# Introduction to Riemannian Manifolds

@inproceedings{Craioveanu2001IntroductionTR, title={Introduction to Riemannian Manifolds}, author={Mircea Craioveanu and Mircea Puta and Themistocles M. Rassias}, year={2001} }

Let M (resp. N) be a connected. smooth (= C x ) n-dimensional manifold without boundary. We denote by C x (M) the ring of smooth real valued functions on M and by x(M) the Lie-algebra of all smooth vector fields on M. Recall that X ∈ x(M) is a smooth map
$$X:M \to TM = \mathop U\limits_{x \in M} {T_x}M$$
such that X (x) = X x , ∈ T x M (= the tangent space of Mat x) for each x ∈ M. T x M may be characterized as the space of all derivations of the algebra of smooth real valued functions…

## 113 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

Isometry groups of Riemannian solvmanifolds

- Mathematics
- 1988

A simply connected solvable Lie group R together with a leftinvariant Riemannian metric g is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds (R,g) and (R',g') may be…

Riemannian Geometry

- Nature
- 1927

THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's…

Differential geometry of geodesic spheres.

- Mathematics
- 1981

This work has originated mainly from two open problems which seem to be very difficult to answer. The first problem is in the theory of harmonic manifolds. The only known examples of such manifolds…

Geometry and Spectra of Compact Riemann Surfaces

- Mathematics
- 1992

Preface.-Chapter 1: Hyperbolic Structures.-Chapter 2: Trigonometry.- Chapter 3: Y-Pieces and Twist Parameters.- Chapter 4:The Collar Theorem.- Chapter 5: Bers' Constant and the Hairy Torus.- Chapter…

The Geometry of Discrete Groups

- Mathematics
- 1995

Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean…

The spectrum of the Laplacian on Riemannian Heisenberg manifolds.

- Mathematics
- 1986

On etudie le spectre du laplacien sur des varietes de Heisenberg riemanniennes compactes, des varietes de la forme (Γ\H n , g) ou H n est le groupe de Heisenberg a (2n+1) dimensions, Γ est un…

Semi-Riemannian Geometry With Applications to Relativity

- Mathematics
- 1983

Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries.…

Elliptic Operators, Topology and Asymptotic Methods

- Mathematics
- 1988

Resume of Riemannian Geometry Connections Riemannian Geometry Differential Forms Exercises Connection, Curvature, and Characteristic Classes Principal Bundles and their Connections Characteristic…

Geometry of Manifolds

- Mathematics
- 1964

Manifolds Lie groups Fibre bundles Differential forms Connexions Affine connexions Riemannian manifolds Geodesics and complete Riemannian manifolds Riemannian curvature Immersions and the second…

Riemannian Geometry: A Modern Introduction

- Mathematics
- 1994

1. Riemannian manifolds 2. Riemannian curvature 3. Riemannian volume 4. Riemannian coverings 5. Surfaces 6. Isoperimetric inequalities (constant curvature) 7. The kinetic density 8. Isoperimetric…