aa r X i v : . [ m a t h . DG ] J un Premanifolds ´A.G.Horv´ath
Abstract
The tangent hyperplanes of the ”manifolds” of this paper equipped aso-called Minkowski product. It is neither symmetric nor bilinear. Wegive a method to handing such an object as a locally hypersurface of ageneralized space-time model and define the main tools of its differen-tial geometry: its fundamental forms, its curvatures and so on. In thecase, when the fixed space-time component of the embedding structure isa continuously differentiable semi-inner product space, we get a naturalgeneralization of some important semi-Riemann manifolds as the hyper-bolic space, the de Sitter sphere and the light cone of a Minkowski-Lorenzspace, respectively.
MSC(2000):
Keywords: arc-length, curvature, generalized space-time model, general-ized Minkowski space, Minkowski product, indefinite-inner product, Riemannmanifold, semi-inner product, semi-indefinite inner product, semi-Riemann ma-nifold
There is no and we will not give a formal definition of an object calling in thispaper premanifold . We use this word for a set if it has a manifold-like struc-ture with high freedom in the choosing of the distance function on its tangenthyperplanes. For example we get premanifolds if we investigate the hypersur-faces of a generalized space-time model. The most important types of manifoldsas Riemannian, Finslerian or semi-Riemannian can be investigated in this way.The structure of our embedding space was introduced in [6] and in this paperwe shall continue the investigations by the build up the differential geometry ofits hypersurfaces. We will give the pre-version of the usual semi-Riemannian orFinslerian spaces, the hyperbolic space, the de Sitter sphere, the light cone andthe unit sphere of the rounding semi-inner product space, respectively. In thecase, when the space-like component of the generalized space-time model is acontinuously differentiable semi-inner product space then we will get back theknown and usable geometrical informations on the corresponding hypersurfacesof a pseudo-Euclidean space, e.g. we will show that a prehyperbolic space hasconstant negative curvature. 1 .1 Terminology concepts without definition: basis, dimension, direct sum of subspaces, hy-perboloid, hyperbolic space and hyperbolic metric, inner (scalar) product,linear and bilinear mapping, real and complex vector spaces, quadraticforms, Riemann, Finsler and semi-Riemann manifolds. acceleration vector field:
See before Definition 16. arc-length:
See section 2.2.
Convexity of a hypersurface:
See Definition 10.
Curvature of a curve:
See Definition 14. de Sitter sphere:
See in paragraph 3.2.
Fundamental forms:
See Definition 11 and 12. generalized Minkowski space:
See Definition 5. generalized space-time model:
Finite dimensional, real, generalized Min-kowski space with one dimensional time-like orthogonal direct compo-nents. geodesic:
See Definition 16. hypersurface:
The definition in a generalized Minkowski space can be foundbefore Lemma 3. imaginary unit sphere:
See Definition 8. i.i.p:
Indefinite inner product (See Definition 3).
Minkowski product:
See Definition 5.
Minkowski-Finsler space:
See Definition 9.
Sectional curvature:
See Definition 15. s.i.i.p:
Semi-indefinite-inner-product (See Definition 4). s.i.p:
Semi-inner product (See Definition 1).
Ricci and scalar curvature:
See Definition 16. tangent vector, tangent hyperplane:
These definitions can be seen beforeLemma 3. velocity vector field:
See before Definition 16.2 .2 Notation C , R , R n , S n : The complex line, the real line, the n -dimensional Euclideanspace and the n -dimensional unit sphere, respectively. h· , ·i : The notion of scalar product and all its suitable generalizations.[ · , · ] − : The notion of s.i.p. corresponding to a generalized Minkowski space.[ · , · ] + : The notion of Minkowski product of a generalized Minkowski space. f ′ : The derivative of a real-valued function f with domain in R . Df : The Frechet derivative of a map between two normed spaces. f ′ e : The directional derivative of a real-valued function of a normed space intothe direction of e .[ x, · ] ′ z ( y ) : The derivative map of an s.i.p. in its second argument, into thedirection of z at the point ( x, y ). See Definition 3. k · k ′ x ( y ) , k · k ′′ x,z ( y ) : The derivative of the norm in the direction of x at thepoint y , and the second derivative of the norm in the directions x and z at the point y . ℜ{·} , ℑ{·} : The real and imaginary part of a complex number, respectively. T v : The tangent space of a Minkowskian hypersurface at its point v . S , T , L : The set of space-like, time-like and light-like vectors respectively. S , T : The space-like and time-like orthogonal direct components of a generalizedMinkowski space, respectively. { e , . . . , e k , e k +1 , . . . , e n } : An Auerbach basis of a generalized Minkowski spacewith { e , . . . , e k } ⊂ S and { e k +1 , . . . , e n } ⊂ T , respectively. All of the e ′ i orthogonal to the another ones with respect to the Minkowski product. G , G + : The unit sphere of a generalized space-time model and its upper sheet,respectively. H , H + : The sphere of radius i and its upper sheet, respectively. K , K + : The unit sphere of the embedding semi-inner product space and itsupper sheet, respectively. L , L + : The light cone of a generalized space-time model and its upper sheet,respectively. g : The function g ( s ) = s + g ( s ) e n with g ( s ) = p − s, s ] defines the pointsof G := { s + g ( s ) | s ∈ S } . 3 : The function h ( s ) = s + h ( s ) e n with h ( s ) = p s, s ] defines the points of H + := { s + h ( s ) | s ∈ S } . k : The function k ( s ) = s + k ( s ) e n with k ( s ) = p − [ s, s ] defines the points of K + := { s + k ( s ) | s ∈ S } . l : The function l ( s ) = s + l ( s ) e n with l ( s ) = p [ s, s ] defines the points of L + := { s + l ( s ) | s ∈ S } . A generalization of the inner product and the inner product spaces (briefly i.pspaces) was raised by G. Lumer in [10].
Definition 1 ([10])
The semi-inner-product (s.i.p) on a complex vector space V is a complex function [ x, y ] : V × V −→ C with the following properties: s1 : [ x + y, z ] = [ x, z ] + [ y, z ] , s2 : [ λx, y ] = λ [ x, y ] for every λ ∈ C , s3 : [ x, x ] > when x = 0 , s4 : | [ x, y ] | ≤ [ x, x ][ y, y ] .A vector space V with a s.i.p. is an s.i.p. space . G. Lumer proved that an s.i.p space is a normed vector space with norm k x k = p [ x, x ] and, on the other hand, that every normed vector space canbe represented as an s.i.p. space. In [7] J. R. Giles showed that the followinghomogeneity property holds: s5 : [ x, λy ] = ¯ λ [ x, y ] for all complex λ .This can be imposed, and all normed vector spaces can be represented as s.i.p.spaces with this property. Giles also introduced the concept of continuouss.i.p. space as an s.i.p. space having the additional property s6 : For any unit vectors x, y ∈ S , ℜ{ [ y, x + λy ] } → ℜ{ [ y, x ] } for all real λ → x, y of the unit sphere S . A characterization of the continuous s.i.p.space is based on the differentiability property of the space.Giles proved in [7] that Theorem 1 ([7])
An s.i.p. space is a continuous (uniformly continuous) s.i.p.space if and only if the norm is Gˆateaux (uniformly Fr`echet) differentiable.
In [6] ´A.G.Horv´ath defined the differentiable s.i.p. as follows:4 efinition 2 A differentiable s.i.p. space is an continuous s.i.p. space wherethe s.i.p. has the additional property s6’: For every three vectors x,y,z and real λ [ x, · ] ′ z ( y ) := lim λ → ℜ{ [ x, y + λz ] } − ℜ{ [ x, y ] } λ does exist. We say that the s.i.p. space is continuously differentiable , if theabove limit, as a function of y , is continuous. First we note that the equality ℑ{ [ x, y ] } = ℜ{ [ − ix, y ] } together with theabove property guarantees the existence and continuity of the complex limit:lim λ → [ x, y + λz ] − [ x, y ] λ . The following theorem was mentioned without proof in [6]:
Theorem 2 ([6])
An s.i.p. space is a (continuously) differentiable s.i.p. spaceif and only if the norm is two times (continuously) Gˆateaux differentiable. Theconnection between the derivatives is k y k ( k · k ′′ x,z ( y )) = [ x, · ] ′ z ( y ) − ℜ [ x, y ] ℜ [ z, y ] k y k . Since the present paper often use this statement, we give a proof for it. Weneed the following useful lemma going back, with different notation, to McShane[15] or Lumer [11].
Lemma 1 ([11])
If E is any s.i.p. space with x, y ∈ E , then k y k ( k · k ′ x ( y )) − ≤ ℜ{ [ x, y ] } ≤ k y k ( k · k ′ x ( y )) + holds, where ( k · k ′ x ( y )) − and ( k · k ′ x ( y )) + denotes the left hand and right handderivatives with respect to the real variable λ . In particular, if the norm isdifferentiable, then [ x, y ] = k y k{ ( k · k ′ x ( y )) + k · k ′− ix ( y ) } . Now we prove Theorem 2.
