} = −Γ̄_{12,2} = −\frac{ρe^{μ}}{2} (2 + ρ∂_{ρ}μ) ;

Γ_{33,1} = −Γ_{13,3} = −ρ sin^{2} θ; Γ̄_{33,1} = −Γ̄_{13,3} = −\frac{ρ sin^{2} θe^{μ}}{2} (2 + ρ∂_{ρ}μ) ;

Γ_{33,2} = −Γ_{23,3} = −ρ sin θ cos θ; Γ̄_{33,2} = −Γ̄_{23,3} = −ρ^{2}e^{μ} sin θ cos θ;

Γ_{44,1} = −Γ_{14,4} = −\frac{1}{2}e^{ν}∂_{ρ}ν; Γ̄_{44,1} = −Γ̄_{14,4} = −\frac{1}{2}e^{μ}∂_{ρ}μ,

as well as

R_{1212} = \frac{1}{2}ρ∂_{ρ}μ; R̄_{1212} = \frac{1}{2}ρe^{μ} (∂_{ρ}μ + ∂_{ρρ}^{2}μ);

R_{1313} = \frac{1}{2}ρ sin^{2} θ∂_{ρ}μ; R̄_{1313} = \frac{1}{2}ρe^{μ} sin^{2} θ (∂_{ρ}μ + ρ∂_{ρρ}^{2}μ);

R_{2323} = ρ^{2} sin^{2} θ(1 − e^{−μ}); R̄_{2323} = −ρ^{3}e^{μ}∂_{ρ}μ sin^{2} θ (1 + \frac{ρ}{4}∂_{ρ}μ);

R_{1414} = \frac{1}{2}e^{ν} \{\frac{1}{2} [∂_{ρ}μ∂_{ρ}ν − (∂_{ρ}ν)^{2}] − ∂_{ρρ}^{2}ν\}; R̄_{1414} = −\frac{1}{2}e^{μ}∂_{ρρ}^{2}μ;

R_{2424} = −\frac{1}{2}ρe^{ν−μ}∂_{ρ}ν; R̄_{2424} = −\frac{1}{2}ρe^{μ}∂_{ρ}μ (1 + \frac{ρ}{2}∂_{ρ}μ) .

By the functional relation (2.8), it follows that:

κ = \frac{1}{R_{1212}g} \sqrt{(R_{1212})^{2} R_{1313}R_{1414}R_{2323}R_{2424}} = \frac{1}{g}\sqrt{R_{1313}R_{1414}R_{2323}R_{2424}},

κ = \frac{\{\frac{1}{8}ρ^{4}(1 − e^{−μ})e^{2ν−μ} sin^{4} θ∂_{ρ}μ∂_{ρ}ν [∂_{ρρ}^{2}ν + \frac{1}{2} (∂_{ρ}ν)^{2} − \frac{1}{2}∂_{ρ}μ∂_{ρ}ν]\}^{\frac{1}{2}}}{ρ^{4}e^{μ+ν} sin^{2} θ} = 2\frac{m^{2}}{ρ^{6}}

and

κ̄ = \frac{1}{R̄_{1212}ḡ}\sqrt{\binom{}{R̄_{1212}}^{2} R̄_{1313}R̄_{1414}R̄_{2323}R̄_{2424}} = \frac{1}{ḡ}\sqrt{R̄_{1313}R̄_{1414}R̄_{2323}R̄_{2424}},

κ̄ = \frac{[\frac{1}{8}ρ^{5}e^{4μ} sin^{4} θ (∂_{ρ}μ)^{2} (∂_{ρ}μ + ρ∂_{ρρ}^{2}μ) ∂_{ρρ}^{2}μ (1 + \frac{ρ}{4}∂_{ρ}μ) (1 + \frac{ρ}{2}∂_{ρ}μ)]^{\frac{1}{2}}}{ρ^{4}e^{4μ} sin^{2} θ} =

= 2\frac{m^{2}}{ρ^{6}} e^{\frac{4m}{ρ}} \sqrt{(1 + \frac{m}{2ρ}) (1 + \frac{m}{ρ})}.▼

4. CONCLUSION

By the functional form (2.8), derived from the general functional form (2.3) generalizing the notion of curvature of Riemannian one and two dimensional spaces to the general notion of curvature of Riemannian vector spaces q^{α} of class C: C = N−M, the notion itself of curvature of Riemannian vector spaces of higher dimensions (M ≥ 2), has been directly related to internal geometry of a space, more exactly, to the components of Reimann-Christoffel’s curvature tensor R_{αδβγ}.

The process of a generalization of the fundamental notions of the differential geometry, presented in this paper, gives the solid base to further generalization of other, whether they are fundamentals or not, notions and theorems of the differential geometry of a surface, and what may be the subject of separated analysis.

The one of such notions is that of Codazzi’s equations (formula) of a surface, [1, 5], which can be, in the general case of a Riemannian space q^{α} of class C: C = N − M, obtained by projection of the vector condition of integrability:

∂_{q^{δ}q^{β}}^{2}g⃗_{α} − ∂_{q^{β}q^{δ}}^{2}g⃗_{α} = 0,

of absolute differentials of the fundamental vectors g⃗_{α}: dg⃗_{α} = ∂_{q^{β}} g⃗_{α}dq^{β}, onto the normal vector space of a Riemannian space q^{α} of class C.

References

[1] “-T. P. Andelic, Tensorial calculus (in Serbian), Scientific book, Belgrade, 1980.

[2] T. Levi-Civita, The absolute differential calculus. Blackie&Son Ltd., Glasgow, 1954.

[3] D. S. Mitrinovic and D. Z. Dokovic, Polynomials and matrices (in Serbian), Civil book, Belgrade, 1986.

[4] V. Pauli, Theory of relativity (in Russian), Science, Moscow, 1983.

[5] M. J. Vigodski, Differential geometry (in Russian), State department of publishing, Moscow, 1949.

The Institute ”Kirilo Savić“, 11000 Belgrade, V. Stepe 51., Serbia E-mail address: bsaric@ptt.yu

Received October, 13, 2000