Lecture 1: Smooth manifolds - deﬁnition and examples; smooth functions, bump functions (smooth urysohn lemma); tangents vectors, vector ﬁelds, tensor ﬁelds - deﬁnition and properties (of tensors).
Lecture 2: Metric tensor, Riemannian manifolds, covariant diﬀerentiation, curvature tensors and curvatures; lengths of curves, distance function, geodesics, parallel transport and exponential map.
Lecture 3: Hopf-Rinow theorem.
Lecture 4: First and second variations of length and energy functionals; Jacobi ﬁelds, Gauss lemma.
Lecture 5: Cartan-Hadamard and Bonnet-Myers theorems.
Lecture 6: Models of constant curvature; Cartan’s theorem on the determination of the metric by (constant) curvatures.
Lecture 7: Rauch and Toponogov comparison theorems (include proof of Rauch but only state Toponogov).
Lectures 8, 9 and 10: Klingenberg’s injectivity radius estimate, Synge’s theorem, Bishop and Bishop-Gromov volume comparison theorems.
Lectures 11, 12, 13 and 14: Preissmann (and Flat-torus) theorems, Eberlein-O’Neill compactiﬁcation, Busemann functions, classiﬁcation of isometries (into elliptic, parabolic and axial or hyperbolic).
Lectures 15 and 16: Riemannian immersions, submersions; immersion and submersion equations; second fundamental form.
Lectures 17 and 18: Symmetric spaces of compact and noncompact type; their curvatures.

Shankar (KS) will give 4 lectures on the Soul theorem, the splitting the- orem, structure of fundamental groups in non-negative and positive curva- ture. About 4 lectures on Bochner technique (BT) (2 by KV and 2 by HS), 4 lectures on Ricci flow (2 by CSA and 2 by HS) and about 4 lectures on Gromov-Hausdorff convergence, by SG.

Besides the above main series lectures, we will have two guest lectures by T N Venkataramana on the construction of compact and finite-volume quotients of real hyperbolic space forms, one guest lecture by Gautam Bharali on uniformization of surfaces and one guest lecture by K Shankar.

Advanced Training in Mathematics Schools