Last edited by Mezizshura

Thursday, May 7, 2020 | History

3 edition of **Conformal transformations in complete Riemannian manifolds.** found in the catalog.

Conformal transformations in complete Riemannian manifolds.

Yoshihiro Tashiro

- 332 Want to read
- 17 Currently reading

Published
**1967**
by Study Group of Geometry] in [Okayama, Japan
.

Written in English

- Riemannian manifolds.,
- Conformal mapping.

**Edition Notes**

Bibliography: leaves 75-77.

Series | Publications of the Study Group of Geometry ;, v. 3 |

Classifications | |
---|---|

LC Classifications | QA646 .T28 |

The Physical Object | |

Pagination | iii, 77l. |

Number of Pages | 77 |

ID Numbers | |

Open Library | OL4232561M |

LC Control Number | 80513012 |

This theorem indicates that non-trivial conformal transformations of non-Riemannian Finsler manifolds may only exist between manifolds with different types of S-curvatures. However, Example in Section 3 means that conformal transformations may even not exist in some special : Bin Shen. One can construct other examples on warped Riemannian manifolds, see [6, Theorem 21]. Remark 2. By Theorem 1, light-line complete pseudo-Riemannian Einstein metrics of indefinite signature do not ad- mit nonhomothetic conformal complete vector fields. The Riemannian version of this result is due to Yano and Nagano [11].Cited by:

Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. 2 Affine Transformations of Riemannian Manifolds. 3 Cartan Connections. 4 Projective and Conformal Connections. Transformation Groups in Differential Geometry. One can construct other examples on warped Riemannian manifolds, see [6, Theorem 21]. Remark 2. By Theorem 1, a light-line complete pseudo-Riemannian Einstein metrics of indefinite signature do not admit nonhomothetic conformal complete vector fields. The Riemannian version of this result is due to Yano and Nagano [9].

curvature R are important in the study of conformal transformations of compact Riemaanniani manifolds. In this direction a basic result is due to Yamabe,1 namely, every Riemannian metric on a compact manifold Mn of dimension n > 2 can be deformed conformally to a Riemanniian rnetric of constant R. Lichnerowicz2. This decomposition is known as the Ricci decomposition, and plays an important role in the conformal geometry of Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of, then the Riemann curvature tensor changes to (seen as a (0, 4)-tensor).

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Nagano proved that if the non-homothetic conformal transformation between complete Riemannian manifolds with parallel Ricci tensor is admitted, then the manifolds are irreducible and isometric to a : Byung Hak Kim.

TASHIRO, Conformai transformations in complete Riemannian manifolds, Publ. Study Group of Geometry Vol. 3 () Google Scholar [T-M] Y. TASHIRO and K. MIYASHITA, On conformai diffeomorphisms of complete Riemannian manifolds with parallel Ricci tensor, J.

Cited by: from the concircular transformation introduced first by K. Yano [17]. A con-circular transformation is by definition a conformal transformation preserving geodesic circles.

Concircular scalar fields and transformations appear often in the theory of transformations in Riemannian manifolds, see for instance [3], [14], [15], [17], [18], [20]. Conformally flat homogeneous pseudo-Riemannian four-manifolds Calvaruso, Giovanni and Zaeim, Amirhesam, Tohoku Mathematical Journal, ; Conformality of Riemannian manifolds to spheres Amur, Krishna and Hegde, V.

S., Journal of Differential Geometry, ; Conformal transformations of Riemannian manifolds Obata, Morio, Journal of Differential Geometry, Cited by: CiteSeerX - Scientific documents that cite the following paper: The conjectures on conformal transformations of Riemannian manifolds. Theorems on conformal mappings of complete Riemannian manifolds 83 we obtain from () that ∆˙ = 0 (see our Lemma).

In this case, ˙ is a harmonic function. Since n 3, we see from () that ˙ is constant. In the other hand, if ˙ is a non-positive function such that s e2˙ s and ˙ 2 LP (M;g) for some 1.

To any pseudo-Riemannian n-manifold (M,g) with n ≥ 3 there is an associated conformal manifold (M,[g]) where [g] is the set of all metrics ˆg which are smooth positive multiples of the metric g. In Riemannian signature (p,q) = (n,0) passing to the conformal manifold means geometrically that we are forgetting the notion of.