Proof: [of Theorem 2] To determine the derivative of the s.i.p., assume thatthe norm is twice differentiable. Then, by Lemma 1 above, we have ℜ{ [ x, y + λz ] } − ℜ{ [ x, y ] } λ = k y + λz k ( k · k ′ x ( y + λz )) − k y k ( k · k ′ x ( y )) λ == k y kk y + λz k ( k · k ′ x ( y + λz )) − k y k ( k · k ′ x ( y )) λ k y k ≥≥ | [ y + λz, y ] | ( k · k ′ x ( y + λz )) − k y k ( k · k ′ x ( y )) λ k y k , k·k ′ x ( y + λz ) λ is positive. Since the deriva-tive of the norm is continuous, this follows from the assumption that k·k ′ x ( y ) λ ispositive. Considering the latter condition, we get ℜ{ [ x, y + λz ] } − ℜ{ [ x, y ] } λ ≥≥ k y k k · k ′ x ( y + λz ) − ( k · k ′ x ( y )) λ k y k + ℜ [ z, y ] k y k k · k ′ x ( y + λz ) . On the other hand, k y + λz k ( k · k ′ x ( y + λz )) − k y k ( k · k ′ x ( y )) λ ≤≤ k y + λz k ( k · k ′ x ( y + λz )) − | [ y, y + λz ] | ( k · k ′ x ( y )) λ k y + λz k == k y + λz k ( k · k ′ x ( y + λz )) − ( k · k ′ x ( y )) λ k y + λz k + λ ℜ [ z, y + λz ] ( k · k ′ x ( y )) λ k y + λz k . Analogously, if k·k ′ x ( y ) λ is negative, then both of the above inequalities are re-versed, and we get that the limitlim λ ℜ{ [ x, y + λz ] } − ℜ{ [ x, y ] } λ exists, and equals to k y k ( k · k ′′ x,z ( y )) + ℜ [ x, y ] ℜ [ z, y ] k y k . Here we note that also in the case k·k ′ x ( y ) λ = 0 there exists a neighborhoodin which the sign of the function k·k ′ x ( y + λz ) λ is constant. Thus we, need notinvestigate this case by itself. Conversely, consider the fraction k y k k · k ′ x ( y + λz ) − ( k · k ′ x ( y )) λ . We assume now that the s.i.p. is differentiable, implying that it is continuous,too. The norm is differentiable by the theorem of Giles. Using again Lemma 1and assuming that ℜ [ x,y ] λ >
0, we have k y k k · k ′ x ( y + λz ) − ( k · k ′ x ( y )) λ = ℜ [ x, y + λz ] k y k − ℜ [ x, y ] k y + λz k λ k y + λz k == ℜ [ x, y + λz ] k y k − ℜ [ x, y ] k y + λz kk y k λ k y kk y + λz k ≤ℜ [ x, y + λz ] k y k − ℜ [ x, y ] | [ y + λz, y ] | λ k y kk y + λz k =6 ℜ{ [ x, y + λz ] } − ℜ{ [ x, y ] } λ k y kk y + λz k − ℜ [ x, y ] ℜ [ z, y ] k y kk y + λz k . On the other hand, using the continuity of the s.i.p. and our assumption ℜ [ x,y ] λ > k y k k · k ′ x ( y + λz ) − ( k · k ′ x ( y )) λ ≥ℜ{ [ x, y + λz ] } − ℜ{ [ x, y ] } λ − ℜ [ x, y + λz ] ℜ [ z, y + λz ] k y + λz k . If we reverse the assumption of signs, then the direction of the inequalities willalso change. Again a limit argument shows that the first differential function isdifferentiable, and the connection between the two derivatives is k y k ( k · k ′′ x,z ( y )) = [ x, · ] ′ z ( y ) − ℜ [ x, y ] ℜ [ z, y ] k y k . (cid:3) From geometric point of view we know that if C is a 0-symmetric, bounded,convex body in the Euclidean n -space R n (with fixed origin O), then it definesa norm whose unit ball is C itself (see [9]). Such a space is called (Minkowskior) normed linear space. Basic results on such spaces are collected in the sur-veys [13], [14], and [12]. In fact, the norm is a continuous function which isconsidered (in geometric terminology, as in [9]) as a gauge function. Combiningthis with the result of Lumer and Giles we get that a normed linear space canbe represented as an s.i.p space. The metric of such a space (called Minkowskimetric), i.e., the distance of two points induced by this norm, is invariant withrespect to translations.Another concept of Minkowski space was also raised by H. Minkowski andused in Theoretical Physics and Differential Geometry, based on the concept ofindefinite inner product. (See, e.g., [8].) Definition 3 ([8])
The indefinite inner product (i.i.p.) on a complex vectorspace V is a complex function [ x, y ] : V × V −→ C with the following properties: i1 : [ x + y, z ] = [ x, z ] + [ y, z ] , i2 : [ λx, y ] = λ [ x, y ] for every λ ∈ C , i3 : [ x, y ] = [ y, x ] for every x, y ∈ V , i4 : [ x, y ] = 0 for every y ∈ V then x = 0 .A vector space V with an i.i.p. is called an indefinite inner product space . The standard mathematical model of space-time is a four dimensional i.i.p.space with signature (+ , + , + , − ), also called Minkowski space in the literature.Thus we have a well known homonymism with the notion of Minkowski space!In [6] the concepts of s.i.p. and i.i.p. was combined in the following one:7 efinition 4 ([6]) The semi-indefinite inner product (s.i.i.p.) on a complexvector space V is a complex function [ x, y ] : V × V −→ C with the followingproperties: [ x + y, z ] = [ x, z ] + [ y, z ] (additivity in the first argument), [ λx, y ] = λ [ x, y ] for every λ ∈ C (homogeneity in the first argument), [ x, λy ] = λ [ x, y ] for every λ ∈ C (homogeneity in the second argument), [ x, x ] ∈ R for every x ∈ V (the corresponding quadratic form is real-valued), if either [ x, y ] = 0 for every y ∈ V or [ y, x ] = 0 for all y ∈ V , then x = 0 (nondegeneracy), | [ x, y ] | ≤ [ x, x ][ y, y ] holds on non-positive and non-negative subspaces of V,respectively (the Cauchy-Schwartz inequality is valid on positive and neg-ative subspaces, respectively).A vector space V with an s.i.i.p. is called an s.i.i.p. space . It was conclude that an s.i.i.p. space is a homogeneous s.i.p. space if andonly if the property s3 holds, too. An s.i.i.p. space is an i.i.p. space if andonly if the s.i.i.p. is an antisymmetric product. In this latter case [ x, x ] = [ x, x ]implies , and the function is also Hermitian linear in its second argument. Infact, we have: [ x, λy + µz ] = [ λy + µz, x ] = λ [ y, x ] + µ [ z, x ] = λ [ x, y ] + µ [ x, z ]. Itis clear that both of the classical ”Minkowski spaces” can be represented eitherby an s.i.p or by an i.i.p., so automatically they can also be represented as ans.i.i.p. space.The following fundamental lemma was proved in [6]: Lemma 2 ([6])
Let ( S, [ · , · ] S ) and ( T, − [ · , · ] T ) be two s.i.p. spaces. Then thefunction [ · , · ] − : ( S + T ) × ( S + T ) −→ C defined by [ s + t , s + t ] − := [ s , s ] − [ t , t ] is an s.i.p. on the vector space S + T . It is possible that the s.i.i.p. space V is a direct sum of its two subspaceswhere one of them is positive and the other one is negative. Then there are twomore structures on V , an s.i.p. structure (by Lemma 2) and a natural thirdone, which was called by Minkowskian structure. Definition 5 ([6])
Let ( V, [ · , · ]) be an s.i.i.p. space. Let S, T ≤ V be posi-tive and negative subspaces, where T is a direct complement of S with respectto V . Define a product on V by the equality [ u, v ] + = [ s + t , s + t ] + =[ s , s ] + [ t , t ] , where s i ∈ S and t i ∈ T , respectively. Then we say that thepair ( V, [ · , · ] + ) is a generalized Minkowski space with Minkowski product [ · , · ] + .We also say that V is a real generalized Minkowski space if it is a real vectorspace and the s.i.i.p. is a real valued function. - of the s.i.i.p.. But in general, property does not hold. (See an example in[6].)By Lemma 2 the function p [ v, v ] − is a norm function on V which can give anembedding space for a generalized Minkowski space. This situation is analogousto the situation when a pseudo-Euclidean space is obtained from a Euclideanspace by the action of an i.i.p.It is easy to see that by the methods of [6], starting with arbitrary twonormed spaces S and T , one can mix a generalized Minkowski space. Of courseits smoothness property is basically determined by the analogous properties of S and T .If now we consider the theory of s.i.p in the sense of Lumer-Giles, we havea natural concept of orthogonality. For the unified terminology we change theoriginal notation of Giles and use instead Definition 6 ([7])
The vector x is orthogonal to the vector y if [ x, y ] = 0 . Since s.i.p. is neither antisymmetric in the complex case nor symmetric in thereal one, this definition of orthogonality is not symmetric in general.Giles proved that in a continuous s.i.p. space x is orthogonal to y in thesense of the s.i.p. if and only if x is orthogonal to y in the sense of Birkhoff-James. (See e.g. [1] and [2].) We note that the s.i.p. orthogonality impliesthe Birkhoff-James orthogonality in every normed space. Lumer pointed outthat a normed linear space can be transformed into an s.i.p. space in a uniqueway if and only if its unit sphere is smooth (i.e., there is a unique supportinghyperplane at each point of the unit sphere). In this case the corresponding(unique) s.i.p. has the homogeneity property [s5] .Let ( V, [ · , · ]) be an s.i.i.p. space, where V is a complex (real) vector space.It was defined the orthogonality of such a space by a definition analogous to thedefinition of the orthogonality of an i.i.p. or s.i.p. space. Definition 7 ([6])
The vector v is orthogonal to the vector u if [ v, u ] = 0 . If U is a subspace of V , define the orthogonal companion of U in V by U ⊥ = { v ∈ V | [ v, u ] = 0 for all u ∈ U } . We note that, as in the i.i.p. case, the orthogonal companion is always asubspace of V . It was proved the following theorem: Theorem 3 ([6])
