The problem of a rigorous theory of singularities in space-times with tor-sion is addressed. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors have done, because their definition of geodesics only involves the Christoffel connection, though studying theories with torsion. We propose a preliminary definition of singularities...

It has been shown by Mabuchi ([18]), Semmes ([20]) and Donaldson ([11]) that the space of Kähler metrics cohomologous to a fixed one has a riemannian structure of infinite dimensional symmetric space of negative curvature. Being its dimension not finite, the usual properties about existence, uniqueness and regularity for geodesics do not hold a priori. They appear crucial in a number of applica...

We specify an angular motion on geodesics to reduce the problem to the case of radial motion elaborated in previous chapters. An appropriate value of entropy for a charged and rotating black hole is obtained by calculating the partition function on thermal geodesics confined under horizons. The quantum aggregation is classified in a similar way to the Reissner–Nordstrøm black hole. A rotation o...

1. Smooth Manifolds: 9/22/14 1 2. Going on a Tangent (Space): 9/24/14 1 3. Immersions and Embeddings: 9/26/14 3 4. Riemannian Metrics: 9/29/14 3 5. Indiana Jones and the Isometries of the Upper Half-Plane: 10/1/14 5 6. That's an Affine Connection You've Got There: 10/3/14 7 7. Missed (Levi-Civita) Connections: 10/6/14 9 8. The Levi-Civita Connection and the Induced Metric: 10/8/14 9 9. Geodesic...

Radial geodesics as a microscopic origin of black hole entropy. Abstract. The entropy of charged black hole is calculated by using the partition function evaluated at radial geodesics confined under horizons. We establish two quantum phase states inside the black hole and a transition between them. The second chapter of present paper is devoted to consequent launching the constructed vehicle to...

We prove that any real doubly periodic geodesic on an n dimensional ellipsoid with distinct semiaxes and caustic parameters is uniquely associated to a real hyperelliptic tangential cover and that the following density property holds: given a real closed geodesic on the ellipsoid Q = {X2 1/a1 + · · ·+X2 n+1/an+1 = 1} with caustic parameters cj, j = 1, . . . , n − 1, for any ǫ > 0 there exist (a...

We specify an angular motion on geodesics to reduce the problem to the case of radial motion elaborated in previous chapters. An appropriate value of entropy for a charged and rotating black hole is obtained by calculating the partition function on thermal geodesics confined under horizons. The quantum aggregation is classified in a similar way to the Reissner–Nordstrøm black hole. A rotation o...

We show that there are uncountably many geodesics between any two non-isometric $n$-dimensional normed spaces. construct explicit can be used to describe all the points of other geodesics.

A general framework is laid out for principal component analysis (PCA) on quotient spaces that result from an isometric Lie group action on a complete Riemannian manifold. If the quotient is a manifold, geodesics on the quotient can be lifted to horizontal geodesics on the original manifold. Thus, PCA on a manifold quotient can be pulled back to the original manifold. In general, however, the q...

It is very hard to find computable examples of geodesics in the space of Kaehler metrics. It seems that they come in two types: (i) geodesics of toric Kaehler metrics, the topic of Lecture 2; (iii) geodesics coming from special test configurations and Hele-Shaw flows. Besides Donaldson’s construction (reviewed in §1) there are not many examples of geodesic rays. Almost the only other explicit e...