DEPARTMENT OF MATHEMATICS

SyllabusMaster of Philosophy Programme

in Mathematics

2013

DEPARTMENT OF MATHEMATICS

Master of Philosophy Programme in Mathematics

Course Overview

The Master of Philosophy Program in Mathematics is being offered based on a creditsystem similar to the other programmes offered by the University. The M. Phil. programhas two semesters with each semester spreading through 15 weeks. The candidate has tosubmit the dissertation within two months after the second semester ends. It can beextended by two more months on specific request. Those who fail to submit thedissertation within the extended time period can register for a further extension of fourmore months with a payment of 50% of the course fee. There will be only two repeatchances (within three years after registration) for the course work papers and norevaluation of papers at any stage of the program. The time taken from the admission tillthe submission of the dissertation shall be considered as the duration of the M. Phil.Program.

This programme is aimed at developing students into mature researchers andpreparing them for higher research degree and teaching.

Department Goal:

To provide all students with training in mathematics that will serve as part offoundation for research and teaching.

Course Content

The course content for the first semester is

Marks credits Duration1) Research Methodology 100 4 60 hrs

The course content for the second semester is

Marks credits Duration

1) Paper 1 100 3 45 hrs2) Paper 2 100 3 45 hrs

Dissertation & Viva-Voce 200 10

The Dissertation work includes presentation on the project proposal (50 Marks),double valuation of the final dissertation (100 marks), and viva-voce examination(50 Marks).

Assessment of course work

Each paper of the semester will be assessed upon 100 marks (Continuous InternalAssessment (CIA) – 45 marks, Attendance –5 marks and End Semester Examination - 50marks).

The internal assessment (comprising various components such as seminar, literaturesurvey, presentation, class test and so on) will be done periodically and the CIA markswill be sent to the Controller of Examination through Dean of Science and a copyforwarded to the General Research Coordinator as per the following guidelines:

CIA 1: 10 Marks assessmentCIA 2: 10 Marks assessmentCIA 3: 25 Marks assessment

The end semester examination will be conducted at the end of second semester forResearch Methodology and for each elective paper. The Maximum marks for each paperwill be 100 and the duration is three hours.

Two sets of independent question papers for each subject, completely sealed, willbe sentto the COE through the Coordinator. The question paper should reach the coordinator atleast 15 days in advance.

There is no minimum mark required for CIA. The minimum mark to pass in ESE for eachpaper is 50%. The minimum mark to pass in each paper is 50% aggregate of CIA marksand ESE marks.

In case a candidate fails due to low marks in CIA, he/she can re-register for that subjectwith a payment of required fee and complete the CIA requirements, by attending theclasses along with the other candidates as directed by the coordinator.

If a candidate fails due to low marks in ESE, he/she can appear for the subject by payingthe prescribed fee, along with the other candidates as scheduled by the office ofexaminations.

Each candidate shall work under the supervision of a guide. Specific guiding for theresearch program/Dissertation may commence from the middle of first semester. TheHOD/Coordinator will allot guides to the candidates approved by the Dean in the middleof the first semester depending upon the area of specialization.

Submission of Dissertation

The title page of dissertation, contents etc. should strictly conform to the format asprescribed by the university and the dissertation (all copies) should carry a declaration bythe candidate and certificate duly signed and issued by the guide. The dissertation shouldbe hard bound. The candidate shall submit five hard bound copies and a soft copy (CD)of his/her dissertation work for assessment.

Adjudication of the M. Phil Dissertation

The dissertation submitted by the candidate under the guidance of the guide will beassessed by two experts (one internal and one external).

The candidates also have to appear for final viva-voce. Assessment based on the viva-voce and the dissertation, along with the assessment of theory papers of both I & IIsemesters will be considered to declare the results.

The candidates will be provided with marks card and a degree certificate. The gradepoints and the class obtained will be entered along with the marks on the marks card.

