By Detlef Laugwitz (Auth.)

Best differential geometry books

Differential Geometry: Theory and Applications (Contemporary Applied Mathematics)

This e-book supplies the fundamental notions of differential geometry, akin to the metric tensor, the Riemann curvature tensor, the basic sorts of a floor, covariant derivatives, and the elemental theorem of floor thought in a self-contained and available demeanour. even supposing the sector is frequently thought of a classical one, it has lately been rejuvenated, because of the manifold functions the place it performs a vital position.

Compactifications of Symmetric and Locally Symmetric Spaces (Mathematics: Theory & Applications)

Introduces uniform structures of lots of the identified compactifications of symmetric and in the community symmetric areas, with emphasis on their geometric and topological constructions quite self-contained reference aimed toward graduate scholars and study mathematicians attracted to the purposes of Lie thought and illustration thought to research, quantity conception, algebraic geometry and algebraic topology

An Introduction to Multivariable Analysis from Vector to Manifold

Multivariable research is a crucial topic for mathematicians, either natural and utilized. except mathematicians, we think that physicists, mechanical engi­ neers, electric engineers, structures engineers, mathematical biologists, mathemati­ cal economists, and statisticians engaged in multivariate research will locate this e-book tremendous necessary.

Additional info for Differential and Riemannian Geometry

Sample text

For every point x(u ) of a surface and for every direction x on the surface through that point, there is one and only one geodesic through the given point in the given direction. Proof. Let the direction on the surface be given by its unit vector ο = v x with υ = g (u )v v = 1. 16) of differential equations, 2 % it % % k ik «*"= - rj (u')ui'u ' k k with the initial conditions w*(0) = u\ u (0) = v\ % From the existence and uniqueness theorem for systems of ordinary differential equations, it follows that there exists one and only one solution u\s) satisfying the initial conditions.

Which leave x, xv x2 invariant, thus t h a t F cannot explicitly depend on the w*. translation in space, χ varies but the Thus F = F(x, xv x2). do not ; hence F = F(xv x2). Now recognizing under a Finally, be means of a suitable parameter transformation the Xi can be transformed into any other pair of vectors. 5 The Curvature of Curves on the Surface Having made use of arc length, we shall next exploit the curvature of curves on the surface in a similar way for studying the geometry of the surface.

Then the angle between v and w is constant along the curve, and the vectors v (t) are all of equal length. x x x f % % 50 2. L O C A L D I F F E R E N T I A L G E O M E T R Y O F S U R F A C E S Proof. 4). This proves first that lengths remain constant (apply the last equation to the special case v = w ) and then that angles remain constant, as was to be proved. % % F I G . 18. Parallel vector field on the sphere. We re-emphasize that geodesic parallelism of vectors is only defined with respect to (or along) a curve.