PROJECTIVE INVARIANTS OF AN ORTHOGONAL ENNUPLE

A NNALES DE L ’ ^INSTITUT F ^OURIER

H. D. P ^ANDE

Projective invariants of an orthogonal ennuple in a Finsler space

Annales de l’institut Fourier, tome 18, n^o2 (1968), p. 337-342

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Ann. Inst. Fourier, Grenoble 18, 2 (1968), 337-342.

PROJECTIVE INVARIANTS OF AN ORTHOGONAL ENNUPLE

IN A FINSLER SPACE

1. Introduction.

We consider an n-dimensional Finsler space F^ with the fundamental metric function F(rc, A). This fundamental function is positive homogeneous of the first degree in x\ it is > 0 for ^(x^^O and the quadratic form (^F^AW)^' is positive definite in the variables ^l. The metric tensor is given by

(1.1) g.,Qr, A)^=-|-6.6,P(a;, x) 0), C)

This tensor is symmetricin the in dices i, / and positive homogeneous of degree zero in A1. The contravariant compo- nents giJ'{x, x) of the metric tensor is determined by

(1.2) ^(x, x)g^x, x) = ^ ^ ^ ^-

The covariant components of the unit vector along the direc- tion of the element of support (x\ x1) are given by

(1.3) l,(x, x) == ^¥{x, A).

The covariant derivative of a vector X^rc, A), depending on the element of support, with respect to xk in the sense of

(*) With the Department of Mathematics, University of Gorakhpur, India, when this work was started.

(1) ^. = b/o^' and 6. == b/W.

(2) Numbers in brackets refer to the references at the end of the paper.

338 H. D. P A N D E

Cartan is given by

(1.4) X\,(x, x) == (^X-) - (WGi + X^i, where

(1.5a) Gi(rr, x) ^ b.G1^, A), (1.5&) 2G\x, x) ^^x)x^x\

Yjk{^ x) being the ChristoffePs symbols of second kind [1] and P^(a;, x) are the Cartan's connection coefficients symmetric in their lower indices and homogeneous of degree zero in their directional arguments. We have [1]

(1.6) G},(^ x)x^ = rji(^ x)x^ = G[{x, x), where G^{x, x) =^= ^G}((a;, A).

Let X(a){a == 1, 2, . . ., n} be the unit tangents of n-con- gruences of an orthogonal ennuple. The subscript « a » in the paranthesis simply distinguishes one congruence from the other. The covariant and contravariant components of X(a) will respectively be denoted by \a)i and X1^). Since yi-con- gruences are mutually orthogonal, we have [2]

(1-7) g^, x)\[^ = ^

where the Kronecker delta ^ == j19 lf a = &. We have (0, if a -=f=- b

the Ricci coefficients of rotation, given by [2, 3]

(1.8) Y,^, x)=^\^\^

where the symbol I denotes the covariant derivative with respect to xk in the sense of Cartan and

(1.9) (x1^, x) ^ S Y^).

h

The geometric entities (A^)(^, x) are called the first curva- ture vector of a curve of congruence in Finsler space [3].

2. Projective transformation.

The equation of a geodesic

/o A\ d2^ . p^./ , ,, dxj dxk ^

al}

^+

^

^

^^

PROJECTIVE INVARIANTS OF AN ORTHOGONAL ENNUPLE 339

assumes the following form by the transformation of its para- meter s to t [4] :

(2.2) x

(^ + r^, x)x^ - x

\^ + r^, x)^} = o,

where

(2.3) r^, x) = r^, A).

The equation (2.2) remains unchanged if we replace the Car- tan's connection coefficient rjj^, x) by a new symmetric coefficient T^x, A), given by [6]

(2.4) r^, x) ^

r^, x) + 2§^ + p^s

where pk{x, A) is a covariant vector, positively homogeneous of degree zero in its directional arguments and

(2.5) p^x, x) ^^L ^jp^ x).

DEFINITION 2.1. — Let F^ and Fn be two spaces with fundamental tensor gij{x, x) and gij{x, x) at the corresponding points. Then the spaces are said to be in geodesic correspondence if their geodesies are the same and we shall call (2.4)a « projectile change » of the Cartan^s function r^(rc, A).

Contracting (2.4) with respect to the indices i and /, we get

(2.6) rfc x) = rfc x) + (n + i)p.(^ ^).

Differentiating (2.6) with respect to x1, we obtain (2.7) ^Ffc x) = W{x, x) + {n + i)p^x, x).

3. Projective invariants.

THEOREM 3.1. — If X^)(^) and \a)i(^) are ^ contrawriant and covariant components of an orthogonal ennuple, then the following geometric entities are invariant under the projectile change:

(3.1) AU^, x) ^= H^ - -^-^ ^ j 2 S \^Ww \.

340 H. D. PANDE

and

(3.2) A^,A)^SW^-n;I.

a

Proof. — If we denote by ^)f/c ^e covariant derivative of X1^) in the sense of Cartan for the connection coefficients P^(^, A), then we have

(3.3) U^ = ^ + W^

Hence we get in consequence of (2.4) (3.4) X^i, - X^i, == 4){2^ + p^1}.

Multiplying (3.4) by \a)i throughout and summing with respect to a and using the orthogonality condition (1.7), we obtain

(3.5) S X^(X^ ~ U)i.) = {ri + l)p,.

a

Eliminating the vector pk(rc, A) from equations (3.4) and (3.5), we get

. i .

(3.6) X^)|fc — ^a)(fc == ^ . ^ ^^a) p) S \ft)m(^r&)lfc — ^^j*)

+ Sfc S \&)m(^)i7 ""' ^)»y)1.

b J

Again, with the help of (2.6), equation (3.5) yields

(3.7) s W^r. - ^)i.)

r^ - r%

a

which gives us (3.2).

THEOREM 3.2. — When F^ and F^ are in geodesic corres- pondence^ we have the following geometric entities which are invariant under the projectile change:

(3.8) C^, x) ^ U^ - ^^{2TO + ^,^1},

and

(3.9) C^'x- x) ===== X^)|fc — —.—.X^)^ S 2X(ft)^S1 l(yX^)jfc) n + 1 ( b

+^Wlj.

PBOJECTIVE INVARIANTS OF AN ORTHOGONAL ENNUPLE 341

Proof. — Using equations (2.6), (2.7) and (3.4), we get

(3.10) \[^ - ^ == ^^ WWl - r%

+ W - r^) + AWI - r%],

which yields the result (3.8).

Again, eliminating p^rc, x) and p^(x^ x) from equations (2.7), (3.4) and (3.5), we obtain

(3.11) x^ifc — x^ifc = ~~rT^) 5 (&)m[^(^^)i/c — ^%ifc)

n ~r JL &

+ wo - w] + ^4 ^ ^a) ' ^J(r ^ ~ ^^

which gives us (3.9).

THEOREM 3.3. — When F^ and F^ ar^ ^ geodesic corres- pondence, we have the following projectile invariant geometric entities:

(3.12) s^, x) =^ Y^ - ^^ (U?or^ + U^r^j)

1

— ^~in Ma)\b^e)^j^

and

(3.13) SUr, A) ^ Y^ - ^-^ [ S (S.Y^ + ^Y^)]

1

~ ^ . ^ ^a)\c)^?d)^r¹^,

an^

(3.14) S,(a;, A) =^ S Y.«» - ^w^L

a

where Y^c are Ricci coefficients of rotation.

Proof. — Multiplying (3.10) by the product X^^c) and using the orthogonality relation (1.7), we get

(3.15) Y^ - ——— (W^l + U^J

n ~r" -i-

+ 4)W^WI) == Y^c - ^-^ (S^^^r^

+ ^W] + X^^^^A^,^!),

u^" H. D. PANDE

where the projectively transformed Ricci coefficients of rota- tion are given by

(3.16) ^^X^Wc).

Similarly, multiplying (3.11) by the product \^ and using the orthogonal relation (1.7), we obtain (3.13).

Again, multiplying (3.7) by X§,) and making use of (1.7), we get

(3.17) S Y,, - Wl = S ^ - Wl,

" a

which shows that S^x, x) are invariant under the projective change.

I am thankful to Professor R. S. Mishra for his kind help in the preparation of this paper and to Professor J. P. 0. Sil- berstein for his valuable suggestions.

BIBLIOGRAPHY

[1] H. RUND, The differential Geometry of Finsler spaces. Springer- VerlasBerlin (1959). r , ^ e 5,

[2] B. B. SINHA, Projective Invariants, Maths. Student, XXXIII, No. 2 and 3 (1965), 121-127.

[3] R. S. MISHRA, On the congruence of curves through points of a subspace imbedded in a Riemann space, Ann. Soc. Sci., Bruxelles (1951), 109-115.

[4] T. Y. THOMAS, Differential Geometry of generalised spaces, Cambridge (1934). ° [5] L. BERWALD, On the projective geometry of paths, Ann. Maths., 37

(1936), 879-898.

[6] H. D. PANDE, The projective transformation in a Finsler space, Atti della Accad. Naz., Lincei. Rendiconti, XLIII, No 6 (1967), 480-484

Manuscrit reQu Ie 7 mars 1968 H. D. PANDE,

Department of Mathematics, The University of Western Australia,

Nedlands, Western Australia.