Cancellation M.Phil. Admission

The admission of the candidate will be cancelled under the following circumstances.

1) Fails to secure 85% attendance.2) Fails to submit the documents/ requirements related to internal assessment.3) Does not pay the course fee within the stipulated time.4) Fails to submit the dissertation within the stipulated time.

Course structure

I Semester

II Semester

(Electives: Choose any one – each elective has two papers)

Paper Code Subjects No. ofHours

Marks Credit

Elective 1. Fluid Mechanics

RMT 261 1. Differential Equations andComputational Methods

45 100 3

RMT 262 2. Advanced Fluid Mechanics 4 100 3

Elective 2. Graph Theory

RMT 263 1. Computational Graph Theory 45 100 3

RMT 264 2. Advanced Graph Theory 45 100 3

Elective 3. Number Theory

RMT 265 1. Advanced Number Theory 45 100 3

RMT 266 2. Algebraic Number Theory 45 100 3

Elective 4. Riemannian Geometry

RMT 267 1. Submanifolds of RiemannianManifolds

45 100 3

RMT 268 2. Riemannian Geometry 45 100 3

Paper Code Subjects Hrs. /week

Marks Credit

RMT 131 Research Methodology 4 100 4

DISSERTATION

Components Marks Credit

Presentation on the researchproposal

50

10Double valuation of thedissertation

100

Viva-Voce examination 50

Total 200

Modular Objective:

SEMESTER I: (General papers)

In the first semester, students are offered two theory papers, viz, General ResearchMethodology and Specific Research Methodology (Mathematical analysis). The modularobjectives of these papers are :

RMT 131: Research Methodology

The research methodology module is intended to assist students in planning andcarrying out research projects. The students are exposed to the principles, procedures andtechniques of implementing a research project.

SEMESTER II: (Electives)

In the second semester, three elective papers are included. Students are supposed tochoose any one of the three electives. Each elective has two papers. The modularobjectives of each elective paper are :

Elective 1. Fluid Mechanics

Paper 1:RMT 261: Differential Equations and Computational Methods

This course will focus on advanced concepts in both ordinary and partialdifferential equations. Special emphasis is given to the computational methods.

Paper 2:RMT 262: Advanced Fluid Mechanics

This paper provides a opportunity to explore various instability problems in fluidmechanics such as Rayleigh-Benard instability, Marangoni instability, Double Diffusiveconvection, convection in porous media and convective instability in non-Newtonianfluids.

Elective 2. Graph Theory

Paper1:RMT 263: Computational Graph Theory

This paper concerns the principal concepts of graph theory such as enumeration

problems, networks, spectral graphs and graph complexity. This paper could be a usefultool for technical research in the area of Graph Theory.

Paper2:RMT 264: Advanced Graph Theory

This paper covers advanced topics in graph theory such as optimization of vertexand edge coloring, matching algorithm, four color conjecture, domination theory and soforth. This paper provides the foundations for advanced research in Graph Theory.

Elective 3. Number theory

Paper1:RMT 265: Advanced Number Theory

This paper focuses on the study of the Perfect Numbers, Mersenne Primes,Amicable Numbers, Fermat Numbers and also Major Contributions of SrinivasaRamanujan, Hardy, Dickson, Erdos and Andrew Wiles.

Paper2:RMT 266: Algebraic Number Theory

This paper focuses on Number Theory and Algebra, Groups, Rings and Fields,Ternary Quadratic Forms, Rational Points on Curves, Polynomials, Algebraic Numbers,Algebraic Number Fields.

Elective 4. Riemannian Geometry

Paper1:RMT 267: Submanifolds of Riemannian Manifolds

This paper focuses on the study of the submanifolds of Riemannian manifolds andhence on hypersurfaces and various other types of submanifolds.

Paper2:RMT 268: Riemannian Geometry

This paper focuses on differentiable manifolds and hence on Riemannian manifolds,a central concept which plays an important role in several different mathematical and/orscientific specialized areas.