Let V be an n -dimensional s.i.i.p. space. Then the orthogo-nal companion of a non-neutral vector u is a subspace having a direct comple-ment of the linear hull of u in V . The orthogonal companion of a neutral vector v is a degenerate subspace of dimension n − containing v . Observe that this proof does not use the property of the s.i.i.p.. So thisstatement is true for any concepts of product satisfying properties - . As wesaw, the Minkowski product is also such a product.9e also note that in a generalized Minkowski space, the positive and negativecomponents S and T are Pythagorean orthogonal to each other. In fact, forevery pair of vectors s ∈ S and t ∈ T , by definition we have [ s − t, s − t ] + =[ s, s ] + [ − t, − t ] = [ s, s ] + + [ t, t ] + .Let V be a generalized Minkowski space. Then we call a vector space-like, light-like, or time-like if its scalar square is positive, zero, or negative,respectively. Let S , L and T denote the sets of the space-like, light-like, andtime-like vectors, respectively.In a finite dimensional, real generalized Minkowski space with dim T = 1 itcan geometrically characterize these sets of vectors. Such a space is called in [6]a generalized space-time model . In this case T is a union of its two parts,namely T = T + ∪ T − , where T + = { s + t ∈ T | where t = λe n for λ ≥ } and T − = { s + t ∈ T | where t = λe n for λ ≤ } . It has special interest, the imaginary unit sphere of a finite dimensional, real,generalized space-time model. (See Def.8 in [6].) It was given a metric on it, andthus got a structure similar to the hyperboloid model of the hyperbolic spaceembedded in a space-time model. In the case when the space S is an Euclideanspace this hypersurface is a model of the n -dimensional hyperbolic space thusit is such-like generalization of it.It was proved in [6] the following: Theorem 4 ([6])
Let V be a generalized space-time model. Then T is an opendouble cone with boundary L , and the positive part T + (resp. negative part T − )of T is convex. We note that if dim
T >
Definition 8 ([6])
The set H := { v ∈ V | [ v, v ] + = − } , is called the imaginary unit sphere of the generalized space-time model. With respect to the embedding real normed linear space ( V, [ · , · ] − ) (seeLemma 2) H is, as we saw, a generalized two sheets hyperboloid correspondingto the two pieces of T , respectively. Usually we deal only with one sheet of thehyperboloid, or identify the two sheets projectively. In this case the space-timecomponent s ∈ S of v determines uniquely the time-like one, namely t ∈ T . Let v ∈ H be arbitrary. Let T v denote the set v + v ⊥ , where v ⊥ is the orthogonalcomplement of v with respect to the s.i.i.p., thus a subspace.10t was also proved that the set T v corresponding to the point v = s + t ∈ H isa positive, (n-1)-dimensional affine subspace of the generalized Minkowski space( V, [ · , · ] + ).Each of the affine spaces T v of H can be considered as a semi-metric space,where the semi-metric arises from the Minkowski product restricted to thispositive subspace of V . We recall that the Minkowski product does not satisfythe Cauchy-Schwartz inequality. Thus the corresponding distance function doesnot satisfy the triangle inequality. Such a distance function is called in theliterature semi-metric (see [17]). Thus, if the set H is sufficiently smooth, thena metric can be adopted for it, which arises from the restriction of the Minkowskiproduct to the tangent spaces of H . Let us discuss this more precisely.The directional derivatives of a function f : S R with respect to a unitvector e of S can be defined in the usual way, by the existence of the limits forreal λ : f ′ e ( s ) = lim λ f ( s + λe ) − f ( s ) λ . Let now the generalized Minkowski space be a generalized space-time model,and consider a mapping f on S to R . Denote by e n a basis vector of T withlength i as in the definition of T + before Theorem 4. The set of points F := { ( s + f ( s ) e n ) ∈ V for s ∈ S } is a so-called hypersurface of this space. Tangent vectors of a hypersurface F in a point p are the vectors associated to the directional derivatives of the coor-dinate functions in the usual way. So u is a tangent vector of the hypersurface F in its point v = ( s + f ( s ) e n ), if it is of the form u = α ( e + f ′ e ( s ) e n ) for real α and unit vector e ∈ S. The linear hull of the tangent vectors translated into the point s is the tangentspace of F in s . If the tangent space has dimension n −
1, we call it tangenthyperplane .We now reformulate Lemma 3 of [6]:
Lemma 3 (See also in [6] as Lemma 3)
Let S be a continuous (complex)s.i.p. space. (So the property s6 holds.) Then the directional derivatives of thereal valued function h : s p s, s ] are h ′ e ( s ) = ℜ [ e, s ] p s, s ] . The following theorem is a consequence of this result.
Theorem 5
Let assume that the s.i.p. [ · , · ] of S is differentiable. (So theproperty s6’ holds.) Then for every two vectors x and z in S we have: [ x, · ] ′ z ( x ) = 2 ℜ [ z, x ] − [ z, x ] , nd k · k ′′ x,z ( x ) = ℜ [ z, x ] − [ z, x ] k x k . If we also assume that the s.i.p. is continuously differentiable (so the norm is a C function), then we also have [ x, · ] ′ x ( y ) = [ x, x ] , and thus k · k ′′ x,x ( y ) = k x k − ℜ [ x, y ] k y k . Proof:
Since1 λ ([ x + λz, x + λz ] − [ x, x ]) = 1 λ ([ x, x + λz ] − [ x, x ]) + 1 λ [ λz, x + λz ] , if λ tends to zero then the right hand side tends to[ x, · ] ′ z ( x ) + [ z, x ] . The left hand side is equal to (cid:16)p x + λz, x + λz ] − p x, x ] (cid:17) (cid:16)p x + λz, x + λz ] + p x, x ] (cid:17) λ thus by Lemma 3 it tends to ℜ [ z, x ] p x, x ] 2 p x, x ] . This implies the first equality[ x, · ] ′ z ( x ) = 2 ℜ [ z, x ] − [ z, x ] . Using Theorem 2 in [6] we also get that k x k ( k · k ′′ x,z ( x )) = [ x, · ] ′ z ( x ) − ℜ [ x, x ] ℜ [ z, x ] k x k , proving the second statement, too.If we assume that the norm is a C function of its argument then the firstderivative of the second argument of the product is a continuous function of itsarguments. So the function A ( y ) : S −→ R defined by the formula A ( y ) = [ x, · ] ′ x ( y ) = lim λ λ ([ x, y + λx ] − [ x, y ])continuous in y = 0. On the other hand for non-zero t ∈ R we use the notation tλ ′ = λ and we get that A ( ty ) = lim λ λ ([ x, ty + λx ] − [ x, y ]) = lim λ ′ ttλ ′ ([ x, y + λ ′ x ] − [ x, y ]) = A ( y ) . x, · ] ′ x ( y ) = A ( y ) = A (0) = [ x, x ]holds for every y . Applying again the formula connected the derivative of theproduct and the norm we get the last statement of the theorem, too. (cid:3) Applying Lemma 3 to H + it was given a connection between the differ-entiability properties and the orthogonality one. The tangent vectors of thehypersurface H + in its point v = s + p s, s ] e n form the orthogonal complement v ⊥ of v with respect to the Minkowski product.It was defined in [6] a Finsler space type structure for a hypersurface of ageneralized space-time model. Definition 9 ([6])
Let F be a hypersurface of a generalized space-time modelfor which the following properties hold:i, In every point v of F , there is a (unique) tangent hyperplane T v for whichthe restriction of the Minkowski product [ · , · ] + v is positive, andii, the function ds v := [ · , · ] + v : F × T v × T v −→ R + ds v : ( v, u , u ) [ u , u ] + v varies differentiably with the vectors v ∈ F and u , u ∈ T v .Then we say that the pair ( F, ds ) is a Minkowski-Finsler space with semi-metric ds embedding into the generalized space-time model V . Naturally ”varies differentiably with the vectors v, u , u ” means that forevery v ∈ T and pairs of vectors u , u ∈ T v the function [ u , u ] v is a differen-tiable function on F . One of the important results on the imaginary unit spherewas Theorem 6 ([6])
Let V be a generalized space-time model. Let S be a contin-uously differentiable s.i.p. space, then ( H + , ds ) is a Minkowski-Finsler space. In present paper we will prefer the name ”pre-hyperbolic space” for thisstructure.
Acknowledgment
The author wish to thank for
G.Moussong who suggested the investigationof H + by the tools of differential geometry and B.Csik´os who also gave helpfulhints. 13
Hypersurfaces as premanifolds
Let S be a continuously differentiable s.i.p. space, V be a generalized space-timemodel and F a hypersurface. We shall say that F is a space-like hypersurface if the Minkowski product is positive on its all tangent hyperplanes. The objectsof our examination are the convexity, the fundamental forms, the concepts ofcurvature, the arc-length and the geodesics. In this section we in a generalizedspace-time model define these that would be a generalizations of the knownconcepts. In a pseudo-Euclidean or semi-Riemann space it can be found in thenotes [4] and the book [5]. Definition 10 ([4])
We say that a hypersurface is convex if it lies on one sideof its each tangent hyperplanes. It is strictly convex if it is convex and its tan-gent hyperplanes contain precisely one points of the hypersurface, respectively.