We mainly follow M. Schottenloher’s book: A Mathematical Introduction to Conformal Field Theory (2nd ed.) Topics to be covered for Part-I are as follows (tentative): Semi-Riemannian manifolds, conformal transformations and conformal Killing fields, classification of conformal transformation, and conformal group.

([MS] Chapter )File Size: KB. Conformal manifolds. A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics g and h are equivalent if and only if.

h = λ 2 g, {\displaystyle h=\lambda ^ {2}g,} where λ is a real-valued smooth function defined on the manifold. The function is called the conformal factor. A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one.

For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map. Moreover, L Z g is the flow of the conformal Killing vector field Z which consists of the conformal transformation of the Riemannian manifold (M, g) and the flow of the conformal Killing vector.

Gaussian curvature under a conformal map. Ask Question Asked 1 year, 8 months ago. Volume of a complete, simply connected Riemannian manifold of constant negative curvature. Question about curvature calculation method in Lee's *Riemannian Manifolds* book. Conformal Transformations of Pseudo-Riemannian manifolds 5 Deﬁnition We call a vector ﬁeld V on a pseudo-Riemannian manifold closed if it is locally a gradient ﬁeld, i.e., if locally there exists a function f such that V = gradf Consequently, from Equation 3 we see that a closed vector ﬁeld V is conformal if and only ∇XV = σX (2)Cited by: Theorems on conformal mappings of complete Riemannian manifolds and their applications Article in Balkan Journal of Geometry and Its Applications 22(1) July with 52 Reads.

Conformal vector ﬂelds on a Riemannian manifold 87 Let (M;g) be an n-dimensional compact Riemannian manifold that admits a non-trivial conformal vector ﬂeld» with potential function denote by ‚1 the ﬂrst nonzero eigenvalue of the Laplacian operator ¢ acting on smooth functions of M and by Ric and S the Ricci tensor ﬂeld and the scalar curvature of M : Sharief Deshmukh.

Under conformal change, P transforms by adding, which is expressed in terms of r2. and (r!)2. If n = 3, the condition W = 0 is automatically true. There is another tensor, the Cotton tensor C, which plays the role of W.

Involves one more di erentiation. C is also relevant for n 4, as we will see. COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.

AN INTRODUCTION TO CONFORMAL GEOMETRY AND TRACTOR CALCULUS, WITH A VIEW TO APPLICATIONS IN Conformal transformations and conformal covariance 9 Conformal Transformations 9 Conformal compacti cation of pseudo-Riemannian manifolds 38 Asymptotic atness and conformal in nity in general relativity 38File Size: KB.

If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact [email protected] for [email protected] for Author: Kotaro Kawai.

Conformal Slant Riemannian Maps to Kähler Manifolds AKYOL, Mehmet Akif and ŞAHIN, Bayram, Tokyo Journal of Mathematics, ; Conformality of Riemannian manifolds to spheres Amur, Krishna and Hegde, V. S., Journal of Differential Geometry, ; Conformal transformations of Riemannian manifolds Obata, Morio, Journal of Differential Geometry, ; On a conformal transformation of a Cited by:.

] CONFORMALITY AND ISOMETRY OF RIEMANNIAN MANIFOLDS 67 For the case a^O, c — 4a2 = 0 and the case a = 0, c — 4a2 #0, Theorem I is due to Yano [10]. Theorem II. A manifold Mn is conformal to an n-sphere if it satisfies condition (C) and any one of the following three sets of conditions:Cited by: 6.manifolds: (I) a locally euclidean manifold, (II,A)a pseudo-hyperbolic space, (II,B) a hyperbolic space, (III) a spherical space, and (IV) a pseudo-euclidean spacei1).

As a consequence, we shall also have Theorem 4(8). If a complete Riemannian manifold M of dimension n ^ 3 admits a complete nonisometric concircular vector field, then M is a. [8] Kato, S., Conformai deformation to prescribed scalar curvature on complete non-compact Riemannian manifolds with nonpositive curvature, Tôhoku Math.

J., 45 (), – [9] Kazdan, J. and Warner, F., Scalar curvature and conformal deformation of Riemannian by: 6.