DISSERTATION

Major emphasis is given to the dissertation work on a chosen research problem. Themodular objective includes research proposal, presentations on the research work done,submission of dissertation and viva-voce examination. The publications of the researchwork in refereed journals and presentation of research work in national/internationalconferences/symposia/seminars will be encouraged.

Course Curriculum

Semester - I (General papers)

Paper 1: RMT 131GENERAL RESEARCH METHODOLOGY

Unit I 15 Hours

Research methodology: An introduction –meaning of research-objectives of research-motivation in research –types of research- research approaches-significance of research-research methods versus methodology-research and scientific method-importance ofknowing how research done-research processes-criteria of good research-definingresearch problem-selecting the problem-necessity of defining the problem-techniqueinvolved in defining a problem-Research design- meaning of research design-need forresearch design-features of good design-different research design-basic principles ofexperimental design

Unit II 15 Hours

Sampling Design: Measurement and Scaling Techniques- Methods of Data Collection, -processing and Analysis of Data,- Sampling Fundamentals, Testing of Hypotheses - I(Parametric or Standard Tests of Hypotheses), Chi-square Test, Analysis of Variance andCovariance, Testing of Hypotheses - II (Nonparametric or Distribution - FreeTest),Multivariate Analysis Techniques

Unit III 15 Hours

Report Writing and Presentation: Interpretation and report writing, technique of reportwriting-precaution in interpretation-significance- different steps of report writing- layoutof research report-oral presentation- mechanics of writing- Exposure to writing tools likeLatex/PDF, Camera Ready Preparation

Unit IV 15 Hours

Role of Scholar, Supervisor and Computer: Originality in research, resources forresearch, Research skills, Time management, Role of supervisor and Scholar, Interactionwith subject expert, The Computer: Its Role in Research, Case study interpretation:minimum 5 case studies

Texts and References:

1. C.R.Kothari, Research Methodology- Methods and Techniques, II edition, Vishwa

Prakashan Publications, New Delhi,2006

2. R.Pannerselvam, Research methodology, PHI 2006.

3. Santosh Gupta, Methodology And Statistical Techniques,(Hardcover - 2000), ISBN-

3:9788171005017, 978-8171005017, Deep & Deep Publications

4. E. B. Wilson Jr: An Introduction to scientific research, Dover publications, Inc. New York

1990.

5. Ram Ahuja: Research Methods, Rawat Publications, New Delhi 2002.

6. Gopal Lal Jain: Research Methodology, Mangal Deep Publications, Jaipur 2003.

7. B. C. Nakra and K. K. Chaudhry: Instrumentation, measurement and analysis, TMH

publishing Co. Ltd., New Delhi 1985.