In an Euclidean space the first fundamental form is a positive definitequadratic form induced by the inner product of the tangent space.In our generalized space-time model the first fundamental form is giving bythe scalar square of the tangent vectors with respect to the Minkowski productrestricted to the tangent hyperplane. If we have a map f : S −→ V then it canbe decomposed to a sum of its space-like and time-like components. We have f = f S + f T where f S : S −→ S and f T : S −→ T , respectively. With respect to theembedding normed space we can compute its Frechet derivative by the low Df = (cid:20) Df S Df T (cid:21) implying that Df ( s ) = Df S ( s ) + Df T ( s ) . Introduce the notation[ f ( c ( t )) , · ] + ′ D ( f ◦ c )( t ) ( f ( c ( t ))) :=:= (cid:16) [( f ) S ( c ( t )) , · ] ′ D (( f ) S ◦ c )( t ) (( f ) S ( c ( t ))) − ( f ) T ( c ( t ))(( f ) T ◦ c ) ′ ( t ) (cid:17) . We need the following technical lemma:
Lemma 4 If f , f : S −→ V are two C maps and c : R −→ S is an arbitrary C curve then ([( f ◦ c )( t )) , ( f ◦ c )( t ))] + ) ′ == [ D ( f ◦ c )( t ) , ( f ◦ c )( t ))] + + [( f ◦ c )( t )) , · ] + ′ D ( f ◦ c )( t ) (( f ◦ c )( t )) . roof: By definition([ f ◦ c, f ◦ c )] + ) ′ | t := lim λ → λ (cid:0) [ f ( c ( t + λ )) , f ( c ( t + λ ))] + − [ f ( c ( t )) , f ( c ( t ))] + (cid:1) = lim λ → λ ([( f ) S ( c ( t + λ )) , ( f ) S ( c ( t + λ ))] − [( f ) S ( c ( t )) , ( f ) S ( c ( t ))]) ++ lim λ → λ ([( f ) T ( c ( t + λ )) , ( f ) T ( c ( t + λ ))] − [( f ) T ( c ( t )) , ( f ) T ( c ( t ))]) . The first part islim λ → λ ([( f ) S ( c ( t + λ )) − ( f ) S ( c ( t )) , ( f ) S ( c ( t + λ ))]++[( f ) S ( c ( t )) , ( f ) S ( c ( t + λ ))] − [( f ) S ( c ( t )) , ( f ) S ( c ( t ))]) == [ D (( f ) S ◦ c ) | t , ( f ) S ( c ( t ))] + [( f ) S ( c ( t )) , · ] ′ D (( f ) S ◦ c )( t ) (( f ) S ( c ( t ))) . To prove this take a coordinate system { e , · · · , e n − } in S and consider thecoordinate-wise representation( f ) S ◦ c = n − X i =1 (( f ) S ◦ c ) i e i of ( f ) S ◦ c . Using Taylor’s theorem for the coordinate functions we have thatthere are real parameters t i ∈ ( t, t + λ ), for which(( f ) S ◦ c )( t + λ ) = (( f ) S ◦ c )( t ) + λD (( f ) S ◦ c )( t ) + 12 λ n − X i =1 (( f ) S ◦ c ) ′′ i ( t i ) e i . Thus we can get[( f ) S ( c ( t )) , ( f ) S ( c ( t + λ ))] − [( f ) S ( c ( t )) , ( f ) S ( c ( t ))] == [( f ) S ( c ( t )) , ( f ) S ( c ( t )) + D (( f ) S ◦ c )( t ) λ ++ 12 λ n − X i =1 (( f ) S ◦ c ) ′′ i ( t i ) e i ] − [( f ) S ( c ( t )) , ( f ) S ( c ( t ))] =([( f ) S ( c ( t )) , ( f ) S ( c ( t )) + D (( f ) S ◦ c )( t ) λ ] − [( f ) S ( c ( t )) , ( f ) S ( c ( t ))]) ++[( f ) S ( c ( t )) , ( f ) S ( c ( t )) + D (( f ) S ◦ c )( t ) λ + 12 λ n − X i =1 (( f ) S ◦ c ) ′′ i ( t i ) e i ] −− [( f ) S ( c ( t )) , ( f ) S ( c ( t )) + D (( f ) S ◦ c )( t ) λ ] . In the second argument of this product, the Lipschwitz condition holds with areal K for enough small λ ’s, so we have that the absolute value of the substrac-tion of the last two terms is less or equal to K " ( f ) S ( c ( t )) , λ n − X i =1 (( f ) S ◦ c ) ′′ i ( t i ) e i . λ → f ) T and ( f ) T are real-real functions, respectively solim λ → λ ([( f ) T ( c ( t + λ )) , ( f ) T ( c ( t + λ ))] − [( f ) T ( c ( t )) , ( f ) T ( c ( t ))]) == − (( f ) T ◦ c ) ′ ( t )( f ) T ( c ( t )) − ( f ) T ( c ( t ))(( f ) T ◦ c ) ′ ( t ) . Hence we have ([( f ◦ c )( t )) , ( f ◦ c )( t ))] + ) ′ == [ D (( f ) S ◦ c )( t ) , (( f ) S ◦ c )( t ))] + [( f ) S ( c ( t )) , · ] ′ D (( f ) S ◦ c )( t ) ((( f ) S ◦ c )( t ))) −− (( f ) T ◦ c ) ′ ( t )( f ) T ( c ( t )) − ( f ) T ( c ( t ))(( f ) T ◦ c ) ′ ( t ) == [ D ( f ◦ c )( t ) , f ( c ( t ))] + ++ (cid:16) [( f ) S ( c ( t )) , · ] ′ D (( f ) S ◦ c )( t ) (( f ) S ( c ( t ))) − ( f ) T ( c ( t ))(( f ) T ◦ c ) ′ ( t ) (cid:17) , and the statement is proved. (cid:3) Let F be a hypersurface defined by the function f : S −→ V . Here f ( s ) = s + f ( s ) e n denotes the point of F . The curve c : R −→ S define a curve on F .We assume that c is a C -curve. The following definition is very important one. Definition 11
The first fundamental form in a point ( f ( c ( t )) of the hypersur-face F is the product I f ( c ( t ) := [ D ( f ◦ c )( t ) , D ( f ◦ c )( t )] + . The variable of the first fundamental form is a tangent vector, the tangentvector of the variable curve c .We can see that it is homogeneous of the second order but (in general) ithas no a bilinear representation.In fact, by the definition of f , if { e i : i = 1 · · · n − } is a basis in S then thecomputation I f ( c ( t )) = [ ˙ c ( t ) + ( f ◦ c ) ′ ( t ) e n , ˙ c ( t ) + ( f ◦ c ) ′ ( t ) e n ] + == [ ˙ c ( t ) , ˙ c ( t )] − [( f ◦ c ) ′ ( t )] = [ ˙ c ( t ) , ˙ c ( t )] − n − X i,j =1 ˙ c i ( t ) ˙ c j ( t ) f ′ e i ( c ( t )) f ′ e j ( c ( t )) == [ ˙ c ( t ) , ˙ c ( t )] − ˙ c ( t ) T h f ′ e i ( c ( t )) f ′ e j ( c ( t )) i n − i,j =1 ˙ c ( t )shows that it is not a quadratic form. It would be a quadratic form if and onlyif the quantity [ ˙ c ( t ) , ˙ c ( t )] − ˙ c ( t ) T ˙ c ( t ) = [ ˙ c ( t ) , ˙ c ( t )] − n − X i =1 ˙ c i ( t )16anishes. Thus if the Minkowski product is an i.p. than we can assume thatthe basis { e , . . . , e n − } in S is orthonormal, the mentioned difference vanishes,and c i ( t ) = h e i , c ( t ) i = h c ( t ) , e i i and ˙ c ( t ) = n − P i =1 ˙ c i ( t ) e i . SoI f ( c ( t )) = ˙ c ( t ) T (cid:18) Id − h f ′ e i ( c ( t )) f ′ e j ( c ( t )) i n − i,j =1 (cid:19) ˙ c ( t ) , and we get back the classical local quadratic representation of the first funda-mental form. Now if c i ( t ) = 0 for i ≥ I = 1 − ( f ′ e ( c ( t ))) − ( f ′ e ( c ( t ))) . We now extend the definition of the second fundamental form take intoconsideration that the product has neither symmetry nor bilinearity properties.If v is a tangent vector and n is a normal vector of the hypersurface at its point f ( c ( t )) then we have0 = [ v, n ] + = [ D ( f ◦ c )( t ) , ( f ◦ c )( t )] + . Using Lemma 4 and the notation follows it, we get0 = ([ D ( f ◦ c )( t ) , ( n ◦ c )( t )] + ) ′ == [ D ( f ◦ c ) , n ( c ( t ))] + + [ D ( f ◦ c )( t ) , · ] + ′ D ( n ◦ c )( t ) ( n ( c ( t ))) . We introduce the unit normal vector fields n by the definition n ( c ( t )) := ( n ( c ( t )) if n light-like vector n ( c ( t )) √ | [ n ( c ( t )) ,n ( c ( t ))] + | otherwise. Definition 12
The second fundamental form at the point f ( c ( t )) defined byone of the equivalent formulas: II := [ D ( f ◦ c )( t ) , ( n ◦ c )( t )] +( f ◦ c )( t ) = − [ D ( f ◦ c )( t ) , · ] + ′ D ( n ◦ c )( t ) (( n ◦ c )( t )) . By the structure of the generalized space-time model assuming that n ( s ) = s + n ( s ) e n we get thatII = [ D ( f ◦ c )( t ) , ( n ◦ c )( t )] +( f ◦ c )( t ) == " D ( ˙ c ( t ) + D ( f ◦ c )( t ) e n ) , c ( t ) + ( n ◦ c )( t ) e n p | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | + == h ¨ c ( t ) + (cid:16) ˙ c ( t ) T h f ′′ e i ,e j | c ( t ) i ˙ c ( t ) + (cid:2) f ′ e i | c ( t ) (cid:3) ¨ c ( t ) (cid:17) e n , c ( t ) + n ( c ( t )) e n i + p | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | =17 (cid:2) ¨ c ( t ) + [ f ′ e i | c ( t ) ]¨ c ( t ) e n , ( n ◦ c )( t ) (cid:3) + − (cid:16) ˙ c ( t ) T h f ′′ e i ,e j | c ( t ) i ˙ c ( t ) (cid:17) ( n ( c ( t )) p | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | == (cid:2) D ( f ) | c ( t ) ¨ c ( t ) , ( n ◦ c )( t ) (cid:3) + − (cid:16) ˙ c ( t ) T h f ′′ e i ,e j | c ( t ) i ˙ c ( t ) (cid:17) ( n ( c ( t )) p | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | == − ˙ c ( t ) T " f ′′ e i ,e j | c ( t ) n ( c ( t )) p | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | n − i,j =1 ˙ c ( t ) . We now can adopt a determinant of this fundamental form. It is the determinantof its quadratic form:det II := det " f ′′ e i ,e j | c ( t ) n ( c ( t )) p | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | n − i,j =1 . If we consider a two-plane in the tangent hyperplane then it has a twodimensional pre-image in S by the regular linear mapping Df . The gettingplane is a normed one and we can consider an Auerbach basis { e , e } in it. Definition 13
