Ricci-calculus: an introduction to tensor analysis and its geometrical applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer
1954
|
| Edition: | 2. ed. |
| Series: | Grundlehren der mathematischen Wissenschaften
10 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 516 S. graph. Darst. |
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MARC
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| 100 | 1 | |a Schouten, Jan A. |d 1883-1971 |e Verfasser |0 (DE-588)117015547 |4 aut | |
| 245 | 1 | 0 | |a Ricci-calculus |b an introduction to tensor analysis and its geometrical applications |c by J. A. Schouten |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin |b Springer |c 1954 | |
| 300 | |a XX, 516 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 10 | |
| 650 | 7 | |a Differentiaalmeetkunde |2 gtt | |
| 650 | 7 | |a Tensoren |2 gtt | |
| 650 | 4 | |a Calculus of tensors | |
| 650 | 4 | |a Geometry, Differential | |
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| 650 | 0 | 7 | |a Tensoranalysis |0 (DE-588)4204323-2 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804116362600120320 |
|---|---|
| adam_text | Contents.
I. Algebraic preliminaries (1).
§ 1. The Bn (1).
affine space (1) — En (l) — affine group (l) — Ga (l) — minor (1) — summation
convention (l) — A1^, A y (1) — point space (l) — allowable coordinate trans¬
formations (1) — rectilinear coordinates (1) — homogeneous linear group (2) —
Gh 0 (2) — centred En (2) — point transformations (2) — = (2) — dragging along
of coordinate system (2) — d% (2) — Kronecker symbol (2, 7) — fix (2) — gene¬
ralized Kronecker symbol (2) — kernel letter (2) — running indices (2) — fixed
indices (2) — kernel index method (3) — flat submanifold (3) — linear submani
fold (3) — C (3) — rank (3) — B% (3) — point (3) — straight line (3) — plane
(3) — hyperplane (3) — coordinate axes (4) — coordinate planes (4) — coordinate
Ep s (4) — net (4) — lie in (4) — contain (4) — translation (4) — parallel (4) —
^ direction (4) — improper Ep_x (4) — Ep i at infinity (4) — direction
(4) — reduction (5) — ^ parallel (4) — section (4, 5) — spanning (4) —
join (4) — projection (4, 5) — screwsense (5) — sense (5) — sense of rotation (5) —
opposite (5) — inner orientation (5) — outer orientation (5).
§2. Quantities in En (6).
quantity (6) — components (6) — kind (6) — sum of quantities (5) — manner
of transformation (6) — scalar (6) — contravariant vector (6) — contragredient
(6, 7) — contravariant basis (7) — dead indices (7) — living indices (7) —
covariant vector (7) — covariant basis (8) — transvection (8) — reciprocal sets
(9) — strangling (10) — tensor (10) — affinor (10) — valence (10) — co and
contravariant tensor (10) — cocontravariant (10) — mixed (10) — unity tensor
(10) — A (10) — intermediate components (11) — equiform (11) — pseudo
tensor (11) — pseudo scalar (11) — pseudo vector (11) — class of pseudo quan¬
tity (11) — tensor density (12) —tensor /( density (12) —weight (12) — W tensor
(12) — a (12) — Gsa (12) — connecting quantities (13)
§3. Invariant processes and relations (13).
addition of quantities (13) — isomer (13) — general multiplication (13) — con¬
traction (13) — transvection (14) — saturated indices (14) — dummy indices (14) —
free indices (14) — mixing (14) — round brackets (14) — symmetric (14) — sym¬
metric part (14) — alternation (14) — square brackets (14) — alternating (14) —
alternating part (14) —concomitant (15) —invariant (15) — rational integral (15)
§ 4. Section and reduction with respect to an Em in En (16).
decomposition with respect to a rigged Em (16) — section (16) — reduction (17I —
rigging (17) — projection (18) — £m part (19)
§ 5. Rank, domain and support of domain with respect to one or more indices (20).
jU rank (20) —/j domain (20) — support of domain (20) — double brackets [[]] (20).
§6. Symmetric tensors (21).
symmetric tensor (21) — symmetric multiplication (21) — divisor of a symmetric
tensor (21) — irreducible symmetric tensors (21).
Contents. XI
§ 7. Multivectors (22).
multivector (22) — ^ vector (22) — bivector, trivector, quadrivector (23) —
alternating multiplication (23) — divisor of a multivector (23) — simple multi
vector (23) — « vector (23) — E*1 ** (25) —*«!,...*, (25) — * (25) — ¦ »
(26) — ex1 xn (26) — identifications (27) — pseudo scalar (27) — tables of
alternating quantities in E3 (28).
§ 8. Tensors of valence 2 (28).
Cocontravariant tensor of valence 2 (28) — trace (29) — spur (29) — eigenvector
(29) — eigenvalue (29) — characteristic equation (29) — first canonical form of
the matrix (33) — elementary divisors (33) — co and contravariant tensors (34) —
s vector tensor (34) — adjoint (34, 38) — symmetric co and contravariant ten¬
sors (34) — index (34) — signature (34) — rank tensors (34) — definite (35) —
indefinite (35) — semi definite (35) — co and contravariant bivectors (35) —
blades (36) — classification of trivectors (36) — reduction number (37).
§ 9. Introduction of a metric in an En (40).
length (40) — distance (40) — indicatrix (40) — fundamental tensor (41) —
euclidean Rn (41) — minkowskian Rn (41, 42) — time like (41) — space like (41) —
^ region (41) — nullcone (41) — unitvector (41) — angle (41) — duration (41) —
isotropic Ep (42) — full isotropic Ep (42) — null Ep (42) — perpendicular (42) —
cartesian coordinate system (42) — minkowskian proper rotations (43) — Lorentz
transformations (43) — minkowskian reflexions (43) — improper minkowskian
rotations (44) — vector (44) — raising and lowering of indices (44) — I* *»
*) ~ (*)
(45) — «a,...^» (45) — i.i (45) — 9 (45) — principal multidirection (46) —
theorem of principal axes (46) — theorem of principal blades (46) — orthogonal
coordinate transformations (47ff.) — rotations (47) — reflexotations (47) —
principal angles (48) — reflexional orthogonal transformations (50) — proper
and improper rotations (50) — ^ parallel (51) — //^ perpendicular (51).
§ 10. Hybrid quantities (51).
quantities of the first and second kind (51) — hybrid quantities (52) — weight
and antiweight (52) — hermitian (symmetric) tensor (52) — hermitian alternating
tensor (52) — hybrid bivector (52) — — invertible (53) — index (53) — signature
(53) — ^ definite (53) — indefinite (53) — semi definite (53) — the auxiliary
E2n (53) — principal En (53) — first and second set of invariant En s (54) — her¬
mitian fundamental tensor (54) — Rn (54) — ordinary Rn (54) — kets and bras (54)
— unitary perpendicular vectors (54) — norm of vector (54) — unitvector (54) —
unitary cartesian coordinate systems (54) — unitary orthogonal group (55) —
fundamental figure in Rn (55) — nullcone in Rn (55) —the auxiliary Rin (55) —
principal multidirections of a hermitian tensor (56) — theorem of principal axes
(56).
§ 11. Abridged notations (57).
collecting indices (57) — representative indices (57) — the notation of Givens (57) —
Vitali s method (57) — Craig s extensors (58) — ideal vectors (58) — systems of
direct calculus (58) — skeleton of a formula (59) — the method of the radius
vector in En and Rn (60) — Cartan s method (61) — abbreviations ad hoc (61).
XII Contents.
II. Analytic preliminaries (61).
§ 1. The arithmetic n dimensional manifold 9ln (61).
arithmetic point (61) — arithmetic manifold 2ln (61) — components of the arith¬
metic point (61) — polycylinder (61) — box (61) — region of 91 n (62) — neighbour¬
hood (62).
§ 2. The geometric n dimensional manifold Xn (62).
coordinate system (62) — analytic functions (62) — class of a function (62) —
coordinate transformation (63) — pseudo group (64) — groupoid (64) — super¬
numerary coordinates (64) — allowable coordinate systems (64) — geometric
manifold (64) — Xn (64) — geometric point (64) — Klein s principle (65) —
cell (65) — region of Xn (65) — imbedding (66) — neighbourhood (66) — analy
ticity in Xn (66) — geometry in the large or global geometry (66) — class in Xn (66).
§ 3. Geometric objects and quantities in Xn (67).
geometric object at a point (67) — { } (67) — components of geometric object (68) —
kind of an object (68) — quantities (68) — connecting objects (68) — pseudo
quantities (69) — pseudo scalar (69) — class of pseudo quantities (69) — tangent
En (69) — local En (69) — object field (70) — analyticity of field (70) — class of
field (70) — natural derivative (70) — gradient (70) — Grad (70) — D (70) —
convention about operators (70) — covariant and contravariant basis vectors in
Xn (70f.) — « ,?;. (71) —A* (71) — 5* (71) —region of constant rank (71) — contra
and covariant measuring vectors (71).
§4. TheXminX, (72).
functional matrix (72) — rank of a set of functions (72) — theorem of adaptation
(73) — rank of a system of equations (73) — null point of a system of equations
(73) — nullmanifold (73) — null form (73) — equivalent systems (73) — minimal
regular (73) — dimension of nullmanifold (73) — imbedding (74 f.) — m dimensional
surface (74) — curve, surface, hypersurface (74) — covariant connecting quantity
(74) — base (74) — first and second base theorem (74f.) — base transformation
(75) — parametric form or representation (75) — minimal regular (75) — contra
variant connecting quantity (75) — coordinate Xm s (76) — normal system of
Xm s (76) — net of a coordinate system (76) — reduction (Zusammenlegung) of
Xn (76) — tangent Em (77) —• section with Xm (77) — reduction with respect to
Xm (77) — rigged (77) — parts of a tensor (77) — prolongation of a field of Xm
over Xn (78).
§ 5. The .Em Held or X}™ in XM. Systems of linear partial differential equations (78).
section with X™ (78) — reduction with respect to X% (78) — rigged X™ (79) —
parts of a tensor (79) — enveloping (79) — outer and inner problem (79) —
Pfaff s problem (79) — solution of a system of partial differential equations (79) —
nullpoint of a system (80) — totally integrable systems (80) — conditions of
integrability (80) — homogeneous linear equations (80) — adjoint systems (80) —
integral function (80) — integral (80) — derived systems (81) — complete systems
(81) — ^fm forming £m field (81) — non homogeneous linear equations (82) —
reduced system (82) — complete non homogeneous systems (82) — gradient
equation (83).
§ 6. The invariant operators Rot and Div (83) •
alternated derivative (83) — natural derivative (83, 84) — rotation (83) — Rot
(83) — D (83) — divergence (84) — Div (84) — gradient product (84) — theorem
of rotation and divergence (85) — other conditions of integrability (86 If •) — section
of a rotation (89).
Contents. XIII
§ 7. Pfafi s problem (89).
class K (89) — rotation class 2g (89) —similarity class k (89) — supports of rotation
(90) — supports (90) — characteristics (90) — S1, S[, S4, S^ (90) — similarity
transformations (91) — gradient transformations (91) — table of results (92) —
canonical form (93) — w^ is ^ enveloping, v = « — q—E (94) — special canonical
forms (94).
§ 8. Theorem of Stokes (95).
generalized theorem of Stokes (97) — differential forms (97 f.) — non invariant
forms (99).
§ 9. Anholonomic coordinates (99).
anholonomic coordinate system (h) (99) — net (100) — Ahx (100) — = (100) —
(di)h (100) — object of anholonomity J2 (100) —Xm in Xn with anholonomic
coordinates (100) — X™ with anholonomic coordinates (101) — intermediate
components Qh a of Qji (101).
§ 10. The Lie derivative (102).
dragging along of a coordinate system (102) — dragging along of an object field
(102) — dragging along over v dt (103) — operator etx (104) — displacement
over rv (104) — infinitesimal point transformation (104) — group generated by
an infinitesimal transformation (104) — Lie derivative, Lie differential (104) —
Lie derivative of sums, products, transvections, contractions (106) — Lie deri¬
vative of scalars, vectors, tensors, ^ vectors, densities (106) — zi densities (107) —
geometric objects (107) — absolutely invariant fields (107) —absolute invariance
with respect to streamlines (107) —¦ Lie derivative of II* x (108) — operator e £
working on linear geometric objects (108) — £{* and £f (109) — commutativity
of £ and D (110).
§ 11. The Lagrange derivative (111).
Lagrange derivative (112) — Lagrange equation (112) — relations for Lagrange
derivative (114).
§ 12. Cartan s symbolical method (117).
alternating differential form (118) — symbolical method of Cart an and Kahler
(119) — symbolical method applied to geometric objects (119) — scheme of trans¬
lation (120) — examples (120f.).
III. Linear connexions (121).
§ 1. Parallel displacement in an En (121).
covariant differential and derivative (122) — parameters of parallel displacement
(122) — d, Fp (122).
§ 2. Parallel displacement in Xn (123).
parallel displacement of quantities (123) — conditions for the covariant differential
(124) — F^x (124) — linear displacement (125) — linear connexion (125) — geo¬
metric interpretation (125) — covariant stationary (125 ff.) —covariant constant
(125f.) — recurrent or parallel field (126) — S^x* (126) — symmetric and semi
symmetric connexion (126) — geometric interpretation (127) — displacement of
points (127) — Cartan displacement (128f.) — covariant differentiation of
densities (129) — covariant differentiation of pseudo quantities (130) — parallel
displaced field (130).
XIV Contents.
§ 3. A linear connexion expressed in terms of S^x , an auxiliary symmetric tensor
field gXx of rank n and V^ gXx (131).
Christoffel symbol (132)— {^ (132) —{fiKx} (132) — metric connexion (132) —
Un (133) — fundamental tensor (133) — vn (133) — riemannian geometry (133) —
ordinary Un and Vn (133) — semi metric connexion (133) — Wn (133) — Weyl s
geometry (133) —conformal transformation of gKx (134) — Umeichung (134) —
Wn dealt with by means of pseudo tensor (135) — necessity of fundamental tensor
of valence 2 (135ff ) — rings of quantities (137)¦
§ 4. Curvature (138).
Rypl* (138) — Riemann Christoffel tensor (139) — curvature tensor (139) —
integrable connexion (139) — V[v P^j and the rule of Leibniz (139f.) — application
to densities and pseudo quantities (140f.) —VvyL (140) —R^, K^^ Ricci tensor
(141) — teleparallelism or absolute parallelism (142) — ^i/H and integrability
conditions (142) —the ideal factors ax, 6, (143) — special semi symmetric and
metric connexions (143) — volume preserving connexions (144).
§ 5. The identities for the curvature tensor (144).
first and second identity (144) — third and fourth identity (145) — principal
directions of Vn (146) — Bianchi s identity (146) — K (148) — Gxk (148) —
scalar curvature k (148) —Gaussian curvature (148) — Ricci space, special (148) —
Einstein space, special (148) —theorem of Herglotz (148) — space of constant
P P+l £.
curvature (148) — Sn (148) — F[^...] (149) — l7^^...] and the rule of Leibniz
(149) — geometric interpretation of F[u...j (150) — field in equilibrium (150) —
geometric interpretation by means of Cart an displacements (151 f ) — Lie deri
vativeof i^ (152)—£ expressed in terms of F7 instead of g (152) —£F^—P^£ (152) —
riemannian curvature with respect to a 2 direction (153) — theorem of Schur
(153).
§ 6. Integrability conditions in Ln (153)
integrability conditions with V (154) •— possibility of covariant constant « vector
fields (155).
§ 7. Geodesies and normal coordinates (155)
geodesic (155) — affine parameter (155) — symmetric connexion determined by
geodesies and their affine parameters (156) — coordinate system geodesic at a
point (157) — ^p...//,A (157) —normal coordinates in An (158) — normal tensors
(159f) — extension (160) — relations between the i? s and the N s (162) — An
symmetric with respect to a point (163) — Rvliji covariant constant in symmetric
^n (163) — differential concomitants (164) — absolute and relative differential
invariants (164) — first and second reduction theorem (164).
§ 8. Fermi coordinates (166).
Fermi coordinates (169) — generalization for Xm (169).
§ 9. Linear connexions expressed in anholonomic coordinates (169).
rft (169) — /] (169) — Qjih (170) — XjHi (170) — r)ih (170) — yihj (171)
coefficients of rotation (171) — i?*;:,:* (172) — Vkj (172).
Contents. XV
§ 10. Cart aim s symbolical method used for connexions (172).
Ah,Qh,lf, Sh, Rfh (172) — structural equations of Ln (173) — thfkjreme de con
P
servation de la courbure et de la torsion (173) — [P ] (174) — mapping of two
sets of n vector fields on each other (174 f.) — mapping of two Ln s on each other
(176ff.).
§ 11. Linear connexions depending on a non symmetric fundamental tensor (179).
non symmetric fundamental tensor (179) — s^x, S *, s *¦*, e, £*¦ ¦, h)lll, kix (180) —
the connexion /£, (180) — ®?*x (180) — the ( h ) differentiation (180) —
gf^ and its divergence (181) — */£} (181 ff.) — the condition for *SX (181) —
the four necessary relations derived by means of the Lie derivative (183) — the
connexion *F^ in special cases (184) — first geometrical interpretation of the
condition (184) — Einstein s interpretation (184) — discussion of results (I84f.).
IV. Lie groups and linear connexions (185).
§ 1. Finite continuous groups (185)
^ parameter finite continuous group (185) — unity element (185) — group germ
(186) — group space (186) — (±) e iuipollent (186) — (± ( constant (187) —
± ±
I^fi (187) — (±) connexion (187) — (i) parallel (187) — R 6^a = 0 (187) — an
holonomic coordinate systems (a) and (A) (!87ff.) — geodesic (188) —Ffy (188) —
(0) connexion (188) — V.V.V (189) S p (189) — c;f, cacb, c*B (190) — cacb =
const., cgB = const. (190) — flj^a (191) — Vt Riy]ia= 0 (191).
§ 2. The parameter groups and the adjoint group of a finite continuous group (191).
continuous group (191) — parameter groups (19lf.) — special anholonomic co¬
ordinate systems in general Xr (195ff.) — application to group space (196) —
structural formulae of the parameter groups after Lie and after Maurer Cartan
(196) — structural constants (196) — £, £ (197) — finite transformations of the
parameter groups (197) — adjoint group (198) — centre (198) — homologous
elements and subgroups (198) — invariant or normal subgroup (198) — linear
adjoint group (199) — normal coordinates rf (199) — series for AaB (200) — linear
transformations of the rf (200).
§ 3. Finite continuous transformation groups (201).
essential parameters (201) — connected and dependent infinitesimal transforma¬
tions (203) — theorem I. 1 (203) — structural formulae of a transformation group
after Lie and their generalization by means of the Lie derivative (205) — struc¬
tural constants (205) — theorems II. 1, III. 1, I. 2 (206) — theorem II. 2 (207) —
generators of a group germ (208) — theorem III. 2 (208).
§ 4. The geometry of group space (209).
relations between general coordinates and the normal coordinates rf (209 ff) —
series for A%, Ag, A$ and A% (211) —series for A%, rc , rc and cJJ,a (212f.) —
reflexions at the point t]a (213).
0
§ 5. Invariants of a transformation group in the Xn of the 5 (214).
invariant functions (214) — invariant subspace (215) — relative invariant sets
of functions (215) — smallest invariant subspace through a point (215) — group
induced in invariant subspace (216) —transitive (216) —simply transitive (217 f.) —
invariant contravariant vector fields (217ff.) — primitive (218).
XVI Contents.
§ 6. Invariants of a group in group space (220).
Sb Sba Scba (22°) — XPS representing subgroups (221 ff.) — the group of gb,
the centre (224) — the derived groups (224f.) — the group of gba (225) — the
group of rcidgda (225).
§ 7. Properties of integrable groups (225).
integrable groups (225) — necessary and sufficient conditions (225) — canonical
form for an integrable group (226).
§ 8. Simple and semi simple groups (227).
the group space of a semi simple group is a Vr (227).
V. Imbedding and Curvature (227).
§ 1. TheXx in Vn (22?j.
linear element (227) — first curvature (228) — curvatura geodetica (228) — cur¬
vature vector (228) — first normal (228) — osculating R2 (228) — stationary point
(228) — second curvature (229) — second normal (229) — osculating R3 (229) —
stationary osculating R2 (229) — higher curvatures (229) — osculating Rm (229) —
Frenet formulae (229 f.) — natural equations (230) — invariance of the proportions
of the curvatures (230) — contact of order u of a curve and a hypersurface (231).
§ 2. The A x in Wn and £„ (231).
X1 in Wn (231) — X1 in Ln (231) — osculating ^ direction (232) — affine para¬
meter (232) — affine length (233).
§ 3. The Xn_! in An (235).
Xn_x in An (235) — rigged Xn_1 (235) — pseudo normal vector (235) — tangent
vector (235) — first normalizing condition (235) — induced connexion (236) —
hcb (236) — principal tangents (237) — asymptotic lines (237) — directions of
principal curvature (237) — equation of Gauss for An_x in An (237) — second
condition of normalization (237) — equations of Codazzi for An_x in An (238) —
different cases of normalization in a volume preserving An (238ff.) — integrability
conditions for an An_x in En (240).
§ 4. The Vn_! in Vn (242).
Vn_1 in Vn (242) — equations of Gauss and Codazzi for Vn_x in Vn and Rn (242) —
second fundamental tensor (243) — integrability conditions for VH_X in Rn (243) —
bending (244).
§ 5. Congruences in Vn (244).
real congruences (244) — canonical congruences (245) — normal congruences
(245) — geodesic congruences (245) — orthogonal net (246)— orthogonal system
(246) — congruences belonging to an orthogonal system (246ff.) — conditions
for the principal directions of a symmetric tensor to be I^1_1 normal (247f) —
conditions for the eigenvectors of a co contravariant tensor to be Xn_1 forming
(248f.).
§ 6. Properties of curvature of a Vn— 1 in Vn (250).
absolute, relative and enforced curvatures and curvature vectors (250) — theorem
of Meusnier (250) —directions of principal curvature (250)—principal curvatures
(250) — mean curvature (250) — theorem of Dupin (250) — principal tangents
(250) — asymptotic lines (250) — geodesic Vn_x in Vn (251) — umbilical point
(251) — umbilical Vn_x (251) — Schur space (252) — Vn_x in Sn (252) — SM_t
in Sn (252) — indicatrix of Dupin (252) — theorem of Euler (253).
Contents. XVII
§ 7. The rigged X™ in Ln and Xn (253).
X™ rigged in Ln (253it) — D symbolism, D^, Dc, Dz (254) — full sets of com¬
ponents (255) — skeleton of a formula (255) — first and second curvature tensors
m m m m m m
H b , L %, H «, L;*b (256) L» (.4™) in Ln (An) (257) Z^*. Z;;a (257)
tn tn m m m
G b. ®%, ®*y. GXzy (258) ri i , r,;;;* (260) £c, 2, (260f.) Rj^*,
tnr in m m m
Ri;yx (261) — equation of Gauss for Z.™ in Ln (261) — R^y*, R^;ba (261 f.) —
equations of Ricci for Z.™ in Ln (261 f.) — equations of Codazzi for Z.™ in Ln
(262 f.) — geodesic Z.™ (263) — Z.™ (^™) rigged in Xn (263 f.) — generalization of
%c and 5)z (264).
§ 8. The rigged Xm in ^4n (265).
•^b* gged in An (265) — equations of Gauss, Codazzi and Ricci for Am in An
(266 f.) —equations for ^Min En (267)—supposition of Schlafli (268) —class of
a Vm (268).
§ 9. The V™ and Vm in Vn and Xn (269).
FnOT in Vn (269) — F^1 in A n.(27O) — Vm in Yn (271) — absolute, relative and enforced
curvature vectors (271) — principal tangent (271) — asymptotic lines (271) —
first curvature region (271) — curvature figure (Kriimmungsgebilde) (271) —
axial, planar and spatial points (272) — umbilical point (272) — umbilical Vm (272)
— mean curvature vector (272) — minimal Vm (272) — transposition of a problem
of variation (273).
§ 10. Higher curvatures of a Vm in Vn (275).
p th curvature region (275) — asymptotic Vm of order p (275) — asymptotic lines
of order p on Vm (27 5) — higher curvature tensors of valence 3 (276) — generalization
of the Frenet formulae for Vm in Vn (277) — equations of Gauss, Codazzi and
Ricci involving higher curvatures (2/Sff.) — imbedding theorems (280ff.)
Wm in Wn and Am in An (285).
§ 11. Product spaces (285).
product spaces (285) — decomposable spaces (285) — breakable and decomposable
objects (285) — covariant constant symmetric tensor fields (286).
VI. Projective and conforrnal transformations of connexions (287).
§ 1. Projective transformation of a symmetric connexion (287).
vector px of projective transformation (287) — preservation of parallelism (287) —
restricted projective transformation (288) — volume preserving connexion (288) —
projectively enclidean connexion (28S ff.) — Dn (288) — projective curvature
tensor (289) — necessary projective transformation of an En into an En (290).
§ 2. Projective transformation of the connexion in a Vn (292).
every Sn a Dn (292) — projective transformation of a Vn into a Vn (292) —¦ equations
of Lame (296).
§ 3. Imbedded spaces in An under projective transformations of the connexion (296).
induced connexion transformed projectively (297) — ^u i n ^n (297f.) — curve
in An (298) — quasi plane curve (298) — affine parameter (299) — affine normal
(299) — affine conic (299).
Schouten, Ricci Calculus, 1. Aufl. II
XVIII Contents.
§ 4. Projective connexions (300).
projective parameters of T. Y. Thomas (301 f.) of Whitehead and of Borto
lotti (302) — connexion of Cartan (302) — connexion of Konig (303) — geometry
of paths (303) — homogeneous coordinates of van Dantzig (303) — if spreads of
Douglas (303).
§ 5. Conformal transformation of a connexion in Vn (304).
geodesic and conformal properties fix a Vn (304) — conformally euclidean Vn
(305) — Cn (305) — every V2 an S2 (306) — conformal curvature tensor (3O6f.) —
orthogonal systems in Cn (307) — umbilical Vn_i s (3O8f.) — conformally geodesic
Vm s (309ff.) — theorem of Thomson and Tait (310) — null geodesic invariant
(310) — umbilical points invariant (310) — theorem of Liouville and Lie (312).
§ 6. Conformal transformations of the connexion in an Einstein space (312).
integrability conditions (313) — conformally invariant tensor density S^ (313) —
theorem of Brinkman (314).
§ 7. Conformal connexions (315).
eonformal parameters of J.M.Thomas (315) — connexions of Cartan, Konig,
T. Y. Thomas and of Schouten Haanjtes (316).
§ 8. Subprojective connexions (317).
A fold projective An (317) — subprojective An (317) — pole direction (317) —
theorem of Rachevski (317) — every subprojective connexion projectively in
variant (319) — auxiliary algebraic theorem (320) — subprojective Dn (321).
§ 9. Adah s problem (321).
torse forming fields (322) — concircular, special concircular, concurrent, recurrent,
parallel and covariant constant fields (322f.) — chief theorem of the subprojective
connexions (324) — special solutions (325 f)
§ 10. Subprojective transformations of a connexion in An (327)
subprojective transformation of a connexion belonging to a direction field (327) —
subgeodesics belonging to a direction field (327).
§ 11. The subprojective Vn (328).
every subprojective Vn, n 2, a special Cn (329) special cases (329).
§ 12. Concircular transformations of a Vn (330).
geodesic circles (330) — concircular transformations (33Off.) — relations with the
principal directions of Vn (321 f.) — umbilical Vm s (321) — necessary and sufficient
conditions for a Ffl to admit a concircular transformation (322f.) — concircular
curvature tensor (334) — concircular euclidean Vn (334).
VII. Variations and deformations (335)
§ 1. General deformation problems (335).
natural value of a field (335) — value after dragging along (335) — value after
pseudoparallel displacement (335) — operators invariant with respect to certain
objects (336) —covariant differential (336) — Lie differential £ dt (336) — appa
n
rent differential D dt (336) — identities (337) — natural variation D dt (335) —
a S
absolute variation D dt (338) — geodesic variation D dt (338) — relations between
these operators (338) a deformation problem (339) — product integrals (340ff.) —
Contents. XIX
n a
transposition of a variation problem (341) — D = D (341) — Bertrand curves
and their generalizations (342 ff.).
§ 2. Groups of motions in Vn and Ln (346).
affine, conformal, homothetic and projective motions (346) — details on affine
motions (346f.) — number of parameters of groups of motions in Vn and An (348) —
Killing s equation (348) — translation (349) — translations in Vn (349) — groups
of motions in Vn and projective mapping (349) — number of parameters of the
group of all motions in a Fn (350).
§ 3. Deformation of subspaces (352).
details on Vm in Vn under infinitesimal transformations (352ff.) — deformation
of V™ in Vn, Lm in Ln and L™ in Ln (352ff.) — method of transposition impossible
for Xm in Xn (354) — method of Nijenhuis for the rigged X% in Xn (354ff.) —
the rigged L™ in Xn (358ff.).
§ 4. The holonomy group of an Ln (361).
the holonomy group (361) — non homogeneous holonomy group (361) — the in¬
finitesimal transformations of the holonomy group deduced by means of product
integrals (362 ff.) — generators of the holonomy group (363 f.) — introduction of
an auxiliary tangent Ef (364) — extension of covariant differentiation (364 ff.) —
covariant constant fields and the holonomy groups (367) — special results for Ln
= Vn (367 f.) — the Vn with a recurrent curvature tensor (368) — product spaces
(370).
§ 5. Affine motions and the holonomy group in a symmetric An (370).
the holonomy group in a symmetric An (3 70) — equations for the affine motions
(371) — extension of the Lie operator (372) — group of isotropy (373) — the
group of affine motions is transitive (374) — number of parameters (374).
§ 6. Cartan s method applied to the holonomy group and the symmetric An (375).
introduction of local coordinate systems (375) — the allowable anholonomic co¬
ordinate system in a region (375) — the infinitesimal transformations of the holo¬
nomy group (376) — the case of the symmetric An (377) — the case of the general
An (377 ff) — group of point transformations in a symmetric An leaving the con¬
nexion invariant (381).
VIII. Miscellaneous Examples (381).
§ 1. The harmonic Vn (38!).
Vn harmonic at a point (381) — characteristic function (382) — completely har¬
monic (382) — centrally harmonic (382) — Ruse s invariant (382) — simply
harmonic (383) — equation of Lichnerowicz (384) — equations of Copson and
Ruse (385) — condition for a completely harmonic Vn to be an Einstein space,
an Sn or an Rn (385ff) — Schur space and centrally harmonic space (385) —
a completely harmonic Vn that is not an Sn (386) — the inequality of Lichnero¬
wicz (388) — imbedding of an Einstein Vn and a completely harmonic Vn in
Sn+1 (388).
§ 2. Connexions for hybrid quantities (388 ff).
analytic and semi analytic fields (389) — the new conditions for the connexion of
hybrid quantities (389) — most general form of connexion (389) — the auxiliary
X2n (390) — the two invariant sets of oc A ,, s in X2n (390) — equipollent figures
(390) — the principal Xn in X2n (390) — most general connexion in X2n (391) —
first invariant conditions: the tangent n directions are parallel (391) — second
invariant condition: the principal Xn is geodesic (391) — third invariant condi¬
tion about equipollence and parallelism (392) — the Ln (393) — the auxiliary X2H
XX Contents.
is an i2n (393) — curvature tensors in Ln (393ff.) — first and second identity
(393) — identity of Bianchi (393) — Vp^ and Vp^ (394) — identities found by
contraction (394f.) — the An (394) — necessary and sufficient conditions for a
semi analytic scalar field to be analytic (395) — necessary and sufficient conditions
for a semi analytic transformation of Xn to be analytic (395)
§ 3. Unitary connexions (395 ff)
hermitian tensor field axx (395) — unitary connexion (396) — the Un (396) —
fundamental tensor of a Un (396) — connexion in a Un derived from a connexion
in an Xin with a symmetric fundamental tensor (397f.) —the auxiliary X2n is a
U2n (397) — the Vn and its auxiliary V2n (397) — the axs in a ^derived from a
scalar (397f.) — Kahler space (397) — the curvature tensors of a Un (398ff.) —
V$fl vanishes and Vp^ is hermetian (398) — the four identities in Un (398f.) —
Bianchi s identity in Un (399) — S^ a gradient in a semi symmetric Un (399) —
Rpx hermitian in a Vn (399) — the identities in Vn (399) — list of formulae in Ln,
A Un and Vn (400 f.) — mutually perpendicular analytic fields of unitvectors in
UH and the integrability and analyticity of the connexion (402 f.) — a Vn with an
analytic connexion is a Rn (403).
§ 4. The Vn of constant curvature (404 ff.).
Vn of constant curvature (404) — Sn (405) — projective transformation of an Sn
into an Rn (405) — the linear element of Fubini and Study in Sn (406) — hermitian
non euclidean geometry (406).
§ 5. Imbedding in an Ln (407ff.).
a rigged Xm in Ln (407) — B£, C (407) — the Xm is an Lm (408) — induced con¬
nexion (408) —a rigged Xmva Ani s, ani m(4O8) — H i ; Li*x; Vc*}, (408) — geodesic
Lm in Ln (409) — two Gauss equations (410) — five Codazzi equations (410) —
two Ricci equations (410) — the Gauss, Codazzi and Ricci equations in Un
(41 If.) — imbedding in Vn (412).
§ 6. Curves in a Vn with a positive definite fundamental tensor (412ff).
the Ut in Un (412) — the dz of Coburn (412) — the R1 in XJn (414) curves with
a real parameter (415)
§ 7. Conformal transformation of a connexion in Vn (413ff).
a conformal transformation is also restricted projective transformation (415) —
a Vn determined by its conformal properties only (416) — conformal euclidean
Vn (416) — Cn (416) — conformal curvature tensor (417) — Cj ^a* (417) — con
formally symmetric Un (417)
§ 8. Conformal unitary connexions (418ff).
3(;.x (418) — conformal connexion I7*x (419) — V (419) — R^ * (419) — Rp^i*
(419).
§ 9. Spaces of recurrent curvature (421f f.).
recurrent curvature tensor (421) — Kn, K* (421) — km always a gradient (421) —
the decomposable K* (421 f.) — flat extension (422) — null extension (422) —
the three cases for the simple K* (423) — classification of simple KJ[ s (423 f•) —
linear elements of Walker (424).
Bibliography 425
Index 512
|
| any_adam_object | 1 |
| author | Schouten, Jan A. 1883-1971 |
| author_GND | (DE-588)117015547 |
| author_facet | Schouten, Jan A. 1883-1971 |
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| author_variant | j a s ja jas |
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| callnumber-raw | QA641 |
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| callnumber-subject | QA - Mathematics |
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| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T15:37:18Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001253913 |
| oclc_num | 530039 |
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| physical | XX, 516 S. graph. Darst. |
| psigel | HUB-ZB011201107 |
| publishDate | 1954 |
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| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Schouten, Jan A. 1883-1971 Verfasser (DE-588)117015547 aut Ricci-calculus an introduction to tensor analysis and its geometrical applications by J. A. Schouten 2. ed. Berlin Springer 1954 XX, 516 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 10 Differentiaalmeetkunde gtt Tensoren gtt Calculus of tensors Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Tensoranalysis (DE-588)4204323-2 gnd rswk-swf Ricci-Kalkül (DE-588)4178086-3 gnd rswk-swf Tensoranalysis (DE-588)4204323-2 s DE-604 Ricci-Kalkül (DE-588)4178086-3 s Differentialgeometrie (DE-588)4012248-7 s Grundlehren der mathematischen Wissenschaften 10 (DE-604)BV000000395 10 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001253913&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schouten, Jan A. 1883-1971 Ricci-calculus an introduction to tensor analysis and its geometrical applications Grundlehren der mathematischen Wissenschaften Differentiaalmeetkunde gtt Tensoren gtt Calculus of tensors Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd Tensoranalysis (DE-588)4204323-2 gnd Ricci-Kalkül (DE-588)4178086-3 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4204323-2 (DE-588)4178086-3 |
| title | Ricci-calculus an introduction to tensor analysis and its geometrical applications |
| title_auth | Ricci-calculus an introduction to tensor analysis and its geometrical applications |
| title_exact_search | Ricci-calculus an introduction to tensor analysis and its geometrical applications |
| title_full | Ricci-calculus an introduction to tensor analysis and its geometrical applications by J. A. Schouten |
| title_fullStr | Ricci-calculus an introduction to tensor analysis and its geometrical applications by J. A. Schouten |
| title_full_unstemmed | Ricci-calculus an introduction to tensor analysis and its geometrical applications by J. A. Schouten |
| title_short | Ricci-calculus |
| title_sort | ricci calculus an introduction to tensor analysis and its geometrical applications |
| title_sub | an introduction to tensor analysis and its geometrical applications |
| topic | Differentiaalmeetkunde gtt Tensoren gtt Calculus of tensors Geometry, Differential Differentialgeometrie (DE-588)4012248-7 gnd Tensoranalysis (DE-588)4204323-2 gnd Ricci-Kalkül (DE-588)4178086-3 gnd |
| topic_facet | Differentiaalmeetkunde Tensoren Calculus of tensors Geometry, Differential Differentialgeometrie Tensoranalysis Ricci-Kalkül |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001253913&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT schoutenjana riccicalculusanintroductiontotensoranalysisanditsgeometricalapplications |