8. S. L. Mayers, Data analysis for Scientists, John Wiley & Sons, 1976.

9. Horaine Blaxter, Christina Hughes, Malcolm Tight How to research, Viva Books Pvt Ltd,

1999

10. J. Bell, Doing your research project, Viva, 1999 (NIAS)

11. A. Thomas, Finding our fast, Vistaar, 1998 (NIAS)

12. P.J.M. Costello, Action research, Continuum, 2005 (NIAS)

13. B. Gilham, Case study research methods, Continuum 2005 (NIAS)

14. S. Kleinman, Emotions and fieldwork, Sage Pub., 1993 (NIAS)

15. I. Gregory, Ethics in research, Continuum, 2005 (NIAS)

16. J. Bennet, Evaluation methods in research, Continuum, 2005 (NIAS)

17. D. L. Morgan, Focus groups as qualitative research, Sage Pub., 1988 (NIAS)

18. Illingham,Jo., Giving presentations, OUP, 2003 (NIAS)

19. M. Denscombe, The good research guide, Viva, 1999 (NIAS)

20. L. Blaxter, How to research, Viva, 2002 (NIAS)

21. D. Ezzy, Qualitative analysis, Routledge, 2002 (NIAS)

22. M. Q. Patton, Qualitative evaluation and research methods, Sage Pub, 1990 (NIAS)

23. J. Kirk, Reliability and validity in qualitative research, Sage Pub, 1986 (NIAS)

Elective I : FLUID MECHANICS

Paper 1: RMT 261DIFFERENTIAL EQUATIONS AND

COMPUTATIONAL METHODS

Unit I: 15 HoursNon Linear Ordinary Differential Equations: Autonomous systems. Thephase plane and its phenomena. Types of critical points. Stability: Criticalpoints and stability for linear systems. Stability of Liapounov’s directmethod. Simple critical points and non-linear systems. Periodic solutions.The poincare Dendixson theorem. Methods of solving the non-lineardifferential equations.

Unit II: 15 HoursComputational Methods: Variational method of Galerkin and RayleighRitz,Homotropy method, Perturbation method, Shooting method, Tanh methodand Modern methods of solving linear and non-linear partial differentialequations. Variational formulation of Boundary value problem.

Unit III: 15 HoursIntroduction to Mathematica Software:Numerical Computation: Numerical solution of differential equations,numerical solution of initial and boundary value problems, numericalintegration, numerical differentiation, matrix manipulations, optimizationtechniques.Graphics: Two- and three-dimensional plots, parametric plots, typesettingcapabilities for labels and text in plots, direct control of final graphics size,resolution etc.

Texts and References:

1. George F. Simmons, Differential Equations with Applications andHistorical Notes, Tata McGraw Hill, 2003.

2. Paul DuChateau and David Zachmann, Partial Differential Equations,Schaum Series, 1986.

3. Carnahan, Luther and Wilkes, Applied Numerical methods, John Wileyand Sons, 1969.

4. Jain, Iyengar and Jain, Numerical methods for Scientific and Engineeringcomputations, Wiley Eastern, 2007.

5. Shijum Liao, Beyond Perturbation, Chapman & Hall / CRC, 2008.

6. Samuel D. Conte & Carl de Boor, Elementary Numerical Analysis, TataMcGraw-Hill, 2005.

7. Tuncer Cebeci & Peter Bradshaw, Physical and Computational Aspect ofConvective Heart Transfer, Springer-Verlag, 2005.

Paper 2 : RMT 262ADVANCED FLUID MECHANICS

Unit I: 15 HoursDimensional Analysis and Similarity: Non-dimensional parametersdetermined from differential equations – Buckingham’s Pi Theorem –Nondimensionalization of the Basic Equations - Non-demensionalparameters and dynamic similarity.Shear Instability: Stability of flow between parallel shear flows - Squire’stheorem for viscous and inviscid theory – Rayleigh stability equation –Derivation of Orr-Sommerfeld equation assuming that the basic flow isstrictly parallel.

Unit II: 15 HoursThermal Instability: Basic concepts of stability theory – Linear and Non-linear theories – Rayleigh Benard Problem – Analysis into normal modes –Principle of Exchange of stabilities – first variation principle – Differentboundary conditions on velocity and temperature – Equations ofhydrodynamics in rotating frame of reference and with Magnetic field.Convective Instability with Surface Tension. Different models to studyconvection problems in porous media - – Principle of exchange of stabilities.Double Diffusive convection.

Unit III: 15 HoursNon – Newtonian Fluids: Constitutive equations of Maxwell, Oldroyd,Ostwald , Ostwald de waele, Reiner – Rivlin and Micropolar fluid.Weissenberg effect and Tom’s effect. Equation of continuity, Conservationof momentum for non-Newtonian fluids. Thermal and Solute concentrationstransport. Boundary conditions on velocity, temperature and concentration.

Assignment Problems:1. Rayleigh Benard problem with rotation for free-free velocity and

isopthermal temperature boundaries.2. Rayleigh Benard problem with magnetic field for free-free velocity

and isopthermal temperature boundaries.