The sectional principal curvature of a 2-section of the tangenthyperplane in the direction of the 2-plane spanned by { u = Df ( e ) and v = Df ( e ) } are the extremal values of the function ρ ( D ( f ◦ c )) := II f ◦ c ( t ) I f ◦ c ( t ) , of the variable D ( f ◦ c ) . We denote them by ρ ( u, v ) max and ρ ( u, v ) min , respec-tively. The sectional (Gauss) curvature κ ( u, v ) (at the examined point c ( t ) ) isthe product κ ( u, v ) := [ n ( c ( t )) , n ( c ( t ))] + ρ ( u, v ) max ρ ( u, v ) min . In the case of a symmetric and bilinear product, both of the fundamentalforms are quadratic and the sectional principal curvatures attained in orthogonaldirections. They are the eigenvalues of the pair of quadratic forms II f ◦ c ( t ) andI f ◦ c ( t ) . This implies that ρ ( u, v ) max and ρ ( u, v ) min are the solutions of theequality:0 = det (cid:0) II f ◦ c ( t ) − λ I f ◦ c ( t ) (cid:1) = det (cid:0) I f ◦ c ( t ) (cid:1) det (cid:0) (I f ◦ c ( t ) ) − II f ◦ c ( t ) − λ Id (cid:1) , showing that κ ( u, v ) := [ n ( c ( t )) , n ( c ( t ))] + ρ ( u, v ) max ρ ( u, v ) min == [ n ( c ( t )) , n ( c ( t ))] + det (cid:16) I − f ◦ c ( t ) II f ◦ c ( t ) (cid:17) = [ n ( c ( t )) , n ( c ( t ))] + det II f ◦ c ( t ) det I f ◦ c ( t ) =18 [ n ( c ( t )) , n ( c ( t ))] + (cid:16) f ′′ e ,e | c ( t ) f ′′ e ,e | c ( t ) − (cid:0) f ′′ e ,e | c ( t ) (cid:1) (cid:17) ( n ( c ( t ))) (cid:0) − ( f ′ e ( c ( t ))) − ( f ′ e ( c ( t ))) (cid:1) | [ c ( t ) , c ( t )] − ( n ( c ( t ))) | . But we can choose for the function nn ( c ( t )) := f ′ e ( c ( t )) e + f ′ e ( c ( t )) e + e n with n ( c ( t )) = 1 and for a 2-plane of the tangent hyperplane which containsonly space-like vectors and has time-like normal vector with absolute value[ n ( c ( t )) , n ( c ( t ))] + = q − ( f ′ e ( c ( t ))) − ( f ′ e ( c ( t ))) getting the well-known formula κ ( u, v ) = − f ′′ e ,e | c ( t ) f ′′ e ,e | c ( t ) + (cid:0) f ′′ e ,e | c ( t ) (cid:1) (cid:0) − ( f ′ e ( c ( t ))) − ( f ′ e ( c ( t ))) (cid:1) (see in [5] p.95.).The Ricci curvature of a Riemannian hypersurface at a point p = ( f ◦ c )( t )in the direction of the tangent vector v = D ( f ◦ c ) is the sum of the sectionalcurvatures in the directions of the planes spanned by the tangent vectors v and u i , where u i are the vectors of an orthonormal basis of the orthogonalcomplement of v . This value is independent from the choosing of the basis.Choose random (by uniform distribution) the orthonormal basis! ([3]) Thecorresponding sectional curvatures κ ( u i , v ) will be random variables with thesame expected values. The sum of them is again a random variable whichexpected value corresponding to the Ricci curvature at p with respect to v .Hence it is equal to n − v . Similarly the scalarcurvature of the hypersurface at a point is the sum of the sectional curvaturesdefined by any two vectors of an orthonormal basis of the tangent space, itis also can be considered as an expected value. This motivates the followingdefinition: Definition 14
The
Ricci curvature Ric( v ) in the direction v at the point f ( c ( t )) is Ric( v ) f ( c ( t )) := ( n − · E ( κ f ( c ( t )) ( u, v )) where κ f ( c ( t )) ( u, v ) is the random variable of the sectional curvatures of thetwo planes spanned by v and a random u of the tangent hyperplane holding theequality [ u, v ] + = 0 . We also say that the scalar curvature of the hypersurface f at its point f ( c ( t )) is Γ f ( c ( t )) := (cid:18) n − (cid:19) · E ( κ f ( c ( t )) ( u, v )) . .2 Arc-length In this section we also assume that the s.i.p. of S is continuously differentiable.If the first fundamental form is positive we can adopt for the curves well-definedarc-lengthes and we can define a metric on the hypersurface.The following definition was used in [6] for the metric of the imaginary unitsphere. We now adopt it for an arbitrary hypersurface. Definition 15
Denote by p, q a pair of points in F where F is a hypersurfaceof the generalized space-time model. Consider the set Γ p,q of equally orientedpiecewise differentiable curves ( f ◦ c )( t ) , a ≤ t ≤ b , of F emanating from p andterminating at q . Then the pre-distance of these points is ρ ( p, q ) = inf b Z a q | I ( f ◦ c )( x ) | dx for f ◦ c ∈ Γ p,q . It is easy to see that pre-distance satisfies the triangle inequality; thus itgives a metric on F (see [17]). On a hypersurface which contains only space-like tangent vectors it is the usual definition of the Minkowski-Finsler distance.Such hypersurfaces were called by space-like ones, we mention for an examplethe imaginary unit sphere. Introduce the arc-length function l a ( τ ) of a curve f ◦ c for which the light-like points gives a closed, zero measured set by thefunction l a ( τ ) = τ Z a q | [ D ( f ◦ c )( x ) , D ( f ◦ c )( x )] + f ( c ( x )) | dx = τ Z a q | I f ( c ( x )) | dx. Give parameters only those points of the curve in which the tangent vector ofthe curve is a non-light-like one. Thus a corresponding reparametrization couldbe well-defined and we get a pair of inverse formulas which are almost all valid;we have that ( l a ( τ )) ′ = q | I f ( c ( τ )) | , and for the inverse function τ ( l a ) : [0 , ε ) −→ [ a, l − a ( ε )) holds( τ ( l a )) ′ = (cid:0) l − a ( τ ) (cid:1) ′ = 1 p | I f ( c ( τ ( l a ))) | . Theorem 7
Consider a curve lying on the hypersurface determining a vectorfield in V . Using the arc-length as parameter, the absolute values of the firstderivative (tangent) vectors are equal to 1, moreover the second derivative vectorfields are orthogonal to the first one. (With respect to the Minkowski product of V .) roof: By definition the tangent vectors are non-light-like. Thus the requireddifferential is D (( f ◦ c ) ◦ ( τ ( l a ))) = D ( f ◦ c ) ◦ ( τ ( l a )) · ( τ ( l a )) ′ == D ( f ◦ c ) ◦ ( τ ( l a )) 1 p | I f ( c ( τ ( l a ))) | , implying that[ D (( f ◦ c ) ◦ ( τ ( l a ))) , D (( f ◦ c ) ◦ ( τ ( l a )))] + = I f ( c ( τ ( l a ))) | I f ( c ( τ ( l a ))) | = sign(I f ( c ( τ ( l a ))) ) . If we would like to consider the derivative of the tangent vector field, we haveto compute D ( D (( f ◦ c ) ◦ ( τ ( l a )))) . By Lemma 4 we get D (cid:16)q | I f ( c ( τ ( l a ))) | (cid:17) == sign(I f ( c ( τ ( l a ))) )2 p | I f ( c ( τ ( l a ))) | ( [ D ( f ◦ c ) ◦ ( τ ( l a )) , D ( f ◦ c ) ◦ ( τ ( l a ))] p | I f ( c ( τ ( l a ))) | ++[ D ( f ◦ c ) ◦ ( τ ( l a )) , · ] ′ D (( Df ◦ c ) ◦ τ ( l a )) ( D ( f ◦ c ) ◦ ( τ ( l a ))) o , and since the s.i.p is continuously differentiable we have by Theorem 5 that[ D ( f ◦ c ) ◦ ( τ ( l a )) , · ] ′ D (( Df ◦ c ) ◦ τ ( l a )) ( D ( f ◦ c ) ◦ ( τ ( l a ))) == [ D ( f ◦ c ) ◦ ( τ ( l a )) , D ( f ◦ c ) ◦ ( τ ( l a ))] p | I f ( c ( τ ( l a ))) | . Thus the complete differential is D D ( f ◦ c ) ◦ ( τ ( l a )) 1 p | I f ( c ( τ ( l a ))) | ! = D ( f ◦ c ) ◦ ( τ ( l a )) | I f ( c ( τ ( l a ))) | −− sign(I f ( c ( τ ( l a ))) ) [ D ( f ◦ c ) ◦ ( τ ( l a )) , D ( f ◦ c ) ◦ ( τ ( l a ))](I f ( c ( τ ( l a ))) ) D ( f ◦ c ) ◦ ( τ ( l a )) . Now we can see that " D D ( f ◦ c ) ◦ ( τ ( l a )) 1 p I f ( c ( τ ( l a ))) ! , D ( f ◦ c ) ◦ ( τ ( l a )) = 0as we stated. (cid:3) Inner metric determines the geodesics of the hypersurface in a standard way,as the curves representing the infinum in Definition 13. Since we use the conceptof arc-length parametrization for a general (not necessary space-like) curve, wecan introduce its velocity and acceleration vectors fields as the first andsecond derivative vector fields of the natural parametrization, respectively. Wecan introduce the concept of geodesics as the solutions of the Euler-Lagrangeequation with respect to the hypersurface. More precisely:21 efinition 16
We say that the C -curve f ◦ c (with almost all non-light-liketangent vectors) is a geodesic of the hypersurface F , if its acceleration vectorfield is orthogonal to the tangent hyperplane of F at each point of the curve. Sothere exists a function α ( τ ( l a )) : R −→ R such that D ( D (( f ◦ c ) ◦ ( τ ( l a )))) = α ( τ ( l a ))( n ◦ c )( τ ( l a )) . The curvature of a curve can be defined as the square root of the absolutevalue of the derivative of its tangent vectors with respect to this parametrization.
