Syllabus For The Subject DIFFERENTIAL GEOMETRY

UNIT - 01

THE THEORY OF SPACE CURVES - ARC LENGTH TANGENT - NORMAL BINORMAL -

DEFINITION OF SPACE CURVE - GEOMETRICAL INTERPRETATION OF - EQUATION OF

THE TANGENT LINE TO A CURVE AT A POINT - TANGENT LINE IN CARTESIAN FORM.

UNIT - 02

OSCULATING PLANE - PRINCIPAL NORMAL AND BINORMAL - CURVATURE OF THE CURVE

IN SPACE - SERRET - FRENET FORMULAE - EXPRESSION FOR TORSION.

UNIT - 03

DIFFERENT OF A SURFACE - CURVES ON A SURFACE - CONTACT BETWEEN CURVES

AND SURFACES - OSCULATING CIRCLE - OSCULATING SPHERE - LOCUS OF THE

CENTRE OF SPHERICAL CURVATURE.

UNIT - 04

TANGENT SURFACES - INVOLUTES AND EVOLUTES - INTRINSIC EQUATION OF SPACE

CURVES - FUNDAMENTALS EXISTENCE THEOREM FOR SPACE CURVES - HELICES - THE

SPHERICAL INDICARICES (OR) SPHERICAL IMAGES.

UNIT - 05

SURFACE REPRESENTATION - REGULAR AND SINGULAR POINTS - CHANGE OF

PARAMETERS - CURVILINEAR EQUATIONS OF THE CURVE ON THE SURFACE - TANGENT

PLANE AND NORMAL - SURFACES OF REVOLUTION.

UNIT - 06

ANCHOR RING - HELICOIDS - METRIC ON A SURFACE - THE FIRST FUNDAMENTAL FORM -

INVARIANCE OF THE METRIC - ELEMENT OF AREA - DIRECTION COEFFICIENTS ON A

SURFACE.

UNIT - 07

FAMILY OF CURVES - DIFFERENTIAL EQUATION OF THE FAMILY OF CURVES -

ORTHOGONAL TRAJECTORIES - DOUBLE FAMILY OF CURVES - ISOMETRIC

CORRESPONDENCE - INTRINSIC PROPERTIES.

UNIT - 08

GEODESICS AND THEIR DIFFERENTIAL EQUATION - CANONICAL GEODESIC EQUATIONS -

SECOND FUNDAMENTAL FORM - FUNDAMENTAL EQUATIONS OF SURFACE THEORY -

GEODESICS ON A SURFACE OF REVOLUTION - CURVATURE OF NORMAL SECTION OF

THE SURFACE - MENNIER`S THEOREM - CLAIRAUT`S THEOREM.

UNIT - 09

NORMAL PROPERTY OF A GEODESIC - EXISTENCE THEOREM - GEODESIC PARALLELS -

GEODESIC POLARS - GEODESIC CURVATURE COMPONENTS OF GEODESIC CURVATURE

- GEODESIC CURVATURE OF THE PARAMETRIC CURVE - LIOUVILLE`S FORMULA GAUSS -

BONNET THEOREM - GAUSSIAN CURVATURE - MINDING`S THEOREM - CONFORMAL

MAPPING - COROLLARY.

Vinayaka Missions University,Directorate of Distance Education

Salem India

MASTER OF SCIENCE IN MATHEMATICS

1 Yr.

DIFFERENTIAL GEOMETRY{MMAT.03}(2030504)

UNIT - 10

GEODESIC MAPPING - RULED SURFACE (DEVELOPABLE AND SKEW) - EQUATION OF THE

RULED SURFACES - NECESSARY CONDITION - LINES OF CURVATURE - DIFFERENTIAL

EQUATION OF LINES OF CURVATURE - PROPERTY OF LINES OF CURVATURES ON

DEVELOPABLE - CONJUGATE DIRECTION - ASYMPTOTIC LINES - ASYMPTOTIC LINES ON

A RULED SURFACE - PARAMETER OF DISTRIBUTION OF A RULED SURFACE -

PROPERTIES OF PARAMETER OF DISTRIBUTION - CENTRAL POINT AND THE EQUATION

OF THE LINE OF STRICTION.

UNIT - 11

JOACHIMSTHAL`S THEOREM - DUPIN`S INDICATRIX - TYPES OF POINT (ELLIPTIC,

HYPERBOLIC AND PARABOLIC) - THIRD FUNDAMENTAL FORM - FAMILY OF SURFACES -

ENVELOPE - THE EDGE OF REGRESSION - DEFINITION OF EDGE OF REGRESSION -

CHARACTERISTIC TOUCHES THE EDGE OF REGRESSION - MINIMAL SURFACES - GAUSS

CHARACTERISTIC EQUATION - MAINARDI - CODAZZI EQUATIONS.

UNIT - 12

COMPACT SURFACES - POINTS ARE UMBILICS - HILBERT`S LEMMA - COMPACT

SURFACES OF CONSTANT GAUSSIAN OR MEAN CURVATURE - COMPLETE SURFACES -

CHARACTERIZATION.