3. Rayleigh – Benard – Marangoni Problem - with rotation / Magneticfield.

4. Rayleigh - Benard convection in Jeffery fluids with free-free andisothermal boundaries.

5. Rayleigh - Benard convection in Micropolar fluids with free-free andisothermal boundaries.

Texts and References :

1. S. Chardrasekhar, Hydrodynamic and hydrodmagnetic stability,Oxford University Press, 2007 (RePrint).

2. Drazin and Reid, Hydrodynamic instability, Cambridge UniversityPress, 2006.

3. J.S. Turner, Buoyancy Effects in Fluids, Cambridge University Press,1973.

4. D.J. Tritton, Physical fluid Dynamics, Van Nostrand ReinholdCompany, England, 1979.

5. P.L. Bhatnagar, A Lecture course on Non-Newtonian fluids, LectureNotes, Department of Applied Mathematics, I. I. Sc., Bangalore.

6. Drazin, Introduction to Hydrodynamic Stability, CambridgeUniversity Press, 2006.

7. Pijush Kundu and Cohen, Fluid Mechanics, Elsevier, 2010.8. M. Frank White, Fluid Mechanics, Tata Mcgraw Hill. 2010.

Research Papers:

1. C. M. Vest and V. S. Arpaci, Over stability of viscous elastic fluid layerheated from below. J. F. M. , Vol. 36, Part III, p-613 (1969)

2. P. G. Siddheshwar and S. Pranesh, Magnetoconvection in a micropolarfluid. Int. J. Engg. Sci., 36, 1173-1181, (1998).

Elective II: GRAPH THEORY

Paper 1: RMT 263COMPUTATIONAL GRAPH THEORY

Unit I: 15 hoursAutomorphisms, Vertex Transitivity, Graph Invariants, Hypercubes,Isometric Subgraphs, Median GraphsIntroduction to Product of Graphs, Commutativity, Associativety andMultiple Factors, Projections and Layers, Classification of Factors.

Unit II: 15 hoursThe Cartesian Product, Prime Factor Decompositions, Cartesian Product andIts Group, Transitive Group Action on Products, Cancellation, S-PrimeGraphsThe Strong Product, Basic Properties and S-Thin Graphs, Cliques and theExtraction of Complete Factors, Unique Prime Factorization for ConnectedGraphs, Automorophisms.

Unit III: 15 hoursThe Direct Product, Nonuniqueness of Prime Factorization, R-Thin Graphs,The Cartesian Skeleton, Factoring Connected, Nonbipartite, R-Thin Graphs,Factoring Connected Nonbipartite Graphs, Automorphisms, Applications tothe Strong Product.The Lexicographic Product, Basic Properties, Self-Complementarity andCancellation Properties, Commutativety, Factorizations and Nonuniqueness,Automorphisms.

Textbook:R. Hammack, W. Imrich and S. Klavzar, Handbook of Product Graphs,2nd

ed. New York: CRC Press, 2011.

References:

1. B. Bollobás, Modern Graph Theory. New York: Springer-Verlag, 1998.

2. C. Berge, Graphs. Oxford: North-Holland, 1985

3. G. Chartrand, Introductory Graph Theory. New York: Dover Publications,1985.

4. J.A. Bondy, and U.S.R. Murty, Graph Theory with Applications. Oxford:North-Holland, 1976.

5. O. Øre, Theory of Graphs. New York: AMS Coloquium Publications 38,1962.

6. R. Diestel, Graph Theory. New York: Springer-Verlag, 19977. R.J. Gould, Graph Theory. San Francisco: Benjamin/Cummings, 19888. R.J. Trudeau, Introduction to Graph Theory. New York: Dover

Publications, 1994.9. R.J. Wilson, and J.J. Watkins, Graphs: An Introductory Approach. John