Definition 17
The curvature of the curve f ◦ c is the non-negative function γ f ◦ c ( τ ( l a )) := p | [ D ( D (( f ◦ c ) ◦ ( τ ( l a )))) , D ( D (( f ◦ c ) ◦ ( τ ( l a ))))] + | == | α ( τ ( l a )) | . If the curvature is non-zero then we can define the vector ( m ◦ c )( τ ( l a )) bythe equality: ( m ◦ c )( τ ( l a )) = D ( D (( f ◦ c ) ◦ ( τ ( l a )))) γ f ◦ c ( τ ( l a )) . From this equality immediately follows that[( m ◦ c )( τ ( l a )) , ( m ◦ c )( τ ( l a ))] + == [ D ( D (( f ◦ c ) ◦ ( τ ( l a )))) , D ( D (( f ◦ c ) ◦ ( τ ( l a ))))] + γ f ◦ c ( τ ( l a )) . Using the equality D ( D (( f ◦ c ) ◦ ( τ ( l a )))) = D D ( f ◦ c ) ◦ ( τ ( l a )) 1 p | I f ( c ( τ ( l a ))) | ! == D ( f ◦ c ) ◦ ( τ ( l a )) | I f ( c ( τ ( l a ))) | −− sign(I f ( c ( τ ( l a ))) ) [ D ( f ◦ c ) ◦ ( τ ( l a )) , D ( f ◦ c ) ◦ ( τ ( l a ))](I f ( c ( τ ( l a ))) ) D ( f ◦ c ) ◦ ( τ ( l a )) , computed in Theorem 7, and the orthogonality property of the vectors D ( f ◦ c )and n ◦ c , we get a connection analogous to the Meusnier’s theorem: γ f ◦ c ( τ ( l a ))[( m ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )] + = (cid:20) D ( f ◦ c ) ◦ ( τ ( l a )) | I f ( c ( τ ( l a ))) | , ( n ◦ c )( τ ( l a ) (cid:21) + meaning that γ f ◦ c ( τ ( l a ))[( m ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )] + = II f ( c ( τ ( l a ))) | I f ( c ( τ ( l a ))) | . γ f ◦ c ( τ ( l a ))[( m ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )] + | I f ( c ( τ ( l a ))) | = II f ( c ( τ ( l a ))) . This for light-like vectors is also valid, if we define their acceleration vectors asvectors of zero length. By definition, for a geodesic curve[( m ◦ c )( τ ( l a )) , ( m ◦ c )( τ ( l a ))] + = ( α ( τ ( l a ))) ( γ f ◦ c ( τ ( l a )) (cid:2) ( n ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )) (cid:3) + , showing that m ◦ c and n ◦ c have the same casual characters and thus m ◦ c = sign( α ( τ ( l a )))( n ◦ c ) . Thus the product form of the Meusnier’s theorem simplified into the equality α ( τ ( l a ))[( n ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )] + | I f ( c ( τ ( l a ))) | = II f ( c ( τ ( l a ))) . Equivalently we get α ( τ ( l a )) = [( n ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )] + II f ( c ( τ ( l a ))) | I f ( c ( τ ( l a ))) | == [( n ◦ c )( τ ( l a )) , ( n ◦ c )( τ ( l a )] + sign(I f ( c ( τ ( l a ))) ) ρ ( D ( f ◦ c )) . If all tangent vectors are space-like vectors and the normal ones are time-like vectors, respectively, then the extremal values of the function α ( τ ( l a )) = − ρ ( D ( f ◦ c )) on a two plane are the negatives of the principal curvatures. Bythe homogeneity properties of the fundamental forms, the investigated functionscan be restricted to such a special subset, on which all of the possible valuesattain, to the unit circle of this plane. This set is compact and thus there aretwo extremal values and at least two corresponding unit vectors, respectively.The convexity of such a hypersurface implies that the signs of the extremalvalues of α ( τ ( l a )) are equals, so the two principal curvatures has the same signsand thus the sectional curvature is negative.On the other hand, the characters of such a tangent plane would be only twotypes; either it is a space-like plane containing only space-like vectors or it hastwo non-paralel light-like vectors partitioning the plane two double cones oneof them contains the space-like vectors and the other one the time-like vectors,respectively.In the second case, we can restrict our function onto the union of the imag-inary unit circle, the de Sitter circle, and the two lines containing the light-likevectors, respectively. We omit the two direction of the light-like vectors and wecan determine the extremal values of the second fundamental form on the deSitter sphere and on the imaginary unit sphere, respectively. For example if thesigns of the functions α ( τ ( l a )) and I f ( c ( τ ( l a ))) are equals, and the normal vec-tors are space-time vectors, then the principal curvatures have the same signs,implying that their product is positive. In this case, the sectional curvature ispositive. 23 Four interesting premanifolds
In this section we give the most important hypersurfaces of a generalized space-time model and determine their geometries, respectively.
Then by Theorem 6 ( H + , ds ) is a Minkowski-Finsler space, where for the vec-tors u and u of T v we have ds v ( u , u ) = [ u , u ] + v with the Minkowski product [ · , · ] + v of the tangent space T v . This gives a pos-sibility to examine the geometric property of H + on the base of the standarddifferential geometry of a space-time hypersurface. First we prove the followingtheorem: Theorem 8 H + is always convex. It is strictly convex if and only if the s.i.p.space S is a strictly convex space. Proof:
Let w = s ′ + t ′ be a point of H + and consider the product[ w − v, v ] + = [ s ′ − s, s ] + [ t ′ − t, t ] = [ s ′ , s ] − [ s, s ] − ( λ ′ − λ ) λ = [ s ′ , s ] − λ ′ λ + 1 , where t ′ = λ ′ e n , t = λe n and s ′ , s ∈ S with positive λ ′ and λ , respectively. Since p s ′ , s ′ ] = λ ′ and p s, s ] = λ thus [ w − v, v ] + = [ s ′ , s ] − p s ′ , s ′ ] p s, s ] + 1 ≤≤ p [ s ′ , s ′ ][ s, s ] − p s ′ , s ′ ] p s, s ] + 1 ≤ , because of the relation[ s ′ , s ′ ][ s, s ] + 2 p [ s ′ , s ′ ][ s, s ] + 1 ≤ [ s ′ , s ′ ][ s, s ] + ([ s ′ , s ′ ] + [ s, s ]) + 1 . (We used here the inequality between the arithmetic and geometric means oftwo positive numbers.) Remark that equality holds if and only if the norms of s ′ and s are equal to each other and thus λ ′ = λ , too. So we have[ s ′ , s ] − [ s, s ] = 0 , or equivalently [ s ′ , s ] = p [ s ′ , s ′ ][ s, s ] . From the characterization of the strict convexity of an s.i.p. space we get H + contains only the point v of the tangent space T v if and only if the s.i.p. space S is strictly convex. (cid:3)
24o determine the first fundamental form consider the map h = s + h ( s ) e n giving the points of H + . (Here h ( s ) = p s, s ] is a real valued function.)Then we get thatI = [ ˙ c ( t ) + ( h ◦ c ) ′ ( t ) e n , ˙ c ( t ) + ( h ◦ c ) ′ ( t ) e n ] + == [ ˙ c ( t ) , ˙ c ( t )] − [( h ◦ c ) ′ ( t )] , where ˙ c ( t ) means the tangent vector of the curve c of S at its point c ( t ). UsingLemma 3 and Theorem 5 we haveI = [ ˙ c, ˙ c ] − (cid:16) [ ˙ c ( t ) , c ( t )] + [ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) c ( t ) , c ( t )]) = [ ˙ c, ˙ c ] − [ ˙ c ( t ) , c ( t )] c ( t ) , c ( t )] . From this formula, by the Cauchy-Schwartz inequality, we can get a newproof for the fact that this form is positive.The second fundamental form of H + isII := [¨ c ( t )+( h ◦ c ) ′′ ( t ) e n , c ( t )+( h ◦ c )( t ) e n ] +( h ◦ c )( t ) = [¨ c ( t ) , c ( t )] − ( h ◦ c ) ′′ ( t ) h ( c ( t )) , since n ◦ c = h ◦ c = c ( t ) + ( h ◦ c )( t ) e n . First we compute the derivative of( h ◦ c ) ′ ( t ) : R −→ R at its point t . We use again the formulas of Lemma 3 and Lemma 4 getting( h ◦ c ) ′′ ( t ) = (( h ◦ c ) ′ ) ′ ( t ) = [ ˙ c ( t ) , c ( t )] p c ( t ) , c ( t )] ! ′ == [ ˙ c ( t ) , c ( t )] ′ p c ( t ) , c ( t )] − [ ˙ c ( t ) ,c ( t )] √ c ( t ) ,c ( t )] [ ˙ c ( t ) , c ( t )](1 + [ c ( t ) , c ( t )])and so ( h ◦ c ) ′′ ( t ) h ( c ( t )) = [ ˙ c ( t ) , c ( t )] ′ − [ ˙ c ( t ) , c ( t )] c ( t ) , c ( t )] = (cid:16) [¨ c ( t ) , c ( t )] + [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) − [ ˙ c ( t ) , c ( t )] c ( t ) , c ( t )] . Thus the second fundamental form isII = − [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) + [ ˙ c ( t ) , c ( t )] c ( t ) , c ( t )] , or using the formula k y kk · k ′′ x,z ( y ) = [ x, · ] ′ z ( y ) − ℜ [ x, y ] ℜ [ z, y ] k y k ,
