Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

r""" 

Group of Tangent-Space Automorphism Fields 

 

Given a differentiable manifold `U` and a differentiable map 

`\Phi: U \rightarrow M` to a differentiable manifold `M` (possibly `U = M` 

and `\Phi=\mathrm{Id}_M`), the *group of tangent-space automorphism fields* 

associated with `U` and `\Phi` is the general linear group 

`\mathrm{GL}(\mathfrak{X}(U,\Phi))` of the module `\mathfrak{X}(U,\Phi)` of 

vector fields along `U` with values on `M\supset \Phi(U)` (see 

:class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldModule`). 

Note that `\mathfrak{X}(U, \Phi)` is a module over 

`C^k(U)`, the algebra of differentiable scalar fields on `U`. 

Elements of `\mathrm{GL}(\mathfrak{X}(U, \Phi))` are fields along `U` 

of automorphisms of tangent spaces to `M`. 

 

Two classes implement `\mathrm{GL}(\mathfrak{X}(U, \Phi))` depending 

whether `M` is parallelizable or not: 

:class:`AutomorphismFieldParalGroup` and :class:`AutomorphismFieldGroup`. 

 

AUTHORS: 

 

- Eric Gourgoulhon (2015): initial version 

- Travis Scrimshaw (2016): review tweaks 

 

REFERENCES: 

 

- Chap. 15 of [God1968]_ 

 

""" 

 

#****************************************************************************** 

# Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr> 

# Copyright (C) 2016 Travis Scrimshaw <tscrimsh@umn.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.structure.parent import Parent 

from sage.categories.groups import Groups 

from sage.misc.cachefunc import cached_method 

from sage.tensor.modules.free_module_linear_group import FreeModuleLinearGroup 

from sage.manifolds.differentiable.vectorfield_module import (VectorFieldModule, 

VectorFieldFreeModule) 

from sage.manifolds.differentiable.automorphismfield import (AutomorphismField, 

AutomorphismFieldParal) 

 

class AutomorphismFieldGroup(UniqueRepresentation, Parent): 

r""" 

General linear group of the module of vector fields along a differentiable 

manifold `U` with values on a differentiable manifold `M`. 

 

Given a differentiable manifold `U` and a differentiable map 

`\Phi: U \rightarrow M` to a differentiable manifold `M` (possibly `U = M` 

and `\Phi = \mathrm{Id}_M`), the *group of tangent-space automorphism 

fields* associated with `U` and `\Phi` is the general linear group 

`\mathrm{GL}(\mathfrak{X}(U,\Phi))` of the module `\mathfrak{X}(U,\Phi)` of 

vector fields along `U` with values on `M \supset \Phi(U)` (see 

:class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldModule`). 

Note that `\mathfrak{X}(U,\Phi)` is a module over 

`C^k(U)`, the algebra of differentiable scalar fields on `U`. 

Elements of `\mathrm{GL}(\mathfrak{X}(U,\Phi))` are fields along `U` of 

automorphisms of tangent spaces to `M`. 

 

.. NOTE:: 

 

If `M` is parallelizable, then :class:`AutomorphismFieldParalGroup` 

*must* be used instead. 

 

INPUT: 

 

- ``vector_field_module`` -- 

:class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldModule`; 

module `\mathfrak{X}(U,\Phi)` of vector fields along `U` with values on `M` 

 

EXAMPLES: 

 

Group of tangent-space automorphism fields of the 2-sphere:: 

 

sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 

sage: U = M.open_subset('U') # complement of the North pole 

sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole 

sage: V = M.open_subset('V') # complement of the South pole 

sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole 

sage: M.declare_union(U,V) # S^2 is the union of U and V 

sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), 

....: intersection_name='W', 

....: restrictions1= x^2+y^2!=0, restrictions2= u^2+v^2!=0) 

sage: uv_to_xy = xy_to_uv.inverse() 

sage: G = M.automorphism_field_group() ; G 

General linear group of the Module X(M) of vector fields on the 

2-dimensional differentiable manifold M 

 

``G`` is the general linear group of the vector field module 

`\mathfrak{X}(M)`:: 

 

sage: XM = M.vector_field_module() ; XM 

Module X(M) of vector fields on the 2-dimensional differentiable 

manifold M 

sage: G is XM.general_linear_group() 

True 

 

``G`` is a non-abelian group:: 

 

sage: G.category() 

Category of groups 

sage: G in Groups() 

True 

sage: G in CommutativeAdditiveGroups() 

False 

 

The elements of ``G`` are tangent-space automorphisms:: 

 

sage: a = G.an_element(); a 

Field of tangent-space automorphisms on the 2-dimensional 

differentiable manifold M 

sage: a.parent() is G 

True 

sage: a.restrict(U).display() 

2 d/dx*dx + 2 d/dy*dy 

sage: a.restrict(V).display() 

2 d/du*du + 2 d/dv*dv 

 

The identity element of the group ``G``:: 

 

sage: e = G.one() ; e 

Field of tangent-space identity maps on the 2-dimensional 

differentiable manifold M 

sage: eU = U.default_frame() ; eU 

Coordinate frame (U, (d/dx,d/dy)) 

sage: eV = V.default_frame() ; eV 

Coordinate frame (V, (d/du,d/dv)) 

sage: e.display(eU) 

Id = d/dx*dx + d/dy*dy 

sage: e.display(eV) 

Id = d/du*du + d/dv*dv 

 

""" 

 

Element = AutomorphismField 

 

def __init__(self, vector_field_module): 

r""" 

See :class:`AutomorphismfieldGroup` for documentation and examples. 

 

TESTS:: 

 

sage: M = Manifold(2, 'M') 

sage: U = M.open_subset('U') ; V = M.open_subset('V') 

sage: M.declare_union(U,V) # M is the union of U and V 

sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart() 

sage: transf = c_xy.transition_map(c_uv, (x+y, x-y), 

....: intersection_name='W', restrictions1= x>0, 

....: restrictions2= u+v>0) 

sage: inv = transf.inverse() 

sage: from sage.manifolds.differentiable.automorphismfield_group \ 

....: import AutomorphismFieldGroup 

sage: G = AutomorphismFieldGroup(M.vector_field_module()) ; G 

General linear group of the Module X(M) of vector fields on the 

2-dimensional differentiable manifold M 

sage: TestSuite(G).run(skip='_test_elements') 

 

``_test_elements`` does not pass due to the failure 

of ``_test_pickling`` in 

:class:`sage.manifolds.differentiable.tensorfield.TensorField`. 

 

""" 

if not isinstance(vector_field_module, VectorFieldModule): 

raise TypeError("{} is not a module of vector fields".format( 

vector_field_module)) 

Parent.__init__(self, category=Groups()) 

self._vmodule = vector_field_module 

 

 

#### Parent methods #### 

 

def _element_constructor_(self, comp=[], frame=None, name=None, 

latex_name=None): 

r""" 

Construct a field of tangent-space automorphisms. 

 

OUTPUT: 

 

- :class:`~sage.manifolds.differentiable.automorphismfield.AutomorphismField` 

 

TESTS:: 

 

sage: M = Manifold(2, 'M') 

sage: U = M.open_subset('U'); V = M.open_subset('V') 

sage: M.declare_union(U,V) # M is the union of U and V 

sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart() 

sage: transf = c_xy.transition_map(c_uv, (x+y, x-y), 

....: intersection_name='W', restrictions1= x>0, 

....: restrictions2= u+v>0) 

sage: inv = transf.inverse() 

sage: G = M.automorphism_field_group() 

sage: a = G(1); a 

Field of tangent-space identity maps on the 2-dimensional 

differentiable manifold M 

sage: a = G([[1+x^2, 0], [0, 1+y^2]], frame=c_xy.frame(), name='a'); a 

Field of tangent-space automorphisms a on the 2-dimensional 

differentiable manifold M 

sage: a.display(c_xy.frame()) 

a = (x^2 + 1) d/dx*dx + (y^2 + 1) d/dy*dy 

 

""" 

if comp == 1: 

return self.one() 

# standard construction 

resu = self.element_class(self._vmodule, name=name, 

latex_name=latex_name) 

if comp != []: 

resu.set_comp(frame)[:] = comp 

return resu 

 

def _an_element_(self): 

r""" 

Construct some specific field of tangent-space automorphisms. 

 

OUTPUT: 

 

- :class:`~sage.manifolds.differentiable.automorphismfield.AutomorphismField` 

 

TESTS:: 

 

sage: M = Manifold(2, 'M') 

sage: U = M.open_subset('U') ; V = M.open_subset('V') 

sage: M.declare_union(U,V) # M is the union of U and V 

sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart() 

sage: transf = c_xy.transition_map(c_uv, (x+y, x-y), 

....: intersection_name='W', restrictions1= x>0, 

....: restrictions2= u+v>0) 

sage: inv = transf.inverse() 

sage: G = M.automorphism_field_group() 

sage: a = G.an_element() ; a 

Field of tangent-space automorphisms on the 2-dimensional 

differentiable manifold M 

sage: a.restrict(U).display() 

2 d/dx*dx + 2 d/dy*dy 

sage: a.restrict(V).display() 

2 d/du*du + 2 d/dv*dv 

sage: a == G.an_element() # indirect doctest 

True 

 

""" 

resu = self.element_class(self._vmodule) 

for dom in resu.domain().subsets(): 

if dom.is_manifestly_parallelizable(): 

fmodule = dom.vector_field_module() 

idm = fmodule.identity_map() 

rst = fmodule.automorphism() 

for frame, comp in idm._components.items(): 

rst._components[frame] = 2 * comp 

resu._restrictions[dom] = rst 

return resu 

 

#### End of parent methods #### 

 

 

#### Monoid methods #### 

 

@cached_method 

def one(self): 

r""" 

Return identity element of ``self``. 

 

The group identity element is the field of tangent-space identity maps. 

 

OUTPUT: 

 

- :class:`~sage.manifolds.differentiable.automorphismfield.AutomorphismField` 

representing the identity element 

 

EXAMPLES: 

 

Identity element of the group of tangent-space automorphism fields of 

the 2-sphere:: 

 

sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 

sage: U = M.open_subset('U') # complement of the North pole 

sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole 

sage: V = M.open_subset('V') # complement of the South pole 

sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole 

sage: M.declare_union(U,V) # S^2 is the union of U and V 

sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), 

....: intersection_name='W', restrictions1= x^2+y^2!=0, 

....: restrictions2= u^2+v^2!=0) 

sage: uv_to_xy = xy_to_uv.inverse() 

sage: G = M.automorphism_field_group() 

sage: G.one() 

Field of tangent-space identity maps on the 2-dimensional differentiable manifold M 

sage: G.one().restrict(U)[:] 

[1 0] 

[0 1] 

sage: G.one().restrict(V)[:] 

[1 0] 

[0 1] 

 

""" 

return self.element_class(self._vmodule, is_identity=True) 

 

#### End of monoid methods #### 

 

def _repr_(self): 

r""" 

Return a string representation of ``self``. 

 

TESTS:: 

 

sage: M = Manifold(2, 'M') 

sage: G = M.automorphism_field_group() 

sage: G._repr_() 

'General linear group of the Module X(M) of vector fields on the 

2-dimensional differentiable manifold M' 

sage: repr(G) # indirect doctest 

'General linear group of the Module X(M) of vector fields on the 

2-dimensional differentiable manifold M' 

sage: G # indirect doctest 

General linear group of the Module X(M) of vector fields on the 

2-dimensional differentiable manifold M 

 

""" 

return "General linear group of the {}".format(self._vmodule) 

 

def _latex_(self): 

r""" 

Return a LaTeX representation of ``self``. 

 

TESTS:: 

 

sage: M = Manifold(2, 'M') 

sage: G = M.automorphism_field_group() 

sage: G._latex_() 

\mathrm{GL}\left( \mathfrak{X}\left(M\right) \right) 

sage: latex(G) # indirect doctest 

\mathrm{GL}\left( \mathfrak{X}\left(M\right) \right) 

 

""" 

from sage.misc.latex import latex 

return r"\mathrm{GL}\left("+ latex(self._vmodule)+ r"\right)" 

 

def base_module(self): 

r""" 

Return the vector-field module of which ``self`` is the general 

linear group. 

 

OUTPUT: 

 

- :class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldModule` 

 

EXAMPLES: 

 

Base module of the group of tangent-space automorphism fields of 

the 2-sphere:: 

 

sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 

sage: U = M.open_subset('U') # complement of the North pole 

sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole 

sage: V = M.open_subset('V') # complement of the South pole 

sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole 

sage: M.declare_union(U,V) # S^2 is the union of U and V 

sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), 

....: intersection_name='W', restrictions1= x^2+y^2!=0, 

....: restrictions2= u^2+v^2!=0) 

sage: uv_to_xy = xy_to_uv.inverse() 

sage: G = M.automorphism_field_group() 

sage: G.base_module() 

Module X(M) of vector fields on the 2-dimensional differentiable 

manifold M 

sage: G.base_module() is M.vector_field_module() 

True 

 

""" 

return self._vmodule 

 

 

#****************************************************************************** 

 

class AutomorphismFieldParalGroup(FreeModuleLinearGroup): 

r""" 

General linear group of the module of vector fields along a differentiable 

manifold `U` with values on a parallelizable manifold `M`. 

 

Given a differentiable manifold `U` and a differentiable map 

`\Phi: U \rightarrow M` to a parallelizable manifold `M` (possibly `U = M` 

and `\Phi = \mathrm{Id}_M`), the *group of tangent-space automorphism 

fields* associated with `U` and `\Phi` is the general linear group 

`\mathrm{GL}(\mathfrak{X}(U, \Phi))` of the module `\mathfrak{X}(U, \Phi)` 

of vector fields along `U` with values on `M \supset \Phi(U)` (see 

:class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldFreeModule`). 

Note that `\mathfrak{X}(U, \Phi)` is a free module over `C^k(U)`, 

the algebra of differentiable scalar fields on `U`. 

Elements of `\mathrm{GL}(\mathfrak{X}(U, \Phi))` are fields along `U` of 

automorphisms of tangent spaces to `M`. 

 

.. NOTE:: 

 

If `M` is not parallelizable, the class 

:class:`AutomorphismFieldGroup` must be used instead. 

 

INPUT: 

 

- ``vector_field_module`` -- 

:class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldFreeModule`; 

free module `\mathfrak{X}(U,\Phi)` of vector fields along `U` 

with values on `M` 

 

EXAMPLES: 

 

Group of tangent-space automorphism fields of a 2-dimensional 

parallelizable manifold:: 

 

sage: M = Manifold(2, 'M') 

sage: X.<x,y> = M.chart() 

sage: XM = M.vector_field_module() ; XM 

Free module X(M) of vector fields on the 2-dimensional differentiable 

manifold M 

sage: G = M.automorphism_field_group(); G 

General linear group of the Free module X(M) of vector fields on the 

2-dimensional differentiable manifold M 

sage: latex(G) 

\mathrm{GL}\left( \mathfrak{X}\left(M\right) \right) 

 

``G`` is nothing but the general linear group of the module 

`\mathfrak{X}(M)`:: 

 

sage: G is XM.general_linear_group() 

True 

 

``G`` is a group:: 

 

sage: G.category() 

Category of groups 

sage: G in Groups() 

True 

 

It is not an abelian group:: 

 

sage: G in CommutativeAdditiveGroups() 

False 

 

The elements of ``G`` are tangent-space automorphisms:: 

 

sage: G.Element 

<class 'sage.manifolds.differentiable.automorphismfield.AutomorphismFieldParal'> 

sage: a = G.an_element() ; a 

Field of tangent-space automorphisms on the 2-dimensional 

differentiable manifold M 

sage: a.parent() is G 

True 

 

As automorphisms of `\mathfrak{X}(M)`, the elements of ``G`` map a vector 

field to a vector field:: 

 

sage: v = XM.an_element() ; v 

Vector field on the 2-dimensional differentiable manifold M 

sage: v.display() 

2 d/dx + 2 d/dy 

sage: a(v) 

Vector field on the 2-dimensional differentiable manifold M 

sage: a(v).display() 

2 d/dx - 2 d/dy 

 

Indeed the matrix of ``a`` with respect to the frame 

`(\partial_x, \partial_y)` is:: 

 

sage: a[X.frame(),:] 

[ 1 0] 

[ 0 -1] 

 

The elements of ``G`` can also be considered as tensor fields of 

type `(1,1)`:: 

 

sage: a.tensor_type() 

(1, 1) 

sage: a.tensor_rank() 

2 

sage: a.domain() 

2-dimensional differentiable manifold M 

sage: a.display() 

d/dx*dx - d/dy*dy 

 

The identity element of the group ``G`` is:: 

 

sage: id = G.one() ; id 

Field of tangent-space identity maps on the 2-dimensional 

differentiable manifold M 

sage: id*a == a 

True 

sage: a*id == a 

True 

sage: a*a^(-1) == id 

True 

sage: a^(-1)*a == id 

True 

 

Construction of an element by providing its components with 

respect to the manifold's default frame (frame associated to 

the coordinates `(x,y)`):: 

 

sage: b = G([[1+x^2,0], [0,1+y^2]]) ; b 

Field of tangent-space automorphisms on the 2-dimensional 

differentiable manifold M 

sage: b.display() 

(x^2 + 1) d/dx*dx + (y^2 + 1) d/dy*dy 

sage: (~b).display() # the inverse automorphism 

1/(x^2 + 1) d/dx*dx + 1/(y^2 + 1) d/dy*dy 

 

We check the group law on these elements:: 

 

sage: (a*b)^(-1) == b^(-1) * a^(-1) 

True 

 

Invertible tensor fields of type `(1,1)` can be converted to 

elements of ``G``:: 

 

sage: t = M.tensor_field(1, 1, name='t') 

sage: t[:] = [[1+exp(y), x*y], [0, 1+x^2]] 

sage: t1 = G(t) ; t1 

Field of tangent-space automorphisms t on the 2-dimensional 

differentiable manifold M 

sage: t1 in G 

True 

sage: t1.display() 

t = (e^y + 1) d/dx*dx + x*y d/dx*dy + (x^2 + 1) d/dy*dy 

sage: t1^(-1) 

Field of tangent-space automorphisms t^(-1) on the 2-dimensional 

differentiable manifold M 

sage: (t1^(-1)).display() 

t^(-1) = 1/(e^y + 1) d/dx*dx - x*y/(x^2 + (x^2 + 1)*e^y + 1) d/dx*dy 

+ 1/(x^2 + 1) d/dy*dy 

 

Since any automorphism field can be considered as a tensor field of 

type-`(1,1)` on ``M``, there is a coercion map from ``G`` to the 

module `T^{(1,1)}(M)` of type-`(1,1)` tensor fields:: 

 

sage: T11 = M.tensor_field_module((1,1)) ; T11 

Free module T^(1,1)(M) of type-(1,1) tensors fields on the 

2-dimensional differentiable manifold M 

sage: T11.has_coerce_map_from(G) 

True 

 

An explicit call of this coercion map is:: 

 

sage: tt = T11(t1) ; tt 

Tensor field t of type (1,1) on the 2-dimensional differentiable 

manifold M 

sage: tt == t 

True 

 

An implicit call of the coercion map is performed to subtract an 

element of ``G`` from an element of `T^{(1,1)}(M)`:: 

 

sage: s = t - t1 ; s 

Tensor field t-t of type (1,1) on 

the 2-dimensional differentiable manifold M 

sage: s.parent() is T11 

True 

sage: s.display() 

t-t = 0 

 

as well as for the reverse operation:: 

 

sage: s = t1 - t ; s 

Tensor field t-t of type (1,1) on the 2-dimensional differentiable 

manifold M 

sage: s.display() 

t-t = 0 

 

TESTS:: 

 

sage: TestSuite(G).run() 

 

""" 

 

Element = AutomorphismFieldParal 

 

def __init__(self, vector_field_module): 

r""" 

See :class:`AutomorphismfieldParalGroup` for documentation and 

examples. 

 

TESTS:: 

 

sage: M = Manifold(2, 'M') ; M 

2-dimensional differentiable manifold M 

sage: X.<x,y> = M.chart() # makes M parallelizable 

sage: G = M.automorphism_field_group(); G 

General linear group of the Free module X(M) of vector fields on 

the 2-dimensional differentiable manifold M 

sage: TestSuite(G).run() 

 

""" 

if not isinstance(vector_field_module, VectorFieldFreeModule): 

raise TypeError("{} is not a free module of vector fields".format( 

vector_field_module)) 

FreeModuleLinearGroup.__init__(self, vector_field_module)