Wiley & Sons, 1990

Paper 2: RMT 264ADVANCED GRAPH THEORY

Unit I: 15 hoursDomination in Graphs, Bounds in terms of Order, Bounds in terms of Order,Degree and Packing, Bounds in terms of Order and Size, Bounds in terms ofDegree, Diameter and Girth, Bounds in terms of Independence and Covering

Unit II: 15 hoursT-Colourings, L(2,1)-colourings, Radio Colourings, Hamiltonian Colourings,Domination and Colourings

Unit III: 15 hoursAcyclic Digraphs, Multipartite Digraphs and Extended Digraphs, TransitiveDigraphs, Line Digraphs, Series-Parallel Digraphs, Quasi-TransitiveDigraphs, Path-Mergeable Digrpahs, Locally Semicomplete Digrraphs,Totally φ-Decomposable Digraphs, Planar Digraphs

Textbooks:1. T.W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of

Domination in Graphs. New York: Marcel Dekker, Inc., 1998.2. G. Chartrand and P. Zhang, Chromatic Graph Theory. New York:

CRC Press, 2009.3. J. Bang-Jensen and G. Gutin, Digraphs. London: Springer, 2009.

References:1. B. Bollobas, Graph Theory: An Introductory Course. New York: Springer,

1979.2. D.B. West, Introduction to Graph Theory. New Jersey: Prentice Hall,

1996.3. F. Harary, Graph Theory. Boston: Addison-Wesley, 1969.4. J. Clark, and D.A. Holton, A first look at Graph Theory. Singapore: World

Scientific, 1995.5. J.L. Gross, and J. Yellen. Graph Theory and its Applications, 2nd ed.

London: Chapman and Hall/CRC, 1998

6. K.R. Parthasarathy, Basic Graph Theory. New Delhi: ata McGraw-Hill,1994.

7. M. Capobianco, and J.C. Molluzzo, Examples and Counterexamples inGraph Theory. Oxford: North-Holland, 1978.

8. R.J. Trudeau, Dots and Lines. Ohio: Kent State University Press, 1976.9. R.J. Wilson, Introduction to Graph Theory. Boston: Addison Wesley

Longman, 1996.

Elective III : NUMBER THEORY

Paper 1: RMT 265ADVANCED NUMBER THEORY

Unit I: 15 hoursThe Order of an Integer Modulo n, Primitive Roots for Primes, CompositeNumbers having Primitive Roots, The Theory of Indices, Perfect Numbers,Mersenne Primes, Amicable Numbers, Fermat Numbers.

Unit II: 15 hoursThe Equation x2+y2=z2, The Fibonacci Sequence, Some Identities involvingFibonacci Numbers, Golden Ratio, Finite Continued Fractions, InfiniteContinued Fractions, Pell’s Equation.

Unit III: 15 hoursMajor Contributions of Srinivasa Ramanujan, Hardy, Dickson, Erdos andAndrew Wiles, Primality Testing and Factorization, An Application toFactoring: Remote Coin Flipping, The Prime Number Theorem and ZetaFunction.

Textbook:D. M. Burton, Elementary Number Theory, 6th ed. New Delhi: Tata McGrawHill Education Private Limited, 2011.

References:1. I. Niven, H.S. Zuckerman and H. L. Montgomery, An Introduction to

the Theory of Numbers, 5th ed. New Delhi: Wiley, 2008.2. M. B. Nathanson, Methods in Number Theory. New Delhi: Springer,

2000.3. C.Y. Hsiung, Elementary Theory of Numbers. Bombay: Allied

Publishers Ltd, 1985.4. N. Robbins, Beginning Number Theory. Sudbury: Jones and Bartlett

Publishers, 2006.

Paper 2: RMT 266ALGEBRAIC NUMBER THEORY

Unit I: 15 hoursNumber Theory and Algebra, Groups, Rings and Fields, Binary QuadraticForms, Equivalence and Reduction of Binary Quadratic Forms, PositiveDefinite Binary Quadratic Forms.

Unit II: 15 hoursTernary Quadratic Forms, Rational Points on Curves, Elliptic Curves,Factorization using Elliptic Curves.

Unit III: 15 hoursPolynomials, Algebraic Numbers, Algebraic Number Fields, QuadraticFields, Units in Quadratic Fields, Primes in Quadratic Fields, UniqueFactorization, Primes in Quadratic Fields having the Unique FactorizationProperty.

Textbook:I. Niven, H.S. Zuckerman and H. L. Montgomery, An Introduction to theTheory of Numbers, 5th ed. New Delhi: Wiley, 2008.

References:1. D. M. Burton, Elementary Number Theory, 6th ed. New Delhi: Tata

McGraw Hill Education Private Limited, 2011.2. L. E. Dickson, Introduction to the Theory of Numbers. New York:

Dover, 1971.3. H. Griffin, Elementary Theory of Numbers. New York: MGH, 1954.4. J. Stillwell, Elements of Number Theory. New Delhi: Springer, 2003.5. K. Ireland and M. Rosen, Classical Introduction to Modern Number

Theory. New Delhi: Springer, 2004.

Elective I : RIEMANNIAN GEOMETRY

Paper 1: RMT 267RIEMANNIAN GEOMETRY

Unit I 15 HoursDIFFERENTIABLE MANIFOLDS: Introduction to Manifolds – Thespace of tangent vectors at a point of Rn – related theorems – Vector fieldson open subsets of Rn – Inverse function theorem – Definition ofDifferentiable manifold – Examples : Euclidean plane – Tangent space at apoint on a manifold - Vector field on a manirfold – Lie algebra of the vectorfields on a manifold – Frobenius theorem.

Unit II 15 HoursTENSOR FIELDS ON MANIFOLDS: Tensor fields – Differential Formsand Lie Differentiation – Covariant Derivative of Tensors – Connections –Torsion Tensor – Symmetric Connections – Curvature tensor field

Unit III 15 HoursRIEMANNIAN MANIFOLDS: Riemannian Manifolds - RiemannianConnection – Fundamental theorem of Riemannian Manifold – Curvaturetensors – Bianchi Identities – Sectional Curvature - Manifold of ConstantCurvature.

Texts and References:

1. S. Kobayashi and K. Nomizu, Foundation of Differential Geometry, JohnWiley and Sons, New York, 1969.

2. N. J. Hicks, Notes on Differential Geometry, Van Nostrand, Princeton,1965.

3. W. Boothby, An introduction to Differentiable Manifolds and RiemannianGeometry, Academic Press, 1984.

4. E. David Blair, Contact Manifolds in Riemannian Geometry, Springer-Verlag, 1976.

Paper 2: RMT 268SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS

Unit I 15 HoursSUBMANIFOLDS OF RIEMANNIAN MANIFOLDS: Submanifolds ofRiemannian manifolds – Induced connection and second fundamental form –Equations of Gauss, Codazzi and Ricci – Mean curvature and Gaussiancurvature.

Unit II 15 HoursUMBILICAL AND MINIMAL SUBMANIFOLDS: Totally umbilicalsubmanifolds, Totally geodesic submanifolds, Minimal submanifolds inEuclidean space.

Unit III 15 HoursHYPERSURFACES: Hyper surfaces of a Riemannian manifolds –Hypersurface of Rn – Fundamental forms on hypersurface of Rn.

Texts and References:

1. Kobayashi S. and Nomizu K., Foundation of Differential Geometry, JohnWiley and Sons, 1969.

2. N. J. Hicks, Notes on Differential Geometry, Van Nostrand, Princeton,1965.

3. B.Y. Chen, Geometry of submanifolds, M. Dekker, New York, 1973

4. E. David Blair: Contact Manifolds in Riemannian Geometry, Springer-Verlag, 1976.