25e have equivalentlyII = −k c ( t ) kk · k ′′ ˙ c ( t ) , ˙ c ( t ) c ( t ) − [ ˙ c ( t ) , c ( t )] k c ( t ) k (1 + k c ( t ) k ) . If we also assume that the norm is a C function of its argument then we canuse Theorem 5 and we getII = − [ ˙ c ( t ) , ˙ c ( t )] + [ ˙ c ( t ) , c ( t )] c ( t ) , c ( t )] = − I . By the positivity of the first fundamental form on H + , we get that the secondfundamental form is negative definite and ρ ( u, v ) max = ρ ( u, v ) min = − . This implies that the sectional curvatures are equal to −
1, the Ricci and scalarcurvatures in any direction at any point is − ( n −
2) and − (cid:0) n − (cid:1) , respectively.We proved: Theorem 9
If the S is a continuously differentiable s.i.p. space then the imag-inary unit sphere has constant negative curvature. Observe that our definitions in the case when the Minkowski product is ani.i.p. go to the usual concepts of hypersurfaces of a semi-Riemann manifolds(see [4], [16] or [18]) so we can regard H + a natural generalization of the usualhyperbolic space. Thus we can say that H is premanifold with constant negativecurvature and H + is a prehyperbolic space. In this subsection we shall investigate the hypersurface of those points of ageneralized space-time model which scalar square is equal to one. In a pseudo-euclidean space this set was called by the de Sitter sphere . The tangenthyperplanes of the de Sitter space are pseudo-euclidean spaces. We will denoteby G this set. G is not a hypersurface of V but we can restrict our investigationto the positive part of G defined by G + = { s + t ∈ G : t = λe n where λ > } . We remark that the local geometry of G + and G is agree by the symmetry of G in the subspace S . G + is already a hypersurface defined by the function g ( s ) = s + g ( s ) e n , where g ( s ) = p − s, s ] for [ s, s ] > . g : s p − s, s ]giving the corresponding tangent vectors of form u = α ( e + g ′ e ( s ) e n ) . Since between g and f : s p s, s ], there is the connection f ( s ) + g ( s ) = 2[ s, s ] , the derivative of g in the direction of the unit vector e ∈ S (by Lemma 1 andLemma 3) can be calculated from the equality2 f ( s ) f ′ e ( s ) + 2 g ( s ) g ′ e ( s ) = 4 k s kk · k ′ e ( s ) = 4[ e, s ] . Thus g ′ e ( s ) = [ e, s ] g ( s ) = [ e, s ] p − s, s ]meaning that[ u, u ] + = α (cid:18) − [ e, s ] ( − s, s ]) (cid:19) = α − s, s ] − [ e, s ] − s, s ] . From this we can see immediately that[ u, u ] + > − s, s ] > [ e, s ] [ u, u ] + = 0 if − s, s ] = [ e, s ] [ u, u ] + < − s, s ] < [ e, s ] . So a vector s ′ of the n − S orthogonal to s determines a space-timetangent vector in the tangent space and a tangent vector corresponding to αs is a time-like one. To determine the light-like tangent vectors consider a unitvector e ∈ S of the form e = ± p − s, s ][ s, s ] s + s ′ , where s ′ ∈ s ⊥ . Such a unit vector lying in the intersection of the unit sphere of S by the unionof n − s ⊥ + ± p − s, s ][ s, s ] s. Since s ⊥ is the orthogonal complement of s in S and ± p − s, s ][ s, s ] ! [ s, s ] = − s, s ][ s, s ] < , n − u = α (cid:16)(cid:16) ± p − s, s ] s + [ s, s ] s ′ (cid:17) ± [ s, s ] e n (cid:17) . Recall that we considered the tangent hyperplane as a subspace of the orig-inal vector space and observe that thus we can admit it an inner Minkowskianstructure, with respect to the positive and negative subspaces S ′ := s ⊥ ∩ S = s ⊥ and T ′ = α (cid:16)p − s, s ] s + [ s, s ] e n (cid:17) . First we note the following:
Theorem 10 G + and its tangent hyperplanes are intersecting, consequentlythere is no point at which G would be convex. Proof:
At an arbitrary point of G + there are two sets lying on G + and havingin distinct halfspaces with respect to the corresponding tangent hyperplane.The first set is the intersection of the 2-plane spanned by e n and s + t ∈ M ;and the other one is an arbitrary curve of the ( n − G and the hyperplane S + ( s + t ). In fact, a normal vectorof the tangent hyperplane at s + t is itself s + t , because we have " e + [ e, s ] p − s, s ] e n , s + p − s, s ] e n + = 0 . Thus with α > √ [ s,s ] we have h(cid:16) αs + p − αs, αs ] e n (cid:17) − (cid:16) s + p − s, s ] e n (cid:17) , s + p − s, s ] e n i + == ( α − s, s ] + ( p − s, s ] − p − αs, αs ]) p − s, s ] == − α [ s, s ] − p ( − αs, αs ])( − s, s ]) == α [ s, s ] − − p − (1 + α )[ s, s ] + α [ s, s ] ≥ α [ s, s ] − > k s k − ≥ . On the other hand if s ′ + t ∈ M arbitrary, then k s ′ k = k s k thus[ s ′ − s + ( t − t ) , s + t ] + = [ s ′ , s ] − [ s, s ] ≤ p [ s ′ , s ′ ] p [ s, s ] − [ s, s ] = 0 , with equality if and only if s ′ = ± s . (cid:3) Continue our investigation with the computation of the fundamental forms.Using the function g the first fundamental form has the formI = [ ˙ c ( t ) + ( g ◦ c ) ′ ( t ) e n , ˙ c ( t ) + ( g ◦ c ) ′ ( t ) e n ] + == [ ˙ c ( t ) , ˙ c ( t )] − [( g ◦ c ) ′ ( t )] . c, ˙ c ] − (cid:16) [ ˙ c ( t ) , c ( t )] + [ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) − c ( t ) , c ( t )]) = [ ˙ c, ˙ c ] − [ ˙ c ( t ) , c ( t )] − c ( t ) , c ( t )] . Furthermore we also have that n ◦ c = g ◦ c = c ( t ) + ( g ◦ c )( t ) e n thusII := [¨ c ( t )+ ( g ◦ c ) ′′ ( t ) e n , c ( t )+ ( g ◦ c )( t ) e n ] +( g ◦ c )( t ) = [¨ c ( t ) , c ( t )] − ( g ◦ c ) ′′ ( t ) g ( c ( t )) . The derivative of the real function( g ◦ c ) ′ ( t ) = D ( g ◦ c )( t ) : R −→ R at its point t is:( g ◦ c ) ′′ ( t ) = [ ˙ c ( t ) , c ( t )] ′ p − c ( t ) , c ( t )] − [ ˙ c ( t ) ,c ( t )] √ − c ( t ) ,c ( t )] [ ˙ c ( t ) , c ( t )]( − c ( t ) , c ( t )])so by Lemma 4( g ◦ c ) ′′ ( t ) g ( c ( t )) = [ ˙ c ( t ) , c ( t )] ′ − [ ˙ c ( t ) , c ( t )] − c ( t ) , c ( t )] == (cid:16) [¨ c ( t ) , c ( t )] + [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) − [ ˙ c ( t ) , c ( t )] − c ( t ) , c ( t )] . Thus we have II = − [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) + [ ˙ c ( t ) , c ( t )] − c ( t ) , c ( t )] . If we assume again that the norm is a C function of its argument then we canuse again Theorem 5 and we getII = − [ ˙ c ( t ) , ˙ c ( t )] + [ ˙ c ( t ) , c ( t )] − c ( t ) , c ( t )] = − I , as in the case of H + . The principal curvatures are equal to −
1. But the scalarsquares of the normal vectors is positive at all points of G + implying that thesectional curvatures are equal to 1. The Ricci curvatures in any directions andat any points are equal to ( n − (cid:0) n − (cid:1) showing that: Theorem 11
The de Sitter sphere G has constant positive curvature if S is acontinuously differentiable s.i.p space. On the basis of this theorem we can say about G as a premanifold of constantpositive curvature and we may say that it is a pre-sphere .29 .3 The light cone The inner geometry of the light cone L can be determined, too. Let L + be thepositive part of this double cone determined by the function: l ( s ) = s + p [ s, s ] e n . If S is a uniformly continuous s.i.p. space, then the tangent vectors at s are ofthe form: u = α ( e + k · k ′ e ( s ) e n ) = α e + [ e, s ] p [ s, s ] e n ! . Thus all tangents orthogonal to l ( s ) which is also a tangent vector. (Choose e = s and α = k s k !) But the orthogonal companion of a neutral (isotropic orlight-like) vector in a s.i.i.p space is an ( n − L + and it is an ( n − V . This also asupport hyperplane of L . In fact, by v = s + t and w = s ′ + t ′ we get[ w − v, v ] + = [ s ′ , s ] + [ t ′ , t ] = [ s ′ , s ] − λ ′ λ where t ′ = λ ′ e n , t = λe n and s ′ , s ∈ S with positive λ ′ and λ , respectively. Since p [ s ′ , s ′ ] = λ ′ and p [ s, s ] = λ thus [ w − v, v ] + = [ s ′ , s ] − p [ s ′ , s ′ ] p [ s, s ] ≤ s ′ = αs meaning that there is only one line of L + in the tangent space T v . Thus the light cone is convex and thus the second fundamental form is semi-definite quadratic form. It also follows that any other vectors of the tangenthyperplane are space-like ones and there are two types of tangent 2-planes; oneof them space-like plane and the other one contains space-like vectors and adoubled line of light-like vectors. In the first case, the corresponding principaland sectional curvatures is well defined and have negative values, respectively.To determine it we compute the fundamental forms.In the case when S is continuously differentiable, the first fundamental formis I = [ ˙ c, ˙ c ] − (cid:16) [ ˙ c ( t ) , c ( t )] + [ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) c ( t ) , c ( t )] = [ ˙ c, ˙ c ] − [ ˙ c ( t ) , c ( t )] [ c ( t ) , c ( t )] , and the second one isII = − [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) + [ ˙ c ( t ) , c ( t )] [ c ( t ) , c ( t )] = − [ ˙ c ( t ) , ˙ c ( t )] + [ ˙ c ( t ) , c ( t )] [ c ( t ) , c ( t )] = − I . Thus the principal curvatures are − − Theorem 12
The light cone L + has zero curvature if S is a continuously dif-ferentiable s.i.p space. Hence L is a premanifold with zero curvature and we may say that it is a pre-Euclidean space. ( V, [ · , · ] − ) In this subsection we shall investigate the hypersurface of those points of thegeneralized space-time model which collects the unit sphere of the embeddings.i.p. space. In a pseudo-euclidean space it is the unit sphere of the embeddingeuclidean space. Its tangent hyperplanes are pseudo-euclidean one. We will de-note by K this set. K is not a hypersurface but we can restrict our investigationto the positive part of K defined by K + = { s + t ∈ K : t = λe n where λ > } .K + is a hypersurface defined by the function k ( s ) = s + k ( s ) e n , where k ( s ) = p − [ s, s ] for [ s, s ] < . The directional derivatives of the function k : s p − [ s, s ] for [ s, s ] < u = α ( e + k ′ e ( s ) e n ) . Since by the function f : s p s, s ] , we have the equality f ( s ) + k ( s ) = 2the derivative in the direction of the unit vector e ∈ S is k ′ e ( s ) = − [ e, s ] p − [ s, s ]31eaning that[ u, u ] + = α (cid:18) − [ e, s ] (1 − [ s, s ]) (cid:19) = α − [ s, s ] − [ e, s ] − [ s, s ] . From this we can see immediately that[ u, u ] + > − [ s, s ] > [ e, s ] [ u, u ] + = 0 if 1 − [ s, s ] = [ e, s ] [ u, u ] + < − [ s, s ] < [ e, s ] . It follows that the vector s ′ of the n − S orthogonal to s gives aspace-time tangent vector and the vector corresponding to αs is a time-like one.As in the case of the imaginary unit sphere we note the following: Theorem 13 K + is convex. If S is a strictly convex space, then K + is alsostrictly convex. Proof:
Let w = s ′ + t ′ be a point of K + and consider the product[ w − v, n v ] + = [ s ′ − s, s ′′ ] + [ t ′ − t, t ′′ ] = [ s ′ , s ′′ ] − [ s, s ′′ ] − ( λ ′ − λ ) λ ′′ , where t ′′ = λ ′′ e n , t ′ = λ ′ e n , t = λe n and s ′′ , s ′ , s ∈ S with positive λ ′′ , λ ′ and λ , respectively. Since p − [ s ′ , s ′ ] = λ ′ and p − [ s, s ] = λ and n v = s − p − [ s, s ] e n thus [ w − v, n v ] + = [ s ′ , s ] + p − [ s ′ , s ′ ] p − [ s, s ] − ≤≤ p [ s ′ , s ′ ][ s, s ] + p − [ s ′ , s ′ ] p − [ s, s ] − ≤ , because 2 p [ s ′ , s ′ ][ s, s ] ≤ [ s ′ , s ′ ] + [ s, s ]) . We remark that equality holds in the inequalities if and only if the norms of s ′ and s are equal to each other. So we have the equality[ s ′ , s ] − [ s, s ] = 0 , or equivalently [ s ′ , s ] = p [ s ′ , s ′ ][ s, s ] . We also get that v is the only point of H + lying on the tangent space T v if andonly if the s.i.p. space S is strictly convex. (cid:3) Using the function k the first fundamental form has the formI = [ ˙ c ( t ) , ˙ c ( t )] − [( k ◦ c ) ′ ( t )] . c, ˙ c ] − (cid:16) [ ˙ c ( t ) , c ( t )] + [ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) − [ c ( t ) , c ( t )]) = [ ˙ c, ˙ c ] − [ ˙ c ( t ) , c ( t )] − [ c ( t ) , c ( t )] , and assuming that 2[ c ( t ) , c ( t )] = 1 we getII = " ¨ c ( t ) + ( k ◦ c ) ′′ ( t ) e n , c ( t ) − ( k ◦ c )( t ) e n p | − c ( t ) , c ( t )] | +( k ◦ c )( t ) == 1 p | − c ( t ) , c ( t )] | ([¨ c ( t ) , c ( t )] + ( k ◦ c ) ′′ ( t ) k ( c ( t ))) . Lemma 4 implies that( k ◦ c ) ′′ ( t ) k ( c ( t )) = − [ ˙ c ( t ) , c ( t )] ′ + [ ˙ c ( t ) , c ( t )] − [ c ( t ) , c ( t )] == − (cid:16) [¨ c ( t ) , c ( t )] + [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) (cid:17) + [ ˙ c ( t ) , c ( t )] − [ c ( t ) , c ( t )] . thus we haveII = 1 p | − c ( t ) , c ( t )] | (cid:18) − [ ˙ c ( t ) , · ] ′ ˙ c ( t ) ( c ( t )) + [ ˙ c ( t ) , c ( t )] − [ c ( t ) , c ( t )] (cid:19) . Assuming that S is continuously differentiable and using Theorem 5 we getII = 1 p | − c ( t ) , c ( t )] | (cid:18) − [ ˙ c ( t ) , ˙ c ( t )] + [ ˙ c ( t ) , c ( t )] − c ( t ) , c ( t )] (cid:19) == − p | − c ( t ) , c ( t )] | I . The principal curvatures at a point k ( c ( t )) are ρ max ( u, v ) = ρ min ( u, v ) = − p | − c ( t ) , c ( t )] | giving the sectional curvatures κ ( u, v ) := [ n ( c ( t )) , n ( c ( t ))] + ρ ( u, v ) max ρ ( u, v ) min = 1 − c ( t ) , c ( t )] . The Ricci curvatures in any directions at the point k ( c ( t )) are equal toRic( v ) k ( c ( t )) := ( n − · E ( κ k ( c ( t )) ( u, v )) = n − − c ( t ) , c ( t )]33nd the scalar curvature of the hypersurface K + at its point k ( c ( t )) isΓ k ( c ( t )) := (cid:18) n − (cid:19) · E ( κ f ( c ( t )) ( u, v )) = (cid:0) n − (cid:1) − c ( t ) , c ( t )] . Finally we remark that at the points of K + having the equality 2[ c ( t ) , c ( t )] = 1all of the curvatures can be defined as in the case of the light cone and can beregard to zero. References [1] Alonso, J., Benitez, C.:
Orthogonality in normed linear spaces: a survey.Part I. Main properties.
Extracta Math. (1988), 1–15.[2] Alonso, J., Benitez, C.: Orthogonality in normed linear spaces: a survey.Part II. Relations between main orthogonalities.
Extracta Math. (1989),121–131.[3] Csik´os, B.: Personal communication. [4] Dawis, M.W., Moussong, G.:
Notes on Nonpositively Curved Polyhedra ,Bolyai Society Mathematical Studies, 8, Budapest, 1999.[5] Dubrovin, B.A., Fomenko A.T., Novikov S.P.:
Modern Geometry- Meth-ods and Applications, Part I. The geometry of Surfaces, TransformationGroups, and Fields. Second Edition , Springer-Verlag, 1992.[6] G.Horv´ath, ´A.:
Semi-indefinite inner product and generalized Minkowskispaces.
Journal of Geometry and Physics (2010) 1190–1208.[7] Giles, J. R.: Classes of semi-inner-product spaces.
Trans. Amer. Math.Soc. (1967), 436–446.[8] Gohberg, I., Lancester, P., Rodman, L.:
Indefinite Linear Algebra andApplications.
Birkh¨auser, Basel-Boston-Berlin 2005.[9] Gruber P. M.- Lekkerkerker C. C.:
Geometry of Numbers.
North-HollandAmsterdam-New York-Oxford-Tokyo 1987.[10] Lumer, G.:
Semi-inner product spaces.
Trans. Amer. Math. Soc. (1961), 29-43.[11] Lumer, G.:
On the isometries of reflexive Orlicz spaces.
Ann. Inst. Fourier,Grenoble (1963) 99–109.[12] Martini, H.: Shadow boundaries of convex bodies.
Discrete Math. (1996), 161-172.[13] Martini, H., Swanepoel, K., Weiss, G.:
The geometry of Minkowski spaces- a survey. Part I.
Expositiones Mathematicae (2001), 97-142.3414] Martini, H., Swanepoel, K.: The geometry of Minkowski spaces - a survey.Part II.
Expositiones Mathematicae (2004), 93-144.[15] McShane, E. J.:
Linear functionals on certain Banach spaces.
Proc. Amer.Math. Soc., Vol. (1950), 402–408.[16] O’Neill, G.: Semi-Riemannian Geometry with Applications to Relativity ,Academic Press, New-York, 1983.[17] Tam´assy, L.:
Finsler spaces corresponding to distance spaces.
Proc. of theConf., Contemporary Geometry and Related Topics, Belgrade, Serbia andMontenegro, June 26–July 2, (2005), 485–495.[18] Verpoort, S.: