5391:
6402:
5156:
4364:, whereas the Laplace–Beltrami operator is typically negative. The sign is merely a convention, and both are common in the literature. The Laplace–de Rham operator differs more significantly from the tensor Laplacian restricted to act on skew-symmetric tensors. Apart from the incidental sign, the two operators differ by a
6165:
1658:
4355:
Up to an overall sign, the
Laplace–de Rham operator is equivalent to the previous definition of the Laplace–Beltrami operator when acting on a scalar function; see the proof for details. On functions, the Laplace–de Rham operator is actually the negative of the Laplace–Beltrami operator, as the
5202:
4900:
3622:
6207:
4958:
1422:
5916:
4550:
3355:
1132:
4118:
1310:
5967:
3719:
697:
4237:
3424:
976:
3824:
311:
5735:
1533:
575:
5386:{\displaystyle \Delta _{S^{2}}f(\theta ,\phi )=(\sin \phi )^{-1}{\frac {\partial }{\partial \phi }}\left(\sin \phi {\frac {\partial f}{\partial \phi }}\right)+(\sin \phi )^{-2}{\frac {\partial ^{2}}{\partial \theta ^{2}}}f}
3152:
841:
2437:
2287:
2187:
4757:
2061:
3191:. Proofs of all these statements may be found in the book by Isaac Chavel. Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the
408:
3912:
5510:
6397:{\displaystyle \Delta _{H^{2}}f(r,\theta )=\sinh(r)^{-1}{\frac {\partial }{\partial r}}\left(\sinh(r){\frac {\partial f}{\partial r}}\right)+\sinh(r)^{-2}{\frac {\partial ^{2}}{\partial \theta ^{2}}}f}
5151:{\displaystyle \Delta _{S^{n-1}}f(\xi ,\phi )=(\sin \phi )^{2-n}{\frac {\partial }{\partial \phi }}\left((\sin \phi )^{n-2}{\frac {\partial f}{\partial \phi }}\right)+(\sin \phi )^{-2}\Delta _{\xi }f}
2599:
3462:
1149:
rather than a form). Neither the gradient nor the divergence actually depends on the choice of orientation, and so the
Laplace–Beltrami operator itself does not depend on this additional structure.
2988:
4706:
3082:
5603:
109:
taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on
4350:
3971:
1331:
192:
3189:
5818:
750:
4441:
1972:
3280:
2917:
4292:
1710:
1023:
3997:
6160:{\displaystyle \Delta _{H^{n-1}}f(t,\xi )=\sinh(t)^{2-n}{\frac {\partial }{\partial t}}\left(\sinh(t)^{n-2}{\frac {\partial f}{\partial t}}\right)+\sinh(t)^{-2}\Delta _{\xi }f}
3233:
2740:
2859:
2798:
2769:
2660:
2631:
2475:
5799:
2830:
1225:
6195:
5186:
3651:
2316:
1896:
507:
3025:
2687:
2542:
1847:
604:
1870:
1817:
1777:
1757:
1198:
4749:
2353:
454:
4751:
is the
Laplacian of the ambient Euclidean space. Concretely, this is implied by the well-known formula for the Euclidean Laplacian in spherical polar coordinates:
4433:
4169:
3360:
1653:{\displaystyle \int _{M}f\,\Delta h\operatorname {vol} _{n}=-\int _{M}\langle df,dh\rangle \operatorname {vol} _{n}=\int _{M}h\,\Delta f\operatorname {vol} _{n}.}
893:
434:
3731:
3272:
2515:
2495:
2088:
1936:
1916:
1797:
1733:
1525:
1505:
1478:
1458:
1218:
1175:
1017:
Combining the definitions of the gradient and divergence, the formula for the
Laplace–Beltrami operator applied to a scalar function ƒ is, in local coordinates
481:
233:
5614:
515:
3094:
3192:
6560:
758:
4605:-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded into
1663:
Because the
Laplace–Beltrami operator, as defined in this manner, is negative rather than positive, often it is defined with the opposite sign.
2365:
4895:{\displaystyle \Delta f=r^{1-n}{\frac {\partial }{\partial r}}\left(r^{n-1}{\frac {\partial f}{\partial r}}\right)+r^{-2}\Delta _{S^{n-1}}f.}
2198:
1799:
allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue
1141:
is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by a
6574:
McIntosh, Alan; Monniaux, Sylvie (2018). "Hodge–Dirac, Hodge–Laplacian and Hodge–Stokes operators in $ L^p$ spaces on
Lipschitz domains".
6418:
2096:
1980:
361:
1487:
As a consequence, the
Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions
3836:
6515:
Chanillo, Sagun, Chiu, Hung-Lin and Yang, Paul C. (2012). "Embeddability for 3-dimensional CR manifolds and CR Yamabe
Invariants".
5423:
3617:{\displaystyle ({\mbox{Hess}}f)_{ij}={\mbox{Hess}}f(X_{i},X_{j})=\nabla _{X_{i}}\nabla _{X_{j}}f-\nabla _{\nabla _{X_{i}}X_{j}}f}
4909:
to define the
Laplace–Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space.
2550:
6632:
6500:
6201: − 2)-sphere. In particular, for the hyperbolic plane using standard notation for polar coordinates we get:
2928:
4623:
3033:
5525:
1484:) characterizes the Laplace–Beltrami operator completely, in the sense that it is the only operator with this property.
1417:{\displaystyle \int _{M}f\,\Delta h\,\operatorname {vol} _{n}=-\int _{M}\langle df,dh\rangle \,\operatorname {vol} _{n}}
3251:
5911:{\displaystyle \Box ={\frac {\partial ^{2}}{\partial x_{1}^{2}}}-\cdots -{\frac {\partial ^{2}}{\partial x_{n}^{2}}}.}
4314:
3927:
6653:
4545:{\displaystyle \Delta f={\frac {1}{\sqrt {|g|}}}\partial _{i}{\sqrt {|g|}}\partial ^{i}f=\partial _{i}\partial ^{i}f}
143:
3157:
723:
6690:
1941:
3350:{\displaystyle \displaystyle {\mbox{Hess}}f\in \mathbf {\Gamma } ({\mathsf {T}}^{*}M\otimes {\mathsf {T}}^{*}M)}
6664:
6674:
3452:
be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the components of
2867:
1127:{\displaystyle \Delta f={\frac {1}{\sqrt {|g|}}}\partial _{i}\left({\sqrt {|g|}}g^{ij}\partial _{j}f\right).}
6695:
4113:{\displaystyle \Delta T=g^{ij}\left(\nabla _{X_{i}}\nabla _{X_{j}}T-\nabla _{\nabla _{X_{i}}X_{j}}T\right)}
3000:-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue
105:. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a
4269:
1677:
1671:
Let M denote a compact
Riemannian manifold without boundary. We want to consider the eigenvalue equation,
6669:
5196:-sphere. In particular, for the ordinary 2-sphere using standard notation for polar coordinates we get:
1305:{\displaystyle \int _{M}df(X)\operatorname {vol} _{n}=-\int _{M}f\nabla \cdot X\operatorname {vol} _{n}}
4140:
3714:{\displaystyle \displaystyle \Delta f:=\mathrm {tr} \nabla \mathrm {d} f\in {\mathsf {C}}^{\infty }(M)}
3206:
2692:
51:
2835:
2774:
2745:
2636:
2607:
2445:
5743:
2803:
6173:
5164:
3435:
2295:
1875:
486:
692:{\displaystyle \nabla \cdot X={\frac {1}{\sqrt {|g|}}}\partial _{i}\left({\sqrt {|g|}}X^{i}\right)}
3003:
2665:
2520:
4564:
4556:
66:
4365:
1826:
4912:
One can also give an intrinsic description of the Laplace–Beltrami operator on the sphere in a
4369:
1823:
are all finite-dimensional. Notice by taking the constant function as an eigenfunction, we get
1852:
1802:
1762:
1742:
1180:
6554:
4734:
4560:
4389:
4232:{\displaystyle \Delta =\mathrm {d} \delta +\delta \mathrm {d} =(\mathrm {d} +\delta )^{2},\;}
4132:
3419:{\displaystyle {\mbox{Hess}}f:=\nabla ^{2}f\equiv \nabla \nabla f\equiv \nabla \mathrm {d} f}
2332:
1146:
439:
27:
4568:
2322:
1666:
6413:
4402:
3247:
971:{\displaystyle \left(\operatorname {grad} f\right)^{i}=\partial ^{i}f=g^{ij}\partial _{j}f}
55:
3819:{\displaystyle \displaystyle \Delta f(x)=\sum _{i=1}^{n}\nabla \mathrm {d} f(X_{i},X_{i})}
3627:
This is easily seen to transform tensorially, since it is linear in each of the arguments
306:{\displaystyle \operatorname {vol} _{n}:={\sqrt {|g|}}\;dx^{1}\wedge \cdots \wedge dx^{n}}
8:
5730:{\displaystyle \Delta _{H^{n-1}}f=\left.\Box f\left(x/q(x)^{1/2}\right)\right|_{H^{n-1}}}
4243:
4156:
4152:
4144:
3976:
provided it is understood implicitly that this trace is in fact the trace of the Hessian
877:
212:
114:
47:
416:
6601:
6583:
6542:
6524:
4913:
3642:
3257:
3243:
2993:
Thus the lower bound in Lichnerowicz's theorem is achieved at least in two dimensions.
2500:
2480:
2073:
2067:
1921:
1901:
1782:
1718:
1510:
1490:
1463:
1443:
1320:
1203:
1160:
570:{\displaystyle (\nabla \cdot X)\operatorname {vol} _{n}:=L_{X}\operatorname {vol} _{n}}
466:
6649:
6628:
6605:
6496:
4932:
of the sphere (the "north pole"), that is geodesic polar coordinates with respect to
4361:
4148:
4136:
1898:. More precisely if we multiply the eigenvalue equation through by the eigenfunction
703:
209:
198:
110:
20:
6593:
6534:
6471:
6423:
5402:
4586:
4579:
4575:
3918:
59:
35:
6546:
5942:
6514:
5414:
4396:
2356:
1011:
118:
106:
43:
3147:{\displaystyle \mathbb {S} ^{n}{\bigg (}{\sqrt {\frac {n-1}{\kappa }}}{\bigg )}}
2922:
we see easily from the formula for the spherical Laplacian displayed below that
2689:
is three dimensional and spanned by the restriction of the coordinate functions
4357:
4251:
3196:
1759:. It can be shown using the self-adjointness proved above that the eigenvalues
1142:
588:
332:
6641:
6495:, Pure and Applied Mathematics, vol. 115 (2nd ed.), Academic Press,
6684:
6538:
5810:
4906:
4380:
Many examples of the Laplace–Beltrami operator can be worked out explicitly.
1736:
986:
858:
718:
457:
340:
117:. The resulting operator is called the Laplace–de Rham operator (named after
6476:
6460:"Certain conditions for a Riemannian manifold to be isometric with a sphere"
6459:
836:{\displaystyle \langle \operatorname {grad} f(x),v_{x}\rangle =df(x)(v_{x})}
592:
5933:
be spherical coordinates on the sphere with respect to a particular point
4928:
be spherical coordinates on the sphere with respect to a particular point
2066:
Performing an integration by parts or what is the same thing as using the
1200:
are formal adjoints, in the sense that for a compactly supported function
3203:. Applications there are to the global embedding of such CR manifolds in
3200:
2432:{\displaystyle \operatorname {Ric} (X,X)\geq \kappa g(X,X),\kappa >0,}
1667:
Eigenvalues of the Laplace–Beltrami operator (Lichnerowicz–Obata theorem)
336:
216:
39:
6597:
4263:
1820:
130:
129:
The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian)
86:
4399:, the metric is reduced to the Kronecker delta, and one therefore has
2282:{\displaystyle \int _{M}|\nabla u|^{2}\ dV=\lambda \int _{M}u^{2}\ dV}
4940:
represents the latitude measurement along a unit speed geodesic from
2182:{\displaystyle -\int _{M}\Delta u\ u\ dV=\int _{M}|\nabla u|^{2}\ dV}
5957:
a parameter representing the choice of direction of the geodesic in
4948:
a parameter representing the choice of direction of the geodesic in
2056:{\displaystyle -\int _{M}\Delta u\ u\ dV=\lambda \int _{M}u^{2}\ dV}
6588:
134:
90:
6529:
5519:
is the subset of the future null cone in Minkowski space given by
3983:
Because the covariant derivative extends canonically to arbitrary
403:{\displaystyle \partial _{i}:={\frac {\partial }{\partial x^{i}}}}
4297:
When computing the Laplace–de Rham operator on a scalar function
352:
4597:
The spherical Laplacian is the Laplace–Beltrami operator on the
4609:
as the unit sphere centred at the origin. Then for a function
3984:
3907:{\displaystyle \Delta f=\sum _{ij}g^{ij}({\mbox{Hess}}f)_{ij}.}
348:
4574:
Similarly, the Laplace–Beltrami operator corresponding to the
6625:
Differential forms with applications to the physical sciences
6197:
is the Laplace–Beltrami operator on the ordinary unit (
713:
The gradient of a scalar function ƒ is the vector field grad
5921:
The operator can also be written in polar coordinates. Let
5188:
is the Laplace–Beltrami operator on the ordinary unit
4905:
More generally, one can formulate a similar trick using the
4723:|) is the degree zero homogeneous extension of the function
16:
Operator generalizing the Laplacian in differential geometry
5649:
5505:{\displaystyle q(x)=x_{1}^{2}-x_{2}^{2}-\cdots -x_{n}^{2}.}
2594:{\displaystyle \lambda _{1}\geq {\frac {n}{n-1}}\kappa .}
4360:
assures that the Laplace–de Rham operator is (formally)
101:
with respect to each vector of an orthonormal basis for
5417:, a real vector space equipped with the quadratic form
3641:. The Laplace–Beltrami operator is then the trace (or
3242:
The Laplace–Beltrami operator can be written using the
2983:{\displaystyle -\Delta _{\mathbb {S} ^{2}}u_{1}=2u_{1}}
2329:-dimensional Riemannian manifold with no boundary with
483:
on the manifold is then defined as the scalar function
3879:
3496:
3470:
3365:
3286:
2544:
of the eigenvalue equation satisfies the lower bound:
6210:
6176:
5970:
5821:
5746:
5617:
5528:
5426:
5205:
5167:
4961:
4760:
4737:
4701:{\displaystyle \Delta _{S^{n-1}}f(x)=\Delta f(x/|x|)}
4626:
4444:
4405:
4317:
4272:
4172:
4000:
3930:
3839:
3735:
3734:
3655:
3654:
3465:
3363:
3284:
3283:
3260:
3209:
3160:
3097:
3077:{\displaystyle \lambda _{1}={\frac {n}{n-1}}\kappa ,}
3036:
3006:
2996:
Conversely it was proved by Morio Obata, that if the
2931:
2870:
2838:
2806:
2777:
2748:
2695:
2668:
2639:
2610:
2604:
This lower bound is sharp and achieved on the sphere
2553:
2523:
2503:
2483:
2448:
2368:
2335:
2298:
2201:
2192:
Putting the last two equations together we arrive at
2099:
2076:
1983:
1944:
1924:
1904:
1878:
1855:
1829:
1805:
1785:
1765:
1745:
1721:
1680:
1536:
1513:
1493:
1466:
1446:
1334:
1228:
1206:
1183:
1163:
1026:
896:
761:
726:
607:
518:
489:
469:
442:
419:
364:
236:
146:
5598:{\displaystyle H^{n}=\{x\mid q(x)=1,x_{1}>1\}.\,}
3987:, the Laplace–Beltrami operator defined on a tensor
6442:
215:. The orientation allows one to specify a definite
6396:
6189:
6159:
5910:
5793:
5729:
5597:
5504:
5385:
5180:
5150:
4894:
4743:
4700:
4544:
4427:
4344:
4286:
4231:
4112:
3965:
3906:
3818:
3713:
3616:
3418:
3349:
3266:
3227:
3183:
3146:
3076:
3019:
2982:
2911:
2853:
2824:
2792:
2763:
2734:
2681:
2654:
2625:
2593:
2536:
2509:
2489:
2469:
2431:
2347:
2310:
2281:
2181:
2082:
2055:
1966:
1930:
1910:
1890:
1864:
1841:
1811:
1791:
1771:
1751:
1727:
1704:
1652:
1519:
1499:
1472:
1452:
1416:
1304:
1212:
1192:
1169:
1126:
970:
880:of the function ƒ; it is a 1-form taking argument
835:
744:
691:
569:
501:
475:
448:
428:
402:
305:
186:
6662:
3139:
3112:
6682:
6573:
4345:{\displaystyle \Delta f=\delta \,\mathrm {d} f.}
3966:{\displaystyle \Delta f=\nabla ^{a}\nabla _{a}f}
3250:associated with the Levi-Civita connection. The
1849:is an eigenvalue. Also since we have considered
187:{\displaystyle \Delta f={\rm {div}}(\nabla f).}
5961:. Then the hyperbolic Laplacian has the form:
3184:{\displaystyle {\sqrt {\frac {n-1}{\kappa }}}}
4952:. Then the spherical Laplacian has the form:
3645:) of the Hessian with respect to the metric:
1319:where the last equality is an application of
745:{\displaystyle \langle \cdot ,\cdot \rangle }
6559:: CS1 maint: multiple names: authors list (
6457:
6419:Laplacian operators in differential geometry
5805:to the interior of the future null cone and
5801:is the degree zero homogeneous extension of
5588:
5542:
1601:
1583:
1400:
1382:
796:
762:
739:
727:
4131:More generally, one can define a Laplacian
4126:
1967:{\displaystyle dV=\operatorname {vol} _{n}}
6646:Riemannian Geometry and Geometric Analysis
4228:
1779:are real. The compactness of the manifold
1152:
267:
77:, the Laplace operator (also known as the
6663:Solomentsev, E.D.; Shikin, E.V. (2001) ,
6587:
6528:
6475:
5594:
4330:
3212:
3100:
2942:
2841:
2780:
2751:
2642:
2613:
1630:
1550:
1403:
1355:
1348:
985:are the components of the inverse of the
223:, given in an oriented coordinate system
6622:
6445:Geometrie des groupes de transformations
5949:represents the hyperbolic distance from
4617:, the spherical Laplacian is defined by
2292:We conclude from the last equation that
1918:and integrate the resulting equation on
2912:{\displaystyle x_{3}=\cos \phi =u_{1},}
706:is implied, so that the repeated index
6683:
6490:
4592:
3690:
3329:
3309:
3087:then the manifold is isometric to the
2497:is any tangent vector on the manifold
1440:for all compactly supported functions
93:vector field, which is the sum of the
2517:. Then the first positive eigenvalue
6640:
4555:which is the ordinary Laplacian. In
4287:{\displaystyle \mathrm {d} +\delta }
1705:{\displaystyle -\Delta u=\lambda u,}
1325:
598:. In local coordinates, one obtains
5396:
3237:
2070:on the term on the left, and since
1872:an integration by parts shows that
13:
6493:Eigenvalues in Riemannian Geometry
6375:
6365:
6320:
6312:
6280:
6276:
6212:
6178:
6145:
6102:
6094:
6049:
6045:
5972:
5884:
5874:
5841:
5831:
5619:
5364:
5354:
5309:
5301:
5275:
5271:
5207:
5169:
5136:
5093:
5085:
5040:
5036:
4963:
4864:
4833:
4825:
4792:
4788:
4761:
4738:
4665:
4628:
4530:
4520:
4504:
4477:
4445:
4383:
4356:conventional normalization of the
4332:
4318:
4274:
4205:
4194:
4180:
4173:
4074:
4069:
4046:
4029:
4001:
3951:
3941:
3931:
3840:
3779:
3775:
3736:
3696:
3677:
3673:
3669:
3666:
3656:
3583:
3578:
3555:
3538:
3409:
3405:
3396:
3393:
3378:
2936:
2217:
2152:
2113:
1997:
1859:
1684:
1631:
1551:
1349:
1283:
1187:
1104:
1059:
1027:
956:
927:
643:
608:
522:
490:
384:
380:
366:
172:
164:
161:
158:
147:
14:
6707:
6576:Revista Matemática Iberoamericana
3228:{\displaystyle \mathbb {C} ^{n}.}
2735:{\displaystyle x_{1},x_{2},x_{3}}
463:The divergence of a vector field
3921:, the operator is often written
3299:
2861:the two dimensional sphere, set
2854:{\displaystyle \mathbb {S} ^{2}}
2793:{\displaystyle \mathbb {S} ^{2}}
2764:{\displaystyle \mathbb {R} ^{3}}
2655:{\displaystyle \mathbb {S} ^{2}}
2626:{\displaystyle \mathbb {S} ^{n}}
2470:{\displaystyle g(\cdot ,\cdot )}
887:. In local coordinates, one has
717:that may be defined through the
5794:{\displaystyle f(x/q(x)^{1/2})}
2825:{\displaystyle (\theta ,\phi )}
1739:associated with the eigenvalue
6567:
6508:
6484:
6451:
6436:
6350:
6343:
6306:
6300:
6262:
6255:
6243:
6231:
6190:{\displaystyle \Delta _{\xi }}
6132:
6125:
6076:
6069:
6028:
6021:
6009:
5997:
5788:
5771:
5764:
5750:
5681:
5674:
5560:
5554:
5436:
5430:
5339:
5326:
5257:
5244:
5238:
5226:
5181:{\displaystyle \Delta _{\xi }}
5123:
5110:
5067:
5054:
5019:
5006:
5000:
4988:
4695:
4691:
4683:
4671:
4659:
4653:
4497:
4489:
4469:
4461:
4435:. Consequently, in this case
4415:
4407:
4216:
4201:
3889:
3875:
3812:
3786:
3748:
3742:
3707:
3701:
3531:
3505:
3480:
3466:
3343:
3303:
2819:
2807:
2800:. Using spherical coordinates
2464:
2452:
2411:
2399:
2387:
2375:
2311:{\displaystyle \lambda \geq 0}
2225:
2213:
2160:
2148:
1891:{\displaystyle \lambda \geq 0}
1251:
1245:
1084:
1076:
1051:
1043:
830:
817:
814:
808:
780:
774:
668:
660:
635:
627:
531:
519:
502:{\displaystyle \nabla \cdot X}
261:
253:
178:
169:
1:
6616:
5405:. Here the hyperbolic space
5401:A similar technique works in
4368:that explicitly involves the
4294:is the Hodge–Dirac operator.
4135:on sections of the bundle of
2325:states that: Given a compact
46:and, even more generally, on
6443:Lichnerowicz, Andre (1958).
4731: − {0}, and
3020:{\displaystyle \lambda _{1}}
2682:{\displaystyle \lambda _{1}}
2537:{\displaystyle \lambda _{1}}
7:
6670:Encyclopedia of Mathematics
6665:"Laplace–Beltrami equation"
6648:, Berlin: Springer-Verlag,
6407:
4388:In the usual (orthonormal)
4375:
4266:. The first order operator
3725:More precisely, this means
2359:satisfies the lower bound:
1938:we get (using the notation
1481:
1430:
97:pure second derivatives of
73:defined on Euclidean space
52:pseudo-Riemannian manifolds
34:is a generalization of the
10:
6712:
4141:pseudo-Riemannian manifold
3830:or in terms of the metric
3274:is the symmetric 2-tensor
1842:{\displaystyle \lambda =0}
124:
18:
6623:Flanders, Harley (1989),
6517:Duke Mathematical Journal
5409:can be embedded into the
4256:(−1)∗d∗
2477:is the metric tensor and
868:of the manifold at point
702:where here and below the
113:using the divergence and
32:Laplace–Beltrami operator
6539:10.1215/00127094-1902154
6429:
5941:(say, the center of the
4914:normal coordinate system
4161:Laplace–de Rham operator
4127:Laplace–de Rham operator
2321:A fundamental result of
1865:{\displaystyle -\Delta }
1812:{\displaystyle \lambda }
1772:{\displaystyle \lambda }
1752:{\displaystyle \lambda }
1312:
1193:{\displaystyle -\nabla }
1157:The exterior derivative
38:to functions defined on
19:Not to be confused with
4744:{\displaystyle \Delta }
4569:alternative expressions
4565:cylindrical coordinates
4557:curvilinear coordinates
4262:-forms, where ∗ is the
3154:, the sphere of radius
2348:{\displaystyle n\geq 2}
2090:has no boundary we get
1153:Formal self-adjointness
449:{\displaystyle \wedge }
197:An explicit formula in
6691:Differential operators
6491:Chavel, Isaac (1984),
6398:
6191:
6161:
5912:
5795:
5731:
5599:
5506:
5387:
5182:
5152:
4896:
4745:
4702:
4546:
4429:
4370:Ricci curvature tensor
4346:
4288:
4233:
4114:
3967:
3908:
3820:
3774:
3715:
3618:
3420:
3351:
3268:
3244:trace (or contraction)
3229:
3185:
3148:
3078:
3021:
2984:
2913:
2855:
2826:
2794:
2765:
2736:
2683:
2656:
2627:
2595:
2538:
2511:
2491:
2471:
2433:
2349:
2312:
2283:
2183:
2084:
2057:
1968:
1932:
1912:
1892:
1866:
1843:
1813:
1793:
1773:
1753:
1729:
1706:
1654:
1521:
1501:
1474:
1454:
1418:
1306:
1214:
1194:
1171:
1128:
972:
837:
746:
693:
571:
503:
477:
450:
430:
413:of the tangent bundle
404:
307:
188:
6477:10.2969/jmsj/01430333
6458:Obata, Morio (1962).
6399:
6192:
6162:
5913:
5796:
5732:
5600:
5507:
5388:
5183:
5153:
4897:
4746:
4703:
4547:
4430:
4428:{\displaystyle |g|=1}
4390:Cartesian coordinates
4347:
4289:
4234:
4133:differential operator
4115:
3968:
3909:
3821:
3754:
3716:
3619:
3436:(exterior) derivative
3421:
3352:
3269:
3230:
3186:
3149:
3079:
3022:
2985:
2914:
2856:
2827:
2795:
2766:
2737:
2684:
2657:
2628:
2596:
2539:
2512:
2492:
2472:
2434:
2350:
2313:
2284:
2184:
2085:
2058:
1969:
1933:
1913:
1893:
1867:
1844:
1814:
1794:
1774:
1754:
1730:
1707:
1655:
1522:
1502:
1475:
1455:
1419:
1307:
1215:
1195:
1172:
1129:
973:
838:
747:
694:
572:
504:
478:
451:
431:
405:
308:
189:
69:real-valued function
28:differential geometry
6414:Covariant derivative
6208:
6174:
5968:
5819:
5744:
5615:
5526:
5424:
5203:
5165:
4959:
4758:
4735:
4624:
4442:
4403:
4366:Weitzenböck identity
4315:
4270:
4246:or differential and
4170:
3998:
3928:
3837:
3732:
3652:
3463:
3361:
3281:
3258:
3248:covariant derivative
3207:
3158:
3095:
3091:-dimensional sphere
3034:
3004:
2929:
2868:
2836:
2804:
2775:
2746:
2693:
2666:
2637:
2608:
2551:
2521:
2501:
2481:
2446:
2366:
2333:
2296:
2199:
2097:
2074:
1981:
1942:
1922:
1902:
1876:
1853:
1827:
1803:
1783:
1763:
1743:
1719:
1678:
1534:
1511:
1491:
1464:
1444:
1332:
1226:
1204:
1181:
1161:
1024:
894:
759:
752:on the manifold, as
724:
605:
516:
487:
467:
440:
417:
362:
234:
144:
133:of the (Riemannian)
56:Pierre-Simon Laplace
54:. It is named after
6696:Riemannian geometry
5901:
5858:
5498:
5474:
5456:
4593:Spherical Laplacian
4244:exterior derivative
4153:Lorentzian manifold
4145:Riemannian manifold
2662:the eigenspace for
1323:. Dualizing gives
878:exterior derivative
213:Riemannian manifold
204:Suppose first that
115:exterior derivative
6394:
6187:
6157:
5908:
5887:
5844:
5791:
5727:
5595:
5502:
5484:
5460:
5442:
5383:
5178:
5148:
4892:
4741:
4698:
4542:
4425:
4342:
4284:
4229:
4137:differential forms
4110:
3963:
3904:
3883:
3861:
3816:
3815:
3711:
3710:
3614:
3500:
3474:
3416:
3369:
3347:
3346:
3290:
3264:
3225:
3181:
3144:
3074:
3017:
2980:
2909:
2851:
2822:
2790:
2761:
2732:
2679:
2652:
2623:
2591:
2534:
2507:
2487:
2467:
2429:
2345:
2323:André Lichnerowicz
2308:
2279:
2179:
2080:
2068:divergence theorem
2053:
1964:
1928:
1908:
1888:
1862:
1839:
1809:
1789:
1769:
1749:
1725:
1702:
1650:
1517:
1497:
1470:
1450:
1414:
1302:
1210:
1190:
1167:
1124:
968:
853:anchored at point
833:
742:
689:
567:
509:with the property
499:
473:
446:
429:{\displaystyle TM}
426:
400:
303:
184:
111:differential forms
6634:978-0-486-66169-8
6523:(15): 2909–2921.
6502:978-0-12-170640-1
6464:J. Math. Soc. Jpn
6389:
6327:
6287:
6109:
6056:
5903:
5860:
5378:
5316:
5282:
5100:
5047:
4840:
4799:
4501:
4474:
4473:
4362:positive definite
4149:elliptic operator
4123:is well-defined.
3882:
3849:
3499:
3473:
3368:
3289:
3267:{\displaystyle f}
3179:
3178:
3135:
3134:
3066:
2583:
2510:{\displaystyle M}
2490:{\displaystyle X}
2272:
2237:
2172:
2127:
2121:
2083:{\displaystyle M}
2046:
2011:
2005:
1931:{\displaystyle M}
1911:{\displaystyle u}
1792:{\displaystyle M}
1728:{\displaystyle u}
1520:{\displaystyle h}
1500:{\displaystyle f}
1473:{\displaystyle h}
1453:{\displaystyle f}
1438:
1437:
1315:
1213:{\displaystyle f}
1170:{\displaystyle d}
1088:
1056:
1055:
704:Einstein notation
672:
640:
639:
476:{\displaystyle X}
398:
265:
199:local coordinates
21:Beltrami operator
6703:
6677:
6658:
6637:
6610:
6609:
6598:10.4171/RMI/1041
6591:
6582:(4): 1711–1753.
6571:
6565:
6564:
6558:
6550:
6532:
6512:
6506:
6505:
6488:
6482:
6481:
6479:
6455:
6449:
6448:
6440:
6424:Laplace operator
6403:
6401:
6400:
6395:
6390:
6388:
6387:
6386:
6373:
6372:
6363:
6361:
6360:
6333:
6329:
6328:
6326:
6318:
6310:
6288:
6286:
6275:
6273:
6272:
6227:
6226:
6225:
6224:
6196:
6194:
6193:
6188:
6186:
6185:
6166:
6164:
6163:
6158:
6153:
6152:
6143:
6142:
6115:
6111:
6110:
6108:
6100:
6092:
6090:
6089:
6057:
6055:
6044:
6042:
6041:
5993:
5992:
5991:
5990:
5932:
5917:
5915:
5914:
5909:
5904:
5902:
5900:
5895:
5882:
5881:
5872:
5861:
5859:
5857:
5852:
5839:
5838:
5829:
5808:
5800:
5798:
5797:
5792:
5787:
5786:
5782:
5760:
5736:
5734:
5733:
5728:
5726:
5725:
5724:
5723:
5707:
5703:
5702:
5698:
5697:
5696:
5692:
5670:
5640:
5639:
5638:
5637:
5604:
5602:
5601:
5596:
5581:
5580:
5538:
5537:
5511:
5509:
5508:
5503:
5497:
5492:
5473:
5468:
5455:
5450:
5403:hyperbolic space
5397:Hyperbolic space
5392:
5390:
5389:
5384:
5379:
5377:
5376:
5375:
5362:
5361:
5352:
5350:
5349:
5322:
5318:
5317:
5315:
5307:
5299:
5283:
5281:
5270:
5268:
5267:
5222:
5221:
5220:
5219:
5195:
5187:
5185:
5184:
5179:
5177:
5176:
5157:
5155:
5154:
5149:
5144:
5143:
5134:
5133:
5106:
5102:
5101:
5099:
5091:
5083:
5081:
5080:
5048:
5046:
5035:
5033:
5032:
4984:
4983:
4982:
4981:
4927:
4901:
4899:
4898:
4893:
4885:
4884:
4883:
4882:
4862:
4861:
4846:
4842:
4841:
4839:
4831:
4823:
4821:
4820:
4800:
4798:
4787:
4785:
4784:
4750:
4748:
4747:
4742:
4707:
4705:
4704:
4699:
4694:
4686:
4681:
4649:
4648:
4647:
4646:
4604:
4584:
4576:Minkowski metric
4551:
4549:
4548:
4543:
4538:
4537:
4528:
4527:
4512:
4511:
4502:
4500:
4492:
4487:
4485:
4484:
4475:
4472:
4464:
4459:
4455:
4434:
4432:
4431:
4426:
4418:
4410:
4351:
4349:
4348:
4343:
4335:
4307:
4293:
4291:
4290:
4285:
4277:
4257:
4238:
4236:
4235:
4230:
4224:
4223:
4208:
4197:
4183:
4119:
4117:
4116:
4111:
4109:
4105:
4101:
4100:
4099:
4098:
4089:
4088:
4087:
4086:
4061:
4060:
4059:
4058:
4044:
4043:
4042:
4041:
4022:
4021:
3972:
3970:
3969:
3964:
3959:
3958:
3949:
3948:
3919:abstract indices
3913:
3911:
3910:
3905:
3900:
3899:
3884:
3880:
3874:
3873:
3860:
3825:
3823:
3822:
3817:
3811:
3810:
3798:
3797:
3782:
3773:
3768:
3720:
3718:
3717:
3712:
3700:
3699:
3694:
3693:
3680:
3672:
3623:
3621:
3620:
3615:
3610:
3609:
3608:
3607:
3598:
3597:
3596:
3595:
3570:
3569:
3568:
3567:
3553:
3552:
3551:
3550:
3530:
3529:
3517:
3516:
3501:
3497:
3491:
3490:
3475:
3471:
3425:
3423:
3422:
3417:
3412:
3386:
3385:
3370:
3366:
3356:
3354:
3353:
3348:
3339:
3338:
3333:
3332:
3319:
3318:
3313:
3312:
3302:
3291:
3287:
3273:
3271:
3270:
3265:
3252:Hessian (tensor)
3246:of the iterated
3238:Tensor Laplacian
3234:
3232:
3231:
3226:
3221:
3220:
3215:
3190:
3188:
3187:
3182:
3180:
3174:
3163:
3162:
3153:
3151:
3150:
3145:
3143:
3142:
3136:
3130:
3119:
3118:
3116:
3115:
3109:
3108:
3103:
3083:
3081:
3080:
3075:
3067:
3065:
3051:
3046:
3045:
3026:
3024:
3023:
3018:
3016:
3015:
2989:
2987:
2986:
2981:
2979:
2978:
2963:
2962:
2953:
2952:
2951:
2950:
2945:
2918:
2916:
2915:
2910:
2905:
2904:
2880:
2879:
2860:
2858:
2857:
2852:
2850:
2849:
2844:
2831:
2829:
2828:
2823:
2799:
2797:
2796:
2791:
2789:
2788:
2783:
2770:
2768:
2767:
2762:
2760:
2759:
2754:
2741:
2739:
2738:
2733:
2731:
2730:
2718:
2717:
2705:
2704:
2688:
2686:
2685:
2680:
2678:
2677:
2661:
2659:
2658:
2653:
2651:
2650:
2645:
2632:
2630:
2629:
2624:
2622:
2621:
2616:
2600:
2598:
2597:
2592:
2584:
2582:
2568:
2563:
2562:
2543:
2541:
2540:
2535:
2533:
2532:
2516:
2514:
2513:
2508:
2496:
2494:
2493:
2488:
2476:
2474:
2473:
2468:
2438:
2436:
2435:
2430:
2354:
2352:
2351:
2346:
2317:
2315:
2314:
2309:
2288:
2286:
2285:
2280:
2270:
2269:
2268:
2259:
2258:
2235:
2234:
2233:
2228:
2216:
2211:
2210:
2188:
2186:
2185:
2180:
2170:
2169:
2168:
2163:
2151:
2146:
2145:
2125:
2119:
2112:
2111:
2089:
2087:
2086:
2081:
2062:
2060:
2059:
2054:
2044:
2043:
2042:
2033:
2032:
2009:
2003:
1996:
1995:
1973:
1971:
1970:
1965:
1963:
1962:
1937:
1935:
1934:
1929:
1917:
1915:
1914:
1909:
1897:
1895:
1894:
1889:
1871:
1869:
1868:
1863:
1848:
1846:
1845:
1840:
1818:
1816:
1815:
1810:
1798:
1796:
1795:
1790:
1778:
1776:
1775:
1770:
1758:
1756:
1755:
1750:
1734:
1732:
1731:
1726:
1711:
1709:
1708:
1703:
1659:
1657:
1656:
1651:
1646:
1645:
1626:
1625:
1613:
1612:
1582:
1581:
1566:
1565:
1546:
1545:
1526:
1524:
1523:
1518:
1506:
1504:
1503:
1498:
1483:
1480:. Conversely, (
1479:
1477:
1476:
1471:
1459:
1457:
1456:
1451:
1432:
1423:
1421:
1420:
1415:
1413:
1412:
1381:
1380:
1365:
1364:
1344:
1343:
1326:
1313:
1311:
1309:
1308:
1303:
1301:
1300:
1279:
1278:
1263:
1262:
1238:
1237:
1219:
1217:
1216:
1211:
1199:
1197:
1196:
1191:
1176:
1174:
1173:
1168:
1133:
1131:
1130:
1125:
1120:
1116:
1112:
1111:
1102:
1101:
1089:
1087:
1079:
1074:
1067:
1066:
1057:
1054:
1046:
1041:
1037:
1003:
977:
975:
974:
969:
964:
963:
954:
953:
935:
934:
922:
921:
916:
912:
846:for all vectors
842:
840:
839:
834:
829:
828:
795:
794:
751:
749:
748:
743:
710:is summed over.
698:
696:
695:
690:
688:
684:
683:
682:
673:
671:
663:
658:
651:
650:
641:
638:
630:
625:
621:
576:
574:
573:
568:
566:
565:
556:
555:
543:
542:
508:
506:
505:
500:
482:
480:
479:
474:
455:
453:
452:
447:
435:
433:
432:
427:
409:
407:
406:
401:
399:
397:
396:
395:
379:
374:
373:
330:
312:
310:
309:
304:
302:
301:
280:
279:
266:
264:
256:
251:
246:
245:
193:
191:
190:
185:
168:
167:
60:Eugenio Beltrami
36:Laplace operator
6711:
6710:
6706:
6705:
6704:
6702:
6701:
6700:
6681:
6680:
6656:
6635:
6619:
6614:
6613:
6572:
6568:
6552:
6551:
6513:
6509:
6503:
6489:
6485:
6456:
6452:
6447:. Paris: Dunod.
6441:
6437:
6432:
6410:
6382:
6378:
6374:
6368:
6364:
6362:
6353:
6349:
6319:
6311:
6309:
6293:
6289:
6279:
6274:
6265:
6261:
6220:
6216:
6215:
6211:
6209:
6206:
6205:
6181:
6177:
6175:
6172:
6171:
6148:
6144:
6135:
6131:
6101:
6093:
6091:
6079:
6075:
6062:
6058:
6048:
6043:
6031:
6027:
5980:
5976:
5975:
5971:
5969:
5966:
5965:
5922:
5896:
5891:
5883:
5877:
5873:
5871:
5853:
5848:
5840:
5834:
5830:
5828:
5820:
5817:
5816:
5806:
5778:
5774:
5770:
5756:
5745:
5742:
5741:
5713:
5709:
5708:
5688:
5684:
5680:
5666:
5662:
5658:
5651:
5648:
5647:
5627:
5623:
5622:
5618:
5616:
5613:
5612:
5576:
5572:
5533:
5529:
5527:
5524:
5523:
5493:
5488:
5469:
5464:
5451:
5446:
5425:
5422:
5421:
5415:Minkowski space
5399:
5371:
5367:
5363:
5357:
5353:
5351:
5342:
5338:
5308:
5300:
5298:
5288:
5284:
5274:
5269:
5260:
5256:
5215:
5211:
5210:
5206:
5204:
5201:
5200:
5189:
5172:
5168:
5166:
5163:
5162:
5139:
5135:
5126:
5122:
5092:
5084:
5082:
5070:
5066:
5053:
5049:
5039:
5034:
5022:
5018:
4971:
4967:
4966:
4962:
4960:
4957:
4956:
4917:
4872:
4868:
4867:
4863:
4854:
4850:
4832:
4824:
4822:
4810:
4806:
4805:
4801:
4791:
4786:
4774:
4770:
4759:
4756:
4755:
4736:
4733:
4732:
4690:
4682:
4677:
4636:
4632:
4631:
4627:
4625:
4622:
4621:
4598:
4595:
4583:(− + + +)
4582:
4533:
4529:
4523:
4519:
4507:
4503:
4496:
4488:
4486:
4480:
4476:
4468:
4460:
4454:
4443:
4440:
4439:
4414:
4406:
4404:
4401:
4400:
4397:Euclidean space
4386:
4384:Euclidean space
4378:
4331:
4316:
4313:
4312:
4302:
4273:
4271:
4268:
4267:
4255:
4242:where d is the
4219:
4215:
4204:
4193:
4179:
4171:
4168:
4167:
4129:
4094:
4090:
4082:
4078:
4077:
4073:
4072:
4068:
4054:
4050:
4049:
4045:
4037:
4033:
4032:
4028:
4027:
4023:
4014:
4010:
3999:
3996:
3995:
3954:
3950:
3944:
3940:
3929:
3926:
3925:
3892:
3888:
3878:
3866:
3862:
3853:
3838:
3835:
3834:
3806:
3802:
3793:
3789:
3778:
3769:
3758:
3733:
3730:
3729:
3695:
3689:
3688:
3687:
3676:
3665:
3653:
3650:
3649:
3640:
3633:
3603:
3599:
3591:
3587:
3586:
3582:
3581:
3577:
3563:
3559:
3558:
3554:
3546:
3542:
3541:
3537:
3525:
3521:
3512:
3508:
3495:
3483:
3479:
3469:
3464:
3461:
3460:
3451:
3408:
3381:
3377:
3364:
3362:
3359:
3358:
3334:
3328:
3327:
3326:
3314:
3308:
3307:
3306:
3298:
3285:
3282:
3279:
3278:
3259:
3256:
3255:
3240:
3216:
3211:
3210:
3208:
3205:
3204:
3199:) on a compact
3164:
3161:
3159:
3156:
3155:
3138:
3137:
3120:
3117:
3111:
3110:
3104:
3099:
3098:
3096:
3093:
3092:
3055:
3050:
3041:
3037:
3035:
3032:
3031:
3011:
3007:
3005:
3002:
3001:
2974:
2970:
2958:
2954:
2946:
2941:
2940:
2939:
2935:
2930:
2927:
2926:
2900:
2896:
2875:
2871:
2869:
2866:
2865:
2845:
2840:
2839:
2837:
2834:
2833:
2805:
2802:
2801:
2784:
2779:
2778:
2776:
2773:
2772:
2755:
2750:
2749:
2747:
2744:
2743:
2726:
2722:
2713:
2709:
2700:
2696:
2694:
2691:
2690:
2673:
2669:
2667:
2664:
2663:
2646:
2641:
2640:
2638:
2635:
2634:
2617:
2612:
2611:
2609:
2606:
2605:
2572:
2567:
2558:
2554:
2552:
2549:
2548:
2528:
2524:
2522:
2519:
2518:
2502:
2499:
2498:
2482:
2479:
2478:
2447:
2444:
2443:
2367:
2364:
2363:
2357:Ricci curvature
2334:
2331:
2330:
2297:
2294:
2293:
2264:
2260:
2254:
2250:
2229:
2224:
2223:
2212:
2206:
2202:
2200:
2197:
2196:
2164:
2159:
2158:
2147:
2141:
2137:
2107:
2103:
2098:
2095:
2094:
2075:
2072:
2071:
2038:
2034:
2028:
2024:
1991:
1987:
1982:
1979:
1978:
1958:
1954:
1943:
1940:
1939:
1923:
1920:
1919:
1903:
1900:
1899:
1877:
1874:
1873:
1854:
1851:
1850:
1828:
1825:
1824:
1804:
1801:
1800:
1784:
1781:
1780:
1764:
1761:
1760:
1744:
1741:
1740:
1720:
1717:
1716:
1679:
1676:
1675:
1669:
1641:
1637:
1621:
1617:
1608:
1604:
1577:
1573:
1561:
1557:
1541:
1537:
1535:
1532:
1531:
1512:
1509:
1508:
1492:
1489:
1488:
1465:
1462:
1461:
1445:
1442:
1441:
1408:
1404:
1376:
1372:
1360:
1356:
1339:
1335:
1333:
1330:
1329:
1321:Stokes' theorem
1296:
1292:
1274:
1270:
1258:
1254:
1233:
1229:
1227:
1224:
1223:
1205:
1202:
1201:
1182:
1179:
1178:
1162:
1159:
1158:
1155:
1107:
1103:
1094:
1090:
1083:
1075:
1073:
1072:
1068:
1062:
1058:
1050:
1042:
1036:
1025:
1022:
1021:
1012:Kronecker delta
1009:
1002:
995:
990:
959:
955:
946:
942:
930:
926:
917:
902:
898:
897:
895:
892:
891:
885:
865:
851:
824:
820:
790:
786:
760:
757:
756:
725:
722:
721:
678:
674:
667:
659:
657:
656:
652:
646:
642:
634:
626:
620:
606:
603:
602:
585:
561:
557:
551:
547:
538:
534:
517:
514:
513:
488:
485:
484:
468:
465:
464:
441:
438:
437:
418:
415:
414:
391:
387:
383:
378:
369:
365:
363:
360:
359:
327:
322:| := |det(
317:
297:
293:
275:
271:
260:
252:
250:
241:
237:
235:
232:
231:
157:
156:
145:
142:
141:
127:
119:Georges de Rham
107:linear operator
44:Euclidean space
24:
17:
12:
11:
5:
6709:
6699:
6698:
6693:
6679:
6678:
6660:
6654:
6638:
6633:
6618:
6615:
6612:
6611:
6566:
6507:
6501:
6483:
6470:(3): 333–340.
6450:
6434:
6433:
6431:
6428:
6427:
6426:
6421:
6416:
6409:
6406:
6405:
6404:
6393:
6385:
6381:
6377:
6371:
6367:
6359:
6356:
6352:
6348:
6345:
6342:
6339:
6336:
6332:
6325:
6322:
6317:
6314:
6308:
6305:
6302:
6299:
6296:
6292:
6285:
6282:
6278:
6271:
6268:
6264:
6260:
6257:
6254:
6251:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6223:
6219:
6214:
6184:
6180:
6168:
6167:
6156:
6151:
6147:
6141:
6138:
6134:
6130:
6127:
6124:
6121:
6118:
6114:
6107:
6104:
6099:
6096:
6088:
6085:
6082:
6078:
6074:
6071:
6068:
6065:
6061:
6054:
6051:
6047:
6040:
6037:
6034:
6030:
6026:
6023:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5989:
5986:
5983:
5979:
5974:
5919:
5918:
5907:
5899:
5894:
5890:
5886:
5880:
5876:
5870:
5867:
5864:
5856:
5851:
5847:
5843:
5837:
5833:
5827:
5824:
5790:
5785:
5781:
5777:
5773:
5769:
5766:
5763:
5759:
5755:
5752:
5749:
5738:
5737:
5722:
5719:
5716:
5712:
5706:
5701:
5695:
5691:
5687:
5683:
5679:
5676:
5673:
5669:
5665:
5661:
5657:
5654:
5650:
5646:
5643:
5636:
5633:
5630:
5626:
5621:
5606:
5605:
5593:
5590:
5587:
5584:
5579:
5575:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5536:
5532:
5513:
5512:
5501:
5496:
5491:
5487:
5483:
5480:
5477:
5472:
5467:
5463:
5459:
5454:
5449:
5445:
5441:
5438:
5435:
5432:
5429:
5398:
5395:
5394:
5393:
5382:
5374:
5370:
5366:
5360:
5356:
5348:
5345:
5341:
5337:
5334:
5331:
5328:
5325:
5321:
5314:
5311:
5306:
5303:
5297:
5294:
5291:
5287:
5280:
5277:
5273:
5266:
5263:
5259:
5255:
5252:
5249:
5246:
5243:
5240:
5237:
5234:
5231:
5228:
5225:
5218:
5214:
5209:
5175:
5171:
5159:
5158:
5147:
5142:
5138:
5132:
5129:
5125:
5121:
5118:
5115:
5112:
5109:
5105:
5098:
5095:
5090:
5087:
5079:
5076:
5073:
5069:
5065:
5062:
5059:
5056:
5052:
5045:
5042:
5038:
5031:
5028:
5025:
5021:
5017:
5014:
5011:
5008:
5005:
5002:
4999:
4996:
4993:
4990:
4987:
4980:
4977:
4974:
4970:
4965:
4903:
4902:
4891:
4888:
4881:
4878:
4875:
4871:
4866:
4860:
4857:
4853:
4849:
4845:
4838:
4835:
4830:
4827:
4819:
4816:
4813:
4809:
4804:
4797:
4794:
4790:
4783:
4780:
4777:
4773:
4769:
4766:
4763:
4740:
4709:
4708:
4697:
4693:
4689:
4685:
4680:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4645:
4642:
4639:
4635:
4630:
4594:
4591:
4567:, one obtains
4553:
4552:
4541:
4536:
4532:
4526:
4522:
4518:
4515:
4510:
4506:
4499:
4495:
4491:
4483:
4479:
4471:
4467:
4463:
4458:
4453:
4450:
4447:
4424:
4421:
4417:
4413:
4409:
4385:
4382:
4377:
4374:
4358:codifferential
4353:
4352:
4341:
4338:
4334:
4329:
4326:
4323:
4320:
4283:
4280:
4276:
4252:codifferential
4240:
4239:
4227:
4222:
4218:
4214:
4211:
4207:
4203:
4200:
4196:
4192:
4189:
4186:
4182:
4178:
4175:
4163:is defined by
4128:
4125:
4121:
4120:
4108:
4104:
4097:
4093:
4085:
4081:
4076:
4071:
4067:
4064:
4057:
4053:
4048:
4040:
4036:
4031:
4026:
4020:
4017:
4013:
4009:
4006:
4003:
3974:
3973:
3962:
3957:
3953:
3947:
3943:
3939:
3936:
3933:
3915:
3914:
3903:
3898:
3895:
3891:
3887:
3877:
3872:
3869:
3865:
3859:
3856:
3852:
3848:
3845:
3842:
3828:
3827:
3814:
3809:
3805:
3801:
3796:
3792:
3788:
3785:
3781:
3777:
3772:
3767:
3764:
3761:
3757:
3753:
3750:
3747:
3744:
3741:
3738:
3723:
3722:
3709:
3706:
3703:
3698:
3692:
3686:
3683:
3679:
3675:
3671:
3668:
3664:
3661:
3658:
3638:
3631:
3625:
3624:
3613:
3606:
3602:
3594:
3590:
3585:
3580:
3576:
3573:
3566:
3562:
3557:
3549:
3545:
3540:
3536:
3533:
3528:
3524:
3520:
3515:
3511:
3507:
3504:
3494:
3489:
3486:
3482:
3478:
3468:
3449:
3438:of a function
3428:
3427:
3415:
3411:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3384:
3380:
3376:
3373:
3345:
3342:
3337:
3331:
3325:
3322:
3317:
3311:
3305:
3301:
3297:
3294:
3263:
3254:of a function
3239:
3236:
3224:
3219:
3214:
3197:Joseph J. Kohn
3193:Kohn Laplacian
3177:
3173:
3170:
3167:
3141:
3133:
3129:
3126:
3123:
3114:
3107:
3102:
3085:
3084:
3073:
3070:
3064:
3061:
3058:
3054:
3049:
3044:
3040:
3014:
3010:
2991:
2990:
2977:
2973:
2969:
2966:
2961:
2957:
2949:
2944:
2938:
2934:
2920:
2919:
2908:
2903:
2899:
2895:
2892:
2889:
2886:
2883:
2878:
2874:
2848:
2843:
2821:
2818:
2815:
2812:
2809:
2787:
2782:
2758:
2753:
2729:
2725:
2721:
2716:
2712:
2708:
2703:
2699:
2676:
2672:
2649:
2644:
2620:
2615:
2602:
2601:
2590:
2587:
2581:
2578:
2575:
2571:
2566:
2561:
2557:
2531:
2527:
2506:
2486:
2466:
2463:
2460:
2457:
2454:
2451:
2440:
2439:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2344:
2341:
2338:
2307:
2304:
2301:
2290:
2289:
2278:
2275:
2267:
2263:
2257:
2253:
2249:
2246:
2243:
2240:
2232:
2227:
2222:
2219:
2215:
2209:
2205:
2190:
2189:
2178:
2175:
2167:
2162:
2157:
2154:
2150:
2144:
2140:
2136:
2133:
2130:
2124:
2118:
2115:
2110:
2106:
2102:
2079:
2064:
2063:
2052:
2049:
2041:
2037:
2031:
2027:
2023:
2020:
2017:
2014:
2008:
2002:
1999:
1994:
1990:
1986:
1961:
1957:
1953:
1950:
1947:
1927:
1907:
1887:
1884:
1881:
1861:
1858:
1838:
1835:
1832:
1808:
1788:
1768:
1748:
1724:
1713:
1712:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1668:
1665:
1661:
1660:
1649:
1644:
1640:
1636:
1633:
1629:
1624:
1620:
1616:
1611:
1607:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1580:
1576:
1572:
1569:
1564:
1560:
1556:
1553:
1549:
1544:
1540:
1516:
1496:
1469:
1449:
1436:
1435:
1426:
1424:
1411:
1407:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1379:
1375:
1371:
1368:
1363:
1359:
1354:
1351:
1347:
1342:
1338:
1317:
1316:
1299:
1295:
1291:
1288:
1285:
1282:
1277:
1273:
1269:
1266:
1261:
1257:
1253:
1250:
1247:
1244:
1241:
1236:
1232:
1209:
1189:
1186:
1166:
1154:
1151:
1143:volume element
1135:
1134:
1123:
1119:
1115:
1110:
1106:
1100:
1097:
1093:
1086:
1082:
1078:
1071:
1065:
1061:
1053:
1049:
1045:
1040:
1035:
1032:
1029:
1005:
998:
993:
979:
978:
967:
962:
958:
952:
949:
945:
941:
938:
933:
929:
925:
920:
915:
911:
908:
905:
901:
883:
863:
849:
844:
843:
832:
827:
823:
819:
816:
813:
810:
807:
804:
801:
798:
793:
789:
785:
782:
779:
776:
773:
770:
767:
764:
741:
738:
735:
732:
729:
700:
699:
687:
681:
677:
670:
666:
662:
655:
649:
645:
637:
633:
629:
624:
619:
616:
613:
610:
589:Lie derivative
583:
578:
577:
564:
560:
554:
550:
546:
541:
537:
533:
530:
527:
524:
521:
498:
495:
492:
472:
445:
425:
422:
411:
410:
394:
390:
386:
382:
377:
372:
368:
333:absolute value
325:
314:
313:
300:
296:
292:
289:
286:
283:
278:
274:
270:
263:
259:
255:
249:
244:
240:
195:
194:
183:
180:
177:
174:
171:
166:
163:
160:
155:
152:
149:
126:
123:
67:differentiable
65:For any twice-
15:
9:
6:
4:
3:
2:
6708:
6697:
6694:
6692:
6689:
6688:
6686:
6676:
6672:
6671:
6666:
6661:
6657:
6655:3-540-42627-2
6651:
6647:
6643:
6639:
6636:
6630:
6626:
6621:
6620:
6607:
6603:
6599:
6595:
6590:
6585:
6581:
6577:
6570:
6562:
6556:
6548:
6544:
6540:
6536:
6531:
6526:
6522:
6518:
6511:
6504:
6498:
6494:
6487:
6478:
6473:
6469:
6465:
6461:
6454:
6446:
6439:
6435:
6425:
6422:
6420:
6417:
6415:
6412:
6411:
6391:
6383:
6379:
6369:
6357:
6354:
6346:
6340:
6337:
6334:
6330:
6323:
6315:
6303:
6297:
6294:
6290:
6283:
6269:
6266:
6258:
6252:
6249:
6246:
6240:
6237:
6234:
6228:
6221:
6217:
6204:
6203:
6202:
6200:
6182:
6154:
6149:
6139:
6136:
6128:
6122:
6119:
6116:
6112:
6105:
6097:
6086:
6083:
6080:
6072:
6066:
6063:
6059:
6052:
6038:
6035:
6032:
6024:
6018:
6015:
6012:
6006:
6003:
6000:
5994:
5987:
5984:
5981:
5977:
5964:
5963:
5962:
5960:
5956:
5952:
5948:
5944:
5943:Poincaré disc
5940:
5936:
5930:
5926:
5905:
5897:
5892:
5888:
5878:
5868:
5865:
5862:
5854:
5849:
5845:
5835:
5825:
5822:
5815:
5814:
5813:
5812:
5811:wave operator
5804:
5783:
5779:
5775:
5767:
5761:
5757:
5753:
5747:
5720:
5717:
5714:
5710:
5704:
5699:
5693:
5689:
5685:
5677:
5671:
5667:
5663:
5659:
5655:
5652:
5644:
5641:
5634:
5631:
5628:
5624:
5611:
5610:
5609:
5591:
5585:
5582:
5577:
5573:
5569:
5566:
5563:
5557:
5551:
5548:
5545:
5539:
5534:
5530:
5522:
5521:
5520:
5518:
5499:
5494:
5489:
5485:
5481:
5478:
5475:
5470:
5465:
5461:
5457:
5452:
5447:
5443:
5439:
5433:
5427:
5420:
5419:
5418:
5416:
5412:
5408:
5404:
5380:
5372:
5368:
5358:
5346:
5343:
5335:
5332:
5329:
5323:
5319:
5312:
5304:
5295:
5292:
5289:
5285:
5278:
5264:
5261:
5253:
5250:
5247:
5241:
5235:
5232:
5229:
5223:
5216:
5212:
5199:
5198:
5197:
5193:
5173:
5145:
5140:
5130:
5127:
5119:
5116:
5113:
5107:
5103:
5096:
5088:
5077:
5074:
5071:
5063:
5060:
5057:
5050:
5043:
5029:
5026:
5023:
5015:
5012:
5009:
5003:
4997:
4994:
4991:
4985:
4978:
4975:
4972:
4968:
4955:
4954:
4953:
4951:
4947:
4943:
4939:
4935:
4931:
4925:
4921:
4915:
4910:
4908:
4907:normal bundle
4889:
4886:
4879:
4876:
4873:
4869:
4858:
4855:
4851:
4847:
4843:
4836:
4828:
4817:
4814:
4811:
4807:
4802:
4795:
4781:
4778:
4775:
4771:
4767:
4764:
4754:
4753:
4752:
4730:
4726:
4722:
4718:
4714:
4687:
4678:
4674:
4668:
4662:
4656:
4650:
4643:
4640:
4637:
4633:
4620:
4619:
4618:
4616:
4612:
4608:
4602:
4590:
4588:
4587:d'Alembertian
4581:
4577:
4572:
4570:
4566:
4562:
4558:
4539:
4534:
4524:
4516:
4513:
4508:
4493:
4481:
4465:
4456:
4451:
4448:
4438:
4437:
4436:
4422:
4419:
4411:
4398:
4394:
4391:
4381:
4373:
4371:
4367:
4363:
4359:
4339:
4336:
4327:
4324:
4321:
4311:
4310:
4309:
4305:
4300:
4295:
4281:
4278:
4265:
4261:
4253:
4249:
4245:
4225:
4220:
4212:
4209:
4198:
4190:
4187:
4184:
4176:
4166:
4165:
4164:
4162:
4158:
4154:
4151:, while on a
4150:
4146:
4142:
4138:
4134:
4124:
4106:
4102:
4095:
4091:
4083:
4079:
4065:
4062:
4055:
4051:
4038:
4034:
4024:
4018:
4015:
4011:
4007:
4004:
3994:
3993:
3992:
3990:
3986:
3981:
3979:
3960:
3955:
3945:
3937:
3934:
3924:
3923:
3922:
3920:
3901:
3896:
3893:
3885:
3870:
3867:
3863:
3857:
3854:
3850:
3846:
3843:
3833:
3832:
3831:
3807:
3803:
3799:
3794:
3790:
3783:
3770:
3765:
3762:
3759:
3755:
3751:
3745:
3739:
3728:
3727:
3726:
3704:
3684:
3681:
3662:
3659:
3648:
3647:
3646:
3644:
3637:
3630:
3611:
3604:
3600:
3592:
3588:
3574:
3571:
3564:
3560:
3547:
3543:
3534:
3526:
3522:
3518:
3513:
3509:
3502:
3492:
3487:
3484:
3476:
3459:
3458:
3457:
3456:are given by
3455:
3448:
3443:
3441:
3437:
3433:
3413:
3402:
3399:
3390:
3387:
3382:
3374:
3371:
3340:
3335:
3323:
3320:
3315:
3295:
3292:
3277:
3276:
3275:
3261:
3253:
3249:
3245:
3235:
3222:
3217:
3202:
3198:
3194:
3175:
3171:
3168:
3165:
3131:
3127:
3124:
3121:
3105:
3090:
3071:
3068:
3062:
3059:
3056:
3052:
3047:
3042:
3038:
3030:
3029:
3028:
3012:
3008:
2999:
2994:
2975:
2971:
2967:
2964:
2959:
2955:
2947:
2932:
2925:
2924:
2923:
2906:
2901:
2897:
2893:
2890:
2887:
2884:
2881:
2876:
2872:
2864:
2863:
2862:
2846:
2816:
2813:
2810:
2785:
2756:
2727:
2723:
2719:
2714:
2710:
2706:
2701:
2697:
2674:
2670:
2647:
2633:. In fact on
2618:
2588:
2585:
2579:
2576:
2573:
2569:
2564:
2559:
2555:
2547:
2546:
2545:
2529:
2525:
2504:
2484:
2461:
2458:
2455:
2449:
2426:
2423:
2420:
2417:
2414:
2408:
2405:
2402:
2396:
2393:
2390:
2384:
2381:
2378:
2372:
2369:
2362:
2361:
2360:
2358:
2355:. Assume the
2342:
2339:
2336:
2328:
2324:
2319:
2305:
2302:
2299:
2276:
2273:
2265:
2261:
2255:
2251:
2247:
2244:
2241:
2238:
2230:
2220:
2207:
2203:
2195:
2194:
2193:
2176:
2173:
2165:
2155:
2142:
2138:
2134:
2131:
2128:
2122:
2116:
2108:
2104:
2100:
2093:
2092:
2091:
2077:
2069:
2050:
2047:
2039:
2035:
2029:
2025:
2021:
2018:
2015:
2012:
2006:
2000:
1992:
1988:
1984:
1977:
1976:
1975:
1959:
1955:
1951:
1948:
1945:
1925:
1905:
1885:
1882:
1879:
1856:
1836:
1833:
1830:
1822:
1806:
1786:
1766:
1746:
1738:
1737:eigenfunction
1722:
1699:
1696:
1693:
1690:
1687:
1681:
1674:
1673:
1672:
1664:
1647:
1642:
1638:
1634:
1627:
1622:
1618:
1614:
1609:
1605:
1598:
1595:
1592:
1589:
1586:
1578:
1574:
1570:
1567:
1562:
1558:
1554:
1547:
1542:
1538:
1530:
1529:
1528:
1514:
1494:
1485:
1467:
1447:
1434:
1427:
1425:
1409:
1405:
1397:
1394:
1391:
1388:
1385:
1377:
1373:
1369:
1366:
1361:
1357:
1352:
1345:
1340:
1336:
1328:
1327:
1324:
1322:
1297:
1293:
1289:
1286:
1280:
1275:
1271:
1267:
1264:
1259:
1255:
1248:
1242:
1239:
1234:
1230:
1222:
1221:
1220:
1207:
1184:
1164:
1150:
1148:
1144:
1140:
1121:
1117:
1113:
1108:
1098:
1095:
1091:
1080:
1069:
1063:
1047:
1038:
1033:
1030:
1020:
1019:
1018:
1015:
1013:
1008:
1001:
996:
988:
987:metric tensor
984:
965:
960:
950:
947:
943:
939:
936:
931:
923:
918:
913:
909:
906:
903:
899:
890:
889:
888:
886:
879:
875:
871:
867:
860:
859:tangent space
856:
852:
825:
821:
811:
805:
802:
799:
791:
787:
783:
777:
771:
768:
765:
755:
754:
753:
736:
733:
730:
720:
719:inner product
716:
711:
709:
705:
685:
679:
675:
664:
653:
647:
631:
622:
617:
614:
611:
601:
600:
599:
597:
594:
590:
586:
562:
558:
552:
548:
544:
539:
535:
528:
525:
512:
511:
510:
496:
493:
470:
461:
459:
458:wedge product
443:
423:
420:
392:
388:
375:
370:
358:
357:
356:
355:to the frame
354:
350:
346:
342:
341:metric tensor
338:
334:
328:
321:
298:
294:
290:
287:
284:
281:
276:
272:
268:
257:
247:
242:
238:
230:
229:
228:
226:
222:
218:
214:
211:
207:
202:
201:is possible.
200:
181:
175:
153:
150:
140:
139:
138:
136:
132:
122:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
63:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
6668:
6645:
6642:Jost, Jürgen
6624:
6579:
6575:
6569:
6555:cite journal
6520:
6516:
6510:
6492:
6486:
6467:
6463:
6453:
6444:
6438:
6198:
6169:
5958:
5954:
5950:
5946:
5938:
5934:
5928:
5924:
5920:
5802:
5739:
5607:
5516:
5514:
5413:dimensional
5410:
5406:
5400:
5191:
5160:
4949:
4945:
4941:
4937:
4933:
4929:
4923:
4919:
4911:
4904:
4728:
4724:
4720:
4716:
4712:
4710:
4614:
4610:
4606:
4600:
4596:
4573:
4554:
4392:
4387:
4379:
4354:
4303:
4298:
4296:
4259:
4254:, acting as
4247:
4241:
4160:
4130:
4122:
3988:
3982:
3977:
3975:
3916:
3829:
3724:
3635:
3628:
3626:
3453:
3446:
3444:
3439:
3434:denotes the
3431:
3429:
3241:
3088:
3086:
2997:
2995:
2992:
2921:
2603:
2441:
2326:
2320:
2291:
2191:
2065:
1714:
1670:
1662:
1486:
1439:
1428:
1318:
1156:
1138:
1136:
1016:
1006:
999:
991:
982:
980:
881:
873:
869:
861:
854:
847:
845:
714:
712:
707:
701:
595:
593:vector field
581:
579:
462:
412:
351:forming the
344:
323:
319:
315:
224:
220:
205:
203:
196:
128:
102:
98:
94:
82:
78:
74:
70:
64:
40:submanifolds
31:
25:
3643:contraction
3201:CR manifold
1821:eigenspaces
1819:, i.e. the
337:determinant
217:volume form
6685:Categories
6617:References
6589:1608.01797
5194:− 2)
4603:− 1)
4559:, such as
4308:, so that
4301:, we have
4264:Hodge star
4157:hyperbolic
989:, so that
591:along the
353:dual frame
343:, and the
131:divergence
87:divergence
48:Riemannian
6675:EMS Press
6627:, Dover,
6606:119123242
6530:1007.5020
6380:θ
6376:∂
6366:∂
6355:−
6341:
6321:∂
6313:∂
6298:
6281:∂
6277:∂
6267:−
6253:
6241:θ
6213:Δ
6183:ξ
6179:Δ
6150:ξ
6146:Δ
6137:−
6123:
6103:∂
6095:∂
6084:−
6067:
6050:∂
6046:∂
6036:−
6019:
6007:ξ
5985:−
5973:Δ
5945:). Here
5885:∂
5875:∂
5869:−
5866:⋯
5863:−
5842:∂
5832:∂
5823:◻
5718:−
5653:◻
5632:−
5620:Δ
5549:∣
5482:−
5479:⋯
5476:−
5458:−
5369:θ
5365:∂
5355:∂
5344:−
5336:ϕ
5333:
5313:ϕ
5310:∂
5302:∂
5296:ϕ
5293:
5279:ϕ
5276:∂
5272:∂
5262:−
5254:ϕ
5251:
5236:ϕ
5230:θ
5208:Δ
5174:ξ
5170:Δ
5141:ξ
5137:Δ
5128:−
5120:ϕ
5117:
5097:ϕ
5094:∂
5086:∂
5075:−
5064:ϕ
5061:
5044:ϕ
5041:∂
5037:∂
5027:−
5016:ϕ
5013:
4998:ϕ
4992:ξ
4976:−
4964:Δ
4877:−
4865:Δ
4856:−
4834:∂
4826:∂
4815:−
4793:∂
4789:∂
4779:−
4762:Δ
4739:Δ
4666:Δ
4641:−
4629:Δ
4580:signature
4561:spherical
4531:∂
4521:∂
4505:∂
4478:∂
4446:Δ
4328:δ
4319:Δ
4282:δ
4213:δ
4191:δ
4185:δ
4174:Δ
4147:it is an
4075:∇
4070:∇
4066:−
4047:∇
4030:∇
4002:Δ
3952:∇
3942:∇
3932:Δ
3851:∑
3841:Δ
3776:∇
3756:∑
3737:Δ
3697:∞
3685:∈
3674:∇
3657:Δ
3584:∇
3579:∇
3575:−
3556:∇
3539:∇
3406:∇
3403:≡
3397:∇
3394:∇
3391:≡
3379:∇
3336:∗
3324:⊗
3316:∗
3300:Γ
3296:∈
3176:κ
3169:−
3132:κ
3125:−
3069:κ
3060:−
3039:λ
3027:one has,
3009:λ
2937:Δ
2933:−
2891:ϕ
2888:
2817:ϕ
2811:θ
2671:λ
2586:κ
2577:−
2565:≥
2556:λ
2526:λ
2462:⋅
2456:⋅
2418:κ
2394:κ
2391:≥
2373:
2340:≥
2303:≥
2300:λ
2252:∫
2248:λ
2218:∇
2204:∫
2153:∇
2139:∫
2114:Δ
2105:∫
2101:−
2026:∫
2022:λ
1998:Δ
1989:∫
1985:−
1883:≥
1880:λ
1860:Δ
1857:−
1831:λ
1807:λ
1767:λ
1747:λ
1694:λ
1685:Δ
1682:−
1632:Δ
1619:∫
1602:⟩
1584:⟨
1575:∫
1571:−
1552:Δ
1539:∫
1401:⟩
1383:⟨
1374:∫
1370:−
1350:Δ
1337:∫
1287:⋅
1284:∇
1272:∫
1268:−
1231:∫
1188:∇
1185:−
1105:∂
1060:∂
1028:Δ
957:∂
928:∂
907:
876:ƒ is the
872:. Here,
797:⟩
769:
763:⟨
740:⟩
737:⋅
731:⋅
728:⟨
644:∂
612:⋅
609:∇
526:⋅
523:∇
494:⋅
491:∇
444:∧
385:∂
381:∂
367:∂
288:∧
285:⋯
282:∧
173:∇
148:Δ
79:Laplacian
6644:(2002),
6408:See also
4936:. Here
4376:Examples
997:= δ
347:are the
210:oriented
135:gradient
91:gradient
81:) takes
5809:is the
4916:. Let
4585:is the
4250:is the
4143:. On a
3985:tensors
3195:(after
1735:is the
1314:(proof)
1147:density
857:in the
587:is the
456:is the
349:1-forms
339:of the
335:of the
331:is the
125:Details
89:of its
85:to the
6652:
6631:
6604:
6547:304301
6545:
6499:
6170:where
5161:where
4944:, and
4711:where
4159:. The
4155:it is
3978:tensor
3454:Hess f
3430:where
2442:where
2271:
2236:
2171:
2126:
2120:
2045:
2010:
2004:
1715:where
1004:with δ
981:where
580:where
316:where
208:is an
30:, the
6602:S2CID
6584:arXiv
6543:S2CID
6525:arXiv
6430:Notes
5740:Here
5608:Then
5515:Then
4578:with
4139:on a
2832:, on
2742:from
6650:ISBN
6629:ISBN
6561:link
6497:ISBN
6338:sinh
6295:sinh
6250:sinh
6120:sinh
6064:sinh
6016:sinh
5953:and
5583:>
3881:Hess
3498:Hess
3472:Hess
3445:Let
3367:Hess
3288:Hess
2421:>
1507:and
1460:and
1177:and
1010:the
904:grad
766:grad
436:and
58:and
50:and
6594:doi
6535:doi
6521:161
6472:doi
5937:of
5330:sin
5290:sin
5248:sin
5114:sin
5058:sin
5010:sin
4727:to
4613:on
4563:or
4395:on
4306:= 0
4258:on
3991:by
3917:In
2885:cos
2771:to
2370:Ric
1974:):
1956:vol
1639:vol
1606:vol
1559:vol
1406:vol
1358:vol
1294:vol
1256:vol
1145:(a
1137:If
559:vol
536:vol
239:vol
227:by
219:on
121:).
42:in
26:In
6687::
6673:,
6667:,
6600:.
6592:.
6580:34
6578:.
6557:}}
6553:{{
6541:.
6533:.
6519:.
6468:14
6466:.
6462:.
5927:,
4922:,
4719:/|
4589:.
4571:.
4372:.
4304:δf
3980:.
3663::=
3634:,
3442:.
3432:df
3375::=
3357:,
2318:.
1527:,
1014:.
994:jk
992:gg
545::=
460:.
376::=
345:dx
329:)|
326:ij
248::=
137::
62:.
6659:.
6608:.
6596::
6586::
6563:)
6549:.
6537::
6527::
6480:.
6474::
6392:f
6384:2
6370:2
6358:2
6351:)
6347:r
6344:(
6335:+
6331:)
6324:r
6316:f
6307:)
6304:r
6301:(
6291:(
6284:r
6270:1
6263:)
6259:r
6256:(
6247:=
6244:)
6238:,
6235:r
6232:(
6229:f
6222:2
6218:H
6199:n
6155:f
6140:2
6133:)
6129:t
6126:(
6117:+
6113:)
6106:t
6098:f
6087:2
6081:n
6077:)
6073:t
6070:(
6060:(
6053:t
6039:n
6033:2
6029:)
6025:t
6022:(
6013:=
6010:)
6004:,
6001:t
5998:(
5995:f
5988:1
5982:n
5978:H
5959:S
5955:ξ
5951:p
5947:t
5939:H
5935:p
5931:)
5929:ξ
5925:t
5923:(
5906:.
5898:2
5893:n
5889:x
5879:2
5855:2
5850:1
5846:x
5836:2
5826:=
5807:□
5803:f
5789:)
5784:2
5780:/
5776:1
5772:)
5768:x
5765:(
5762:q
5758:/
5754:x
5751:(
5748:f
5721:1
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5711:H
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5700:)
5694:2
5690:/
5686:1
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5678:x
5675:(
5672:q
5668:/
5664:x
5660:(
5656:f
5645:=
5642:f
5635:1
5629:n
5625:H
5592:.
5589:}
5586:1
5578:1
5574:x
5570:,
5567:1
5564:=
5561:)
5558:x
5555:(
5552:q
5546:x
5543:{
5540:=
5535:n
5531:H
5517:H
5500:.
5495:2
5490:n
5486:x
5471:2
5466:2
5462:x
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5448:1
5444:x
5440:=
5437:)
5434:x
5431:(
5428:q
5411:n
5407:H
5381:f
5373:2
5359:2
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5340:)
5327:(
5324:+
5320:)
5305:f
5286:(
5265:1
5258:)
5245:(
5242:=
5239:)
5233:,
5227:(
5224:f
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5213:S
5192:n
5190:(
5146:f
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5124:)
5111:(
5108:+
5104:)
5089:f
5078:2
5072:n
5068:)
5055:(
5051:(
5030:n
5024:2
5020:)
5007:(
5004:=
5001:)
4995:,
4989:(
4986:f
4979:1
4973:n
4969:S
4950:S
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4942:p
4938:ϕ
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4926:)
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4890:.
4887:f
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4852:r
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4844:)
4837:r
4829:f
4818:1
4812:n
4808:r
4803:(
4796:r
4782:n
4776:1
4772:r
4768:=
4765:f
4729:R
4725:f
4721:x
4717:x
4715:(
4713:f
4696:)
4692:|
4688:x
4684:|
4679:/
4675:x
4672:(
4669:f
4663:=
4660:)
4657:x
4654:(
4651:f
4644:1
4638:n
4634:S
4615:S
4611:f
4607:R
4601:n
4599:(
4540:f
4535:i
4525:i
4517:=
4514:f
4509:i
4498:|
4494:g
4490:|
4482:i
4470:|
4466:g
4462:|
4457:1
4452:=
4449:f
4423:1
4420:=
4416:|
4412:g
4408:|
4393:x
4340:.
4337:f
4333:d
4325:=
4322:f
4299:f
4279:+
4275:d
4260:k
4248:δ
4226:,
4221:2
4217:)
4210:+
4206:d
4202:(
4199:=
4195:d
4188:+
4181:d
4177:=
4107:)
4103:T
4096:j
4092:X
4084:i
4080:X
4063:T
4056:j
4052:X
4039:i
4035:X
4025:(
4019:j
4016:i
4012:g
4008:=
4005:T
3989:T
3961:f
3956:a
3946:a
3938:=
3935:f
3902:.
3897:j
3894:i
3890:)
3886:f
3876:(
3871:j
3868:i
3864:g
3858:j
3855:i
3847:=
3844:f
3826:,
3813:)
3808:i
3804:X
3800:,
3795:i
3791:X
3787:(
3784:f
3780:d
3771:n
3766:1
3763:=
3760:i
3752:=
3749:)
3746:x
3743:(
3740:f
3721:.
3708:)
3705:M
3702:(
3691:C
3682:f
3678:d
3670:r
3667:t
3660:f
3639:j
3636:X
3632:i
3629:X
3612:f
3605:j
3601:X
3593:i
3589:X
3572:f
3565:j
3561:X
3548:i
3544:X
3535:=
3532:)
3527:j
3523:X
3519:,
3514:i
3510:X
3506:(
3503:f
3493:=
3488:j
3485:i
3481:)
3477:f
3467:(
3450:i
3447:X
3440:f
3426:,
3414:f
3410:d
3400:f
3388:f
3383:2
3372:f
3344:)
3341:M
3330:T
3321:M
3310:T
3304:(
3293:f
3262:f
3223:.
3218:n
3213:C
3172:1
3166:n
3140:)
3128:1
3122:n
3113:(
3106:n
3101:S
3089:n
3072:,
3063:1
3057:n
3053:n
3048:=
3043:1
3013:1
2998:n
2976:1
2972:u
2968:2
2965:=
2960:1
2956:u
2948:2
2943:S
2907:,
2902:1
2898:u
2894:=
2882:=
2877:3
2873:x
2847:2
2842:S
2820:)
2814:,
2808:(
2786:2
2781:S
2757:3
2752:R
2728:3
2724:x
2720:,
2715:2
2711:x
2707:,
2702:1
2698:x
2675:1
2648:2
2643:S
2619:n
2614:S
2589:.
2580:1
2574:n
2570:n
2560:1
2530:1
2505:M
2485:X
2465:)
2459:,
2453:(
2450:g
2427:,
2424:0
2415:,
2412:)
2409:X
2406:,
2403:X
2400:(
2397:g
2388:)
2385:X
2382:,
2379:X
2376:(
2343:2
2337:n
2327:n
2306:0
2277:V
2274:d
2266:2
2262:u
2256:M
2245:=
2242:V
2239:d
2231:2
2226:|
2221:u
2214:|
2208:M
2177:V
2174:d
2166:2
2161:|
2156:u
2149:|
2143:M
2135:=
2132:V
2129:d
2123:u
2117:u
2109:M
2078:M
2051:V
2048:d
2040:2
2036:u
2030:M
2019:=
2016:V
2013:d
2007:u
2001:u
1993:M
1960:n
1952:=
1949:V
1946:d
1926:M
1906:u
1886:0
1837:0
1834:=
1787:M
1723:u
1700:,
1697:u
1691:=
1688:u
1648:.
1643:n
1635:f
1628:h
1623:M
1615:=
1610:n
1599:h
1596:d
1593:,
1590:f
1587:d
1579:M
1568:=
1563:n
1555:h
1548:f
1543:M
1515:h
1495:f
1482:2
1468:h
1448:f
1433:)
1431:2
1429:(
1410:n
1398:h
1395:d
1392:,
1389:f
1386:d
1378:M
1367:=
1362:n
1353:h
1346:f
1341:M
1298:n
1290:X
1281:f
1276:M
1265:=
1260:n
1252:)
1249:X
1246:(
1243:f
1240:d
1235:M
1208:f
1165:d
1139:M
1122:.
1118:)
1114:f
1109:j
1099:j
1096:i
1092:g
1085:|
1081:g
1077:|
1070:(
1064:i
1052:|
1048:g
1044:|
1039:1
1034:=
1031:f
1007:k
1000:k
983:g
966:f
961:j
951:j
948:i
944:g
940:=
937:f
932:i
924:=
919:i
914:)
910:f
900:(
884:x
882:v
874:d
870:x
866:M
864:x
862:T
855:x
850:x
848:v
831:)
826:x
822:v
818:(
815:)
812:x
809:(
806:f
803:d
800:=
792:x
788:v
784:,
781:)
778:x
775:(
772:f
734:,
715:f
708:i
686:)
680:i
676:X
669:|
665:g
661:|
654:(
648:i
636:|
632:g
628:|
623:1
618:=
615:X
596:X
584:X
582:L
563:n
553:X
549:L
540:n
532:)
529:X
520:(
497:X
471:X
424:M
421:T
393:i
389:x
371:i
324:g
320:g
318:|
299:n
295:x
291:d
277:1
273:x
269:d
262:|
258:g
254:|
243:n
225:x
221:M
206:M
182:.
179:)
176:f
170:(
165:v
162:i
159:d
154:=
151:f
103:R
99:f
95:n
83:f
75:R
71:f